FRACTAL ANALYSIS APPLIED TO ANCIENT EGYPTIAN MONUMENTAL ART
by
Jessica Robkin
A Thesis Submitted to the Faculty of
The Dorothy F. Schmidt College of Arts and Letters
in Partial Fulfillment of the Requirements for the Degree of
Master of Arts
Florida Atlantic University
Boca Raton, Florida
December 2012
Copyright by Jessica Robkin 2012
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ACKNOWLEDGEMENTS
The following research paper would not be possible without the patience and understanding of my thesis committee. I am thankful for their invaluable insights, education, and support during my studies. I am particularly grateful to Dr. Clifford T.
Brown for helping me to find a path of research that allowed me to do credible and important work. Thank you for introducing me to fractals, for showing me a way to bridge the gap between science and subjectivity, and for helping me every step of the way. Lastly, great thanks go out to my loved ones, whose faith has never wavered.
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ABSTRACT
Author: Jessica Robkin
Title: Fractal Analysis Applied to Ancient Egyptian Monumental Art Institution: Florida Atlantic University
Thesis Advisor: Dr. Clifford T. Brown
Degree: Master of Arts
Year: 2012
The study of ancient Egyptian monumental art is based on subjective and qualitative analyses by art historians and Egyptologists who use the change in stylistic trends as Dynastic chronological markers. The art of the ancient Egyptians is recognized the world over due to its specific and consistent style that lasted the whole of Dynastic
Egypt. This artwork exhibits fractal qualities that support the applicability of applying fractal analysis as a quantitative and statistical tool to be used in this field. In this thesis, I show the fractality of ancient Egyptian monumental art by analyzing black and white line drawings of twenty-eight separate bas-reliefs with three separate programs: Benoit 1.3,
ImageJ, and Fractal3e. After preparing the images with GIMP2 software – used to remove non-original lines – I analyzed each image using the fractal box-counting analysis function in the above programs and calculated their fractal dimension, D. The resulting fractal dimension supported the consistency visually identified in the artwork from ancient Egypt, both chronologically and geographically.
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FRACTAL ANALYSIS APPLIED TO ANCIENT EGYPTIAN MONUMENTAL ART
LIST OF TABLES ...... viii
LIST OF FIGURES ...... ix
CHAPTER 1. INTRODUCTION AND PROBLEM STATEMENT ...... 1
CHAPTER 2. ESSENTIAL CHARACTERISTICS OF ANCIENT EGYPTIAN ART4
CHAPTER 3. ARTISTIC CHANGES IN ANCIENT EGYPTIAN ART ...... 14
Old Kingdom, First Intermediate Period Artistic Changes ...... 14
Middle Kingdom, Second Intermediate Period Artistic Changes ...... 16
New Kingdom, Amarna Period Artistic Changes ...... 20
Third Intermediate Period, Late Period Artistic Changes ...... 23
Graeco-Roman Period Artistic Changes ...... 24
CHAPTER 4. GEOGRAPHY AND GEOGRAPHIC VARIATION ...... 27
CHAPTER 5. QUANTITATIVE APPROACHES TO ART ...... 30
CHAPTER 6. MATERIALS AND METHODS ...... 37
CHAPTER 7. RESULTS ...... 43 vi
CHAPTER 8. DISCUSSION ...... 124
CHAPTER 9. CONCLUSION...... 134
REFERENCES CITED ...... 136
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LIST OF TABLES
1. Ancient Egyptian Chronology ...... 5
2. Analysis Results Listed Chronologically ...... 123
3. Benoit 1.3, ImageJ, Fractal 3e Data Correlation ...... 126
4. Benoit 1.3 ANOVA Single Factor Results ...... 129
5. ImageJ ANOVA Single Factor Results ...... 129
6. Fractal3e ANOVA Single Factor Results ...... 129
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LIST OF FIGURES
1. Ramesses III Battle with the Maritime Nations, Medinet Habu ...... 3
2. Limestone Relief from the Chapel of Seti I, 19th Dynasty ...... 6
3. Ancient Egyptian Artist Gridlines ...... 8
4. Mastaba of Ptah-Hotep, 5th Dynasty, Relief Showing Funerary Offerings ...... 10
5. Black Granite Statue of Amenemhat I, 12th Dynasty ...... 13
6. Examples of Theban Art Style ...... 18
7. Canonical Line Grid ...... 19
8. Archaeological Map of Ancient Egypt ...... 29
9. Illustration of the First Six Iterations of the Middle Third Cantor Set ...... 34
10. The Narmer Palette, Predynastic Period, 0/1st Dynasty ...... 44
11. Scatter Plot of Benoit Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Narmer Palette Obverse ...... 45
12. Scatter Plot of ImageJ Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Narmer Palette Obverse ...... 46
13. Scatter Plot of Frac3 Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Narmer Palette Obverse ...... 46
14. Scatter Plot of Benoit Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes - Narmer Palette Reverse...... 47
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15. Scatter Plot of ImageJ Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes - Narmer Palette Reverse...... 48
16. Scatter Plot of Frac3 Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes - Narmer Palette Reverse...... 48
17. Menes Tablet, Predynastic Period, 1st Dynasty ...... 49
18. Scatter Plot of Benoit Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Menes Tablet ...... 50
19. Scatter Plot of ImageJ Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Menes Tablet ...... 50
20. Scatter Plot of Frac3 Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Menes Tablet ...... 51
21. Mastaba of Meresankh III, Old Kingdom, 4th Dynasty - Main Room, East Wall, South of Entrance ...... 52
22. Scatter Plot of Benoit Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Mastaba of Meresankh III, Main Room East Wall, South of Entrance ...... 53
23. Scatter Plot of ImageJ Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Mastaba of Meresankh III, Main Room East Wall, South of Entrance ...... 54
24. Scatter Plot of Frac3 Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Mastaba of Meresankh III, Main Room East Wall, South of Entrance ...... 54
25. Mastaba of Meresankh III, Old Kingdom, 4th Dynasty - South Door Jamb...... 55
26. Scatter Plot of Benoit Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Mastaba of Meresankh III, South Door Jamb...... 56
27. Scatter Plot of ImageJ Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Mastaba of Meresankh III, South Door Jamb...... 57
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28. Scatter Plot of Frac3 Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Mastaba of Meresankh III, South Door Jamb...... 57
29. Tomb Chapel of Werirenptah, Old Kingdom, 5th Dynasty ...... 58
30. Scatter Plot of Benoit Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Werirenptah ...... 59
31. Scatter Plot of ImageJ Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Werirenptah ...... 59
32. Scatter Plot of Frac3 Data Showing the Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Werirenptah ...... 60
33. Mastaba of Sekhemka, Old Kingdom, 5th Dynasty - East Wall, South Section...... 61
34. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Sekhemka, East Wall, South Section ...... 62
35. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Sekhemka, East Wall, South Section ...... 62
36. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Sekhemka, East Wall, South Section ...... 63
37. Mastaba of Sekhemka, Old Kingdom, 5th Dynasty - South Wall ...... 63
38. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Sekhemka, South Wall ...... 64
39. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Sekhemka, South Wall ...... 65
40. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Sekhemka, South Wall ...... 65
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41. Mastaba of Ptah-Hotep, Old Kingdom, 5th Dynasty ...... 66
42. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Ptah-Hotep ...... 67
43. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Ptah-Hotep ...... 67
44. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Ptah-Hotep ...... 68
45. Mastaba of Senedjemib Inti, Old Kingdom, 5th Dynasty - Room II, South Wall ...... 68
46. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Senedjemib Inti...... 70
47. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Senedjemib Inti...... 70
48. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Senedjemib Inti...... 71
49. Mastaba of Qar, Old Kingdom, 6th Dynasty – Court C, Lower Half of North Wall ...... 71
50. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Qar ...... 72
51. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Qar ...... 73
52. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Qar ...... 73
53. Mastaba of Idu, Old Kingdom, 6th Dynasty - West Wall, South of Stela Niche Mastaba of Idu ...... 74
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54. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Idu ...... 75
55. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Idu ...... 76
56. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes - Mastaba of Idu ...... 76
57. Temple of Mentuhotep II, Middle Kingdom, 11th Dynasty ...... 77
58. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Mentuhotep II ...... 78
59. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Mentuhotep II ...... 78
60. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Mentuhotep II ...... 79
61. Tomb of Antefoker, Middle Kingdom, 12th Dynasty ...... 80
62. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Antefoker ...... 81
63. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Antefoker ...... 81
64. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Antefoker ...... 82
65. Amenemhat I Pyramid Temple, Middle Kingdom, 12th Dynasty ...... 82
66. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Amenemhat I Pyramid Temple ...... 83
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67. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Amenemhat I Pyramid Temple ...... 84
68. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Amenemhat I Pyramid Temple ...... 84
69. Temple of Senwosret I, Middle Kingdom, 12th Dynasty ...... 85
70. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Senwosret I ...... 86
71. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Senwosret I ...... 87
72. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Senwosret I ...... 87
73. Pyramid Complex of Senwosret I, Middle Kingdom, 12th Dynasty ...... 88
74. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Pyramid Complex of Senwosret I ...... 89
75. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Pyramid Complex of Senwosret I ...... 90
76. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Pyramid Complex of Senwosret I ...... 90
77. Temple of Amenemhat-Sobekhotep at Medmud, Middle Kingdom, 13th Dynasty: Temple Gates ...... 91
78. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Amenemhat-Sobekhotep ...... 92
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79. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Amenemhat-Sobekhotep ...... 93
80. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Amenemhat-Sobekhotep ...... 93
81. Tomb of Rekhmire, New Kingdom, 18th Dynasty ...... 94
82. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Rekhmire ...... 95
83. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Rekhmire ...... 95
84. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Rekhmire ...... 96
85. Tomb of Meryra, New Kingdom, 18th Dynasty (Amarna Period) ...... 97
86. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Meryra ...... 98
87. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Meryra ...... 99
88. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Meryra ...... 99
89. Tomb of Panehesy, New Kingdom, 18th Dynasty (Amarna Period) ...... 100
90. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Panehesy ...... 101
91. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Panehesy ...... 101
92. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Panehesy ...... 102
93. Tomb of Meryra II, New Kingdom, 18th Dynasty (Amarna Period) ...... 103
94. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Meryra II ...... 104 xv
95. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Meryra II ...... 105
96. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Meryra II ...... 105
97. Tomb of Apy, New Kingdom, 18th Dynasty (Amarna Period) ...... 106
98. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Apy ...... 107
99. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Apy ...... 108
100. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Apy ...... 108
101. Tomb of Neferhotep, New Kingdom, 18th/19th Dynasty ...... 109
102. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Nefer-hotep ...... 110
103. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Nefer-hotep ...... 111
104. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Nefer-hotep ...... 111
105. Temple of Beit el-Wali, New Kingdom, 19th Dynasty ...... 112
106. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Beit el-Wali ...... 113
107. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Beit el-Wali ...... 114
108. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Beit el-Wali ...... 114
109. Tomb of Merenptah, New Kingdom, 19th Dynasty ...... 115
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110. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Merenptah ...... 116
111. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Merenptah ...... 117
112. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Tomb of Merenptah ...... 117
113. Temple of Merenptah, New Kingdom, 19th Dynasty ...... 118
114. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Merenptah ...... 119
115. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Merenptah ...... 119
116. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Temple of Merenptah ...... 120
117. Great Triumphal Stela of Piye, Late Period, 25th Dynasty ...... 120
118. Scatter Plot of Benoit Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Great Triumphal Stela of Piye ...... 121
119. Scatter Plot of ImageJ Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Great Triumphal Stela of Piye ...... 122
120. Scatter Plot of Frac3 Data Showing Logarithms of the Box Sizes and Number of Occupied Boxes – Great Triumphal Stela of Piye ...... 122
121. Change in Fractal Dimension of Ancient Egyptian Art Over the Course of Dynastic Egypt - Benoit 1.3, ImageJ, and Frac3 Combined Results ...... 125
122. Change in Fractal Dimension of Ancient Egyptian Art Over the Course of Dynastic Egypt - Benoit 1.3 ...... 127
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123. Change in Fractal Dimension of Ancient Egyptian Art Over the Course of Dynastic Egypt - ImageJ ...... 128
124. Change in Fractal Dimension of Ancient Egyptian Art Over the Course of Dynastic Egypt - Fractal3e ...... 128
125. Scatter Plot Showing the Geographic Variation of the Fractal Dimension – Benoit 1.3 ...... 131
126. Scatter Plot Showing the Geographic Variation of the Fractal Dimension – ImageJ ...... 131
127. Scatter Plot Showing the Geographic Variation of the Fractal Dimension – Fractal3e ...... 132
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CHAPTER 1
INTRODUCTION AND PROBLEM STATEMENT
The purpose of the following research is to evaluate the use of fractal analysis to quantify the style and composition of ancient Egyptian art. Fractal analysis is a set of mathematical methods designed to measure the geometric and topological qualities of highly irregular forms. The goal of the research is to investigate whether fractal analysis provides an objective, quantitative, and useful means of studying the stylistic variation in ancient Egyptian art. Archaeologists and art historians are naturally interested in the stylistic variation in Egyptian art for many reasons. The aesthetic and cultural genius that gave rise to the art is innately interesting, but archaeologists in particular have more practical reasons for studying the way the art style varies. Archaeologists normally identify and describe the variation in material culture 1) to create chronological sequences, 2) to map geographical patterns of culture, and 3) to understand functional variation in artifacts. The standard archaeological approach has several weaknesses.
Particularly with more complex art styles, archaeologists (and art historians) may rely upon subjective criteria to assign works to a particular period, region, or type. At present, the only objective method used for this purpose that has a sound mathematical foundation is symmetry analysis, but it only works with a specific range of “symmetrical,” mostly abstract, designs (Brown et al. 2005: 54; Washburn 1977: 17-22). The figurative and
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narrative art of Egypt is not symmetrical. Fractal analysis, however, can quantify significant qualities of non-symmetrical patterns provided they are fractal. Therefore, I will be evaluating whether fractal analysis can provide an objective measure of the variation in Egyptian art that will prove useful in understanding its chronological, regional, and/or functional variation. For example, if the fractality of Egyptian art varies chronologically, then fractal analysis might help to date monuments of unknown age. I will use fractal analysis to evaluate whether Egyptian art has fractal characteristics, and then I will explore how that fractality varies: geographically, temporally, or functionally.
Fractal analysis can only provide a meaningful measure of the scaling in a pattern if the pattern is at least approximately or statistically fractal. So, first I must – and will – show that each work is fractal, but it would only make sense to attempt this research if we had some premonition or suspicion that Egyptian art were fractal. One of the essential qualities of fractals is they exhibit scaling in which the elements that make up the composition increase dramatically in number as their size diminishes. In fact, the composition of most Egyptian bas-reliefs suggests a scaling of elements by size, which qualitatively supports the theory that artwork from Dynastic Egypt is fractal. For example, in the Battle with the Maritime Nations from the mortuary temple of Ramesses
III, we see a single large human figure, several small human figures, and scores of yet smaller figures representing the sea people, their ships, slaves, and hieroglyphs (Figure
1). Thus, most of the major iconographic techniques employed to communicate symbolic meaning—the use of relative size to denote importance, the use of registers, the incorporation of hieroglyphic textual elements, and the use of hachure and incision to supply differentiating details—contribute to a composition whose elements scale in size 2
with a small number of large ones and ever-increasing numbers of smaller ones. This is the qualitative description of a fractal pattern. Therefore, I felt justified in assaying the use of fractal analysis to study Egyptian art. I focused on bas-relief sculpture. Egyptian sculpture in the round appears less fractal in composition than the reliefs and therefore seemed like a poor candidate, not to mention the fact that analysis of three-dimensional datasets is more challenging. Paintings look quite as fractal as the bas-reliefs, but the use of colors presents serious complications that I thought would be best to avoid in this first, pioneering study.
If fractal dimension varies systematically with the age of Egyptian art (because the style changed through time), then it ought to be possible to use fractal analysis as an independent line of evidence to date works that are found out of context or for which the dates are ambiguous for any other reason. Similarly, if fractal dimension varies with the geographical origin of the art works, then fractal analysis could be used to assign a regional provenience to works of uncertain provenance. In either case, fractal analysis would supply archaeologists and art historians with a new set of tools for understanding the temporal and spatial distributions of ancient Egyptian art.
Figure 1. Ramesses III Battle with the Maritime Nations, Medinet Habu (Erman 1971: 541) 3
CHAPTER 2
ESSENTIAL CHARACTERISTICS OF ANCIENT EGYPTIAN ART
I will first explain the salient qualities of ancient Egyptian art so that the reader may understand what exactly the fractal analysis is measuring and why it works. I will also explain how ancient Egyptian art changed through time. Ancient Egyptian art is stylistically unified – no one would confuse an Egyptian relief with an Assyrian or Greek one – so chronological changes in the style took place within well-defined boundaries.
The canons of Egyptian art, which were well established by the 4th Dynasty, c. 2600 BC
(Table 1), are usually described in terms of a number of essential elements or formal qualities: balance and symmetry, measured proportions, color, naturalistic details, animal imagery, multiple points of view, scale, surface contrasts and geometry (Davis 1989: 9;
Robins 1997: 21). I describe each of these below before turning to an explanation of how they changed through time.
I begin with one of the most easily recognized characteristics, balance and symmetry. Balanced forms can be seen when artists depict the same or a similar number of focal points on either side of the vertical axis, as can be seen in Figure 2. In this relief from the 19th Dynasty chapel of King Seti I, c. 1306 – 1290 BC, in Abydos, the king is seated with the goddesses Nekhbet and Wadjit while the gods Thoth and Horus stand on either side of the three seated figures. As you can see from the limestone relief, if you
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were to draw a line along the vertical axis of the scene, there is a balanced number of figures on either side – almost mirroring each other.
Table 1. Ancient Egyptian Chronology (Digital Egypt for Universities) Date Periods Duration 4000 - 3500 BC Naqada I (Predynastic) 500 years 3500 - 3200 BC Naqada II (Predynastic) 300 years 3200 - 3100 BC Naqada III (Predynastic) 100 years 3100 - 2686 BC Early Dynastic 400 years 2686 - 2181 BC Old Kingdom 500 years 2181 - 2025 BC First Intermediate Period 150 years 2025 - 1700 BC Middle Kingdom 325 years 1700 - 1550 BC Second Intermediate Period 150 years 1550 - 1069 BC New Kingdom 500 years 1069 - 664 BC Third Intermediate Period 400 years 664 - 525 BC Late Period 139 years 525 - 404 BC First Persian Period 121 years 404 - 343 BC Late Dynastic Period 61 years 343 - 332 BC Second Persian Period 11 years 332 - 305 BC Macedonian Period 27 years 323 - 30 BC Ptolemaic Period 293 years 30 BC - 640 AD Roman/Byzantine Period 670 years
This symmetry can repeatedly be seen in temples and tombs where the works of art served their primary purpose: religion. Egyptian artists utilized symmetry as a method of maintaining balance in their work. By creating this standard of balanced elements, artists created order within each scene that was pleasing to the viewer in its clarity and simplicity. Keeping with the theme of simplicity, the subjects of Egyptian art were kept relatively consistent throughout the whole of Dynastic Egypt, with only small aberrations in subject matter showing up in the New Kingdom. Art showed scenes of everyday life,
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the world that surrounded the Egyptians, the great battles that enlarged their borders.
These scenes were exemplified by the use of bold colors and simple well-defined subjects and shapes, all of which kept the art easy to replicate and relatively consistent for over four thousand years.
Figure 2. Limestone relief from the Chapel of Seti I, 19th Dynasty (Robins 1997: 174)
While Egyptian artisans strove to maintain balance through the use of aesthetically pleasing symmetrical images, they also adhered to a precise set of proportions when portraying figures and scenery. This well-established procedure allowed the artists to maintain standardized proportions. Artists would use a standard grid
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pattern, as can be seen in Figure 3, left and right, for art that was produced from the Old
Kingdom through Graeco-Roman times, in varying forms before beginning actual carving or wall decoration. Both examples show how this grid, which had both horizontal and vertical guidelines, was used to keep figures true to the established canon (Davis
1989: 19-21). The overlaid grid was one of the primary reasons for the consistency of
Egyptian art throughout antiquity, regardless of size or medium. By using the width of the palm as a guideline, artisans were able to define the conventional proportions used for the human figures that were the focal point in most reliefs.
Since the primary use of art was religious, the main subject of the work would commonly be the gods, Pharaoh or the owners of the given tomb. Egyptian artists would convey the importance of the figures in a given scene through the use of size, representing this main figure in a larger scale than ancillary figures. Following the primary personage in size were the spouse, then the children. Smallest in scale were slaves and defeated enemies. This size variance is one of the most easily recognizable attributes of Egyptian art and was one of the simplest ways in which artists expressed meaning.
In addition to using size to convey meaning, artists also used color. The colors that Egyptian artists used had very specific meanings to the Egyptians. For instance, artists represented the Nile with the colors of blue and green, the sun with the color yellow, the desert was represented by the color gold and the color red was reserved for portraying power (Robins 1997: 14-19). When grandiose scenes were on painted surfaces they were always in outstanding colors representing not only beauty and abundance, but also conveying deep significance as well. The basic color palette showed 7
how deeply Egyptian culture was rooted in the Nile and surrounding geography. Egyptian artists would use these colors to lend realism and depth to the work.
Figure 3. Ancient Egyptian artist gridlines: Above left, 18 Grid square, Old Kingdom (Davis 1989: 22); right, Amarna Period grid (Davis 1989: 23).
Artists usually found ways to include details of their environment in their art, as can be seen in Figure 4, a wall scene depicting funerary offerings from Ptah-Hotep, 5th
Dynasty c. 2465 – 2323 BC. Numerous bas reliefs and funerary paintings have been found that exhibit the Egyptians’ ability to capture the fine details of the world around them. While the artists were limited by strict guidelines regarding both the subject matter and the method of representation, they would use color to lend realism and depth to their work and they employed subtle touches to portray musculature which showed that artists had at least a cursory knowledge of anatomy (Davis 1989: 7-8; Robins 1997: 12-30). 8
Artists would utilize details of color and would use different scenery in relation to subject matter, as can be seen in the “details of anatomy, costume, scale or pose [that] were sometimes rendered quite differently for women, children, the lower classes, and foreigners” (Schafer 1974: 17, 19-60 as quoted in Davis 1989:7). Animal depictions were also a very common subject in Egyptian art as animals were believed to represent many aspects of life and the afterlife, furthering the use of naturalistic details in artistic renderings. Representations of animals are an integral part of Egyptian art, appearing as early as the Predynastic Period, where “pottery was often decorated with painted scenes which included people as well as animals” (Malek 1999: 43). Figure 4 is an excellent example in which artists utilized these depictions of animals to further demonstrate how the artists used the world around them for content.
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Figure 4. Mastaba of Ptah-Hotep, 5th Dynasty, relief showing funerary offerings (Maspero 2004[1895]: 121).
The mastaba of Ptah-Hotep (Figure 4) also illustrates the way artists represented the human form by using multiple points of view. It is important to remember that while the art always served a specific purpose, it was expected to be aesthetically pleasing
(Zauzich 1992: 3-5). Egyptian artists wanted to show the human figure in the most flattering way possible. To achieve this they used multiple viewpoints when they were arranging the subjects of the work. Artists would depict the human form in two dimensional representations, by rotating parts of the body to present several viewing
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angles simultaneously. This technique allowed the artists to enhance the central figure and make use of the entire two dimensional surface.
This technique of viewpoint manipulation is shown in both Figures 2 and 4. Both reliefs are perfect examples of how artists made use of different points of view. The shoulders of Seti I and Ptah-Hotep are shown from the front, while their legs, face, and feet are shown in profile. Both figures turn at the waist, so that the chest appears frontal.
The most notable view point manipulation can be seen in the artist’s portrayal of the face, which is shown in a profile, side view. It is very rare – with the exception of statuary and during the Graeco-Roman era – to show the face in a frontal view.
These diverse viewpoints were only further exemplified by the orientation of the registers which were used to maintain order. Registers could be either horizontally or vertically oriented, depending on how they made the composition look. Artists would often manipulate register orientation in order to have a well-organized finished piece.
Registers were defined by their upper and lower boundary lines which allowed for further micro-organization. Within the registers, social importance and status were again defined by size and orientation. While importance was defined by size, distance was defined by layers of figures on top of each other. The images to the front would be interpreted as closer while the figures at the back of the series would be interpreted as farther away
(Robins 1997: 21). The registers in both Figures 1 and 3 are not oriented to frontal viewing, but instead face the main subject of the relief, further accentuating him as the central figure.
The use of multiple points of view was not transferable to three dimensional surfaces such as sculpture in the round. As the primary focus of this study is bas reliefs, I 11
will only briefly discuss statuary as it relates to the overall geometry of Egyptian art.
Statues exhibit what Egyptologists call frontality, which is defined as the main character facing straight ahead, without twisting or turning, an example of which can be seen in
Figure 5. This black granite statue represents the high official Amenemhat and dates to the 12th Dynasty, c. 1991-1962 BC. It shows the high official as a man in the prime of his health with broad shoulders and substantial musculature, which is very common among statuary from this period. This statue and others like it from Upper to Lower Egypt display the aspect of frontality that was so common in three-dimensional representations for thousands of years in Dynastic Egypt (Robins 1997: 19). Statuary provided the medium for surface contrasts – another feature typical of Egyptian art. Even thousands of years after it was originally chiseled, the statue of Amenemhat still has a gloss and luster.
To properly execute the techniques required to create the statues and reliefs that were traditional to Dynastic history, artists would utilize the main structural elements recognizable in Egyptian art - the cube, horizontal axes, and vertical axes.
The geometric cube most commonly used in sculpture and statuary (Figure 5) was used to keep art true to established canons. We know from grid patterns found on unfinished artwork that ancient Egyptians had a set of well-defined structural elements, both the horizontal and vertical axes. These elements, along with the concept of the cube, can be found in wide and consistent use throughout the whole of ancient Egyptian history, whether the medium was temple and tomb walls or sculptures. Laying the well- known grid on the three dimensional media used for sculpture, the artists would carve the emerging form, constantly replacing the grid on the material, so as to assure conformity
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to the standard. This same method was used by Egyptian artists when creating tomb and temple bas reliefs.
Figure 5. Black granite statue of Amenemhat I, 12th Dynasty (Robins 1997: 107)
The cube can be seen throughout Dynastic history, showing up as early as
Predynastic times in pottery exhibiting cross hatching. By basing organization within a solid geometric standard and establishing a set of canons to which all art was to be held,
Egyptian artists were able to keep artwork consistent over long periods of time, resulting in art that is known the world over even thousands of years after the decline of their civilization. Aspects of these lasting qualities are found as early as Predynastic times in pottery fragments recovered and dated to this Period. In the next section, I will discuss the evolution that the above characteristics undergo throughout Dynastic history. 13
CHAPTER 3
ARTISTIC CHANGES IN ANCIENT EGYPTIAN ART
The characteristic qualities of Egyptian art that we just discussed evolved through time. Archaeologists have traced the manner in which these qualities changed through time through the study of dated monuments and works of art. Thus, the change in sequence is known and can be used to estimate the dates of works of unknown age. These same changes in balance, symmetry, proportion, and composition mean that works of art might susceptible to dating by objective and quantitative methods such as fractal analysis.
The following section provides a brief overview of how the essential qualities of
Egyptian art changed through time to indicate why fractal analysis should function to discriminate works of different periods.
Old Kingdom, First Intermediate Period Artistic Changes
Throughout ancient Egyptian history changes in artistic style are used in identifying chronology. Religious motifs and monumental art begin in earnest during the
Old Kingdom. It is from this Period that the fundamentals of Egyptian art originate.
Motifs that began during this Period would be continued through the whole of Dynastic history. Artists displayed knowledge of anatomy and of the world around them. These
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subjects would be incorporated into works of monumental art. The artists would convey a summarized event – the reign of a king, a great battle, the conquering of a nation – on a massive scale (Robins 1997: 29; Smith 1952: 33). Depictions of animals in scenes of everyday activities began as early as the Old Kingdom, where they were utilized to express the “simple, but highly productive” farming techniques used by early Egyptians
(Malek 2003: 10). During the majority of this Period, the standard size for the human figure was eighteen palms, not including the top of the head, with the face equaling two palms; “shoulders are aligned at sixteen palms from the base of the figure, the elbows align at twelve from the base, and the knees at six” (Metropolitan Museum of Art 2012).
During the 5th Dynasty artists developed a system of guidelines used to portray the human figure in a standardized way. The system used a vertical axial line that passed through the ear (the head was shown in profile) and separated the torso in half, while a series of horizontal lines were laid out where the hairline, knees and lower border of the buttocks would be placed. These reference points would go through minor modifications during the 6th and 7th Dynasties resulting in a narrowing of the male figure as the Old
Kingdom came to a close. As the Dynasties of the Old Kingdom changed into the ruling families of the First Intermediate Period there is a notable decline in craftsmanship. This decline was attributed to the disintegration of the established principles of First
Intermediate Period artists (Terrace 1967: 42). While researchers now see these changes in style as gateways to new artistic freedoms in the Middle Kingdom, in the past this was seen as a major failing of First Intermediate Period art.
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Middle Kingdom, Second Intermediate Period Artistic Changes
By the end of the Old Kingdom and First Intermediate Period, Egypt was undergoing a time of internal struggle. The relative prosperity of Old Kingdom rulers was replaced by a divided Egypt. As the First Intermediate Period came to a close, Lower
Egypt would come under the control of a succession of rulers from Heracleopolis while
Upper Egypt would be controlled by rulers from Thebes. During the 11th Dynasty, Egypt was firmly divided at Abydos and art would evolve differently in both regions. In
Heracleopolis controlled Lower Egypt artists continued the traditional themes created during the 5th and 6th Dynasties. Wall paintings in the Middle Kingdom exhibit many modified features of the Old Kingdom, namely the blending of sharp angles into more fluid, angular proportions. This art is not as uniform as Old Kingdom artwork, but there are still strong bonds between the evolving styles. Middle Kingdom art drew from the preceding periods for inspiration, and the changes in techniques and subject matter used in the Middle Kingdom paralleled this so-called decline in artistic principles during the
First Intermediate Period. Due to the usage of poor quality materials and the reuse of materials during later Dynasties, little artwork from this period in Egyptian history remains. Materials of substandard quality start showing up during this time and reflect the real world the artists were living in, one in which their country was divided and northern
Egypt was sinking into decline.
At the same time northern Egypt was floundering, southern Egypt was seeing a time of prosperity. Thebes soon became the royal capital, but before being absorbed into
Theban rule the various districts south of Abydos had time, and freedom from an overbearing central government, to create very distinct styles. These styles were allowed 16
to flourish under Theban rule and soon artists began experimenting with bright colors and stylistic changes. The strict adherence to registers was loosened and, in some cases, completely abandoned. While the styles were changing under Theban rule, the topics and uses of art stayed consistent with the Old Kingdom religious and funerary practices and scenes of everyday life, including farming and animal husbandry scenes continued to appear (Malek 1999: 156-160; Robins 1997: 92-105).
It is from Thebes that the highest quality artwork from the Middle Kingdom can be found; yet both Lower and Upper Egypt would look to the 5th and 6th Dynasties for inspiration. The Middle Kingdom years of reunification shows the human form returning to early Old Kingdom proportions with male figures having “broad shoulders, a low small of the back and thick, muscled limbs” (Robins 1997: 106) and female figures being represented with little musculature, a high small of the back and a slender form, a fine example of which can be seen in Figure 6, left, a limestone relief from the temple of
Sahure, c. 2440 BC. While it was rare for all of the above characteristics to occur in the same image, even one of these aspects is in strong contrast to the weak muscled, large eyed form with a high small of the back that characterized the First Intermediate Period reunification Theban style that can be seen in the limestone relief of the High Official
Mereriqer at Dendera, dated to the First Intermediate Period, Figure 6, right (Malek
1999:101). At this time artists in the north were still using the Old Kingdom grid made up of vertical and horizontal lines, but by the 12th Dynasty, under Theban rule the grid was replaced by a new squared grid. During the middle of the 11th Dynasty, Theban rulers overtook Heracleopolitan rulers in the North and Egypt was again reunified and the squared grid became standard throughout the kingdom. This new squared grid was 17
created based off of the old lined grid pattern (Figure 7). The major body lines at the knee, the lower border of the buttocks and the elbow would be transferred to specific squares, grids 6, 9 and 12, respectively, as evidenced in the grids seen in Figure 2 (Robins
1997: 107). These grids allowed the artists to add greater detail than ever before. The new grid was used more as a guideline to acceptable proportions rather than as a steadfast rule. This gradual movement towards a more realistic, naturalistic style can be seen as early as the Middle Kingdom reign of Senwosret II, c. 1897-1887 BC (Malek 1999: 271;
Robins 1997: 118-119; Smith 1952: 72-3).
Figure 6. Examples of Theban art style: Above left, Sahure's Temple, c. 2410 BC (Malek 1999: 101); right, Tomb of Mereriqer, 1st Intermediate Period (Robins 1997: 83).
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Figure 7. Canonical line grid (Davis 1989:12)
The 12th Dynasty, with a newly reunified Egypt, was a time of great prosperity and this is reflected in the increased quality of materials dated to this period; however, the
13th Dynasty would again see a time of struggle during which Egypt was ruled by over seventy kings during a one hundred and fifty year period. There is little discernible change to the art from this time during the Middle Kingdom, but the weakened internal structure of Egypt made way for the successful Hyksos invasion (Malek 1999: 161-163;
Robins 1997: 109-112). The end of the Middle Kingdom was marked by the encroachment of the Hyksos, foreign invaders from Western Asia, which again resulted in the dividing of Upper from Lower Egypt. Lower Egypt would show increasing changes brought about from the Hyksos presence, while much of Upper Egypt would
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remain influenced by Theban rule. The internal division would exhibit itself as an increased variation in artistic quality during the Second Intermediate Period. This variation can be attributed not only to the growing Hyksos presence in the north, but also to the growing resources available to low-status Egyptians. The increase in resources allowed Egyptians of non-royal status to commission monuments, which in turn increased the demand for low priced artisans. The cheaper quality artwork turned out by these artists littered the Second Intermediate Period with low-quality compositions.
While much of the art from this period would have been considered substandard to
Egyptian elite, the kings ruling Upper Egypt were able to commission high quality artists and materials. This high quality work combined with the emerging low quality work created an environment of artistic variety that influenced New Kingdom artists.
New Kingdom, Amarna Period Artistic Changes
The range of standards formed in the previous periods would guide artists towards stylistics changes in New Kingdom art. This art is radically different from previous dynasties in formality and figure representation. Beginning with the reign of Hatshepsut, c. 1473-1458 BC, the severity of style lessened and proportions were used as a tool to portray elegance (Aldred 1961: 11). The most drastic changes attributed to the New
Kingdom are dated to the reign of Amenhotep IV, who changed his name to Akhenaten, c. 1353-1335 BC, and moved the royal capital to the desert city of Akhetaten, modern day el-Amarna. Akhenaten would also attempt to change the religion of Egypt from polytheism to monotheism, and he heavily involved himself in changing the artistic
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norms. This period of stylistic change and dynastic rule is usually referred to as the
Amarna Period.
The themes used in the Amarna Period are still very religious; however, the idealized lines of the previous dynasties are completely replaced by hyper-realistic portrayals. This period is known for new methods of portraying reality and can most readily be identified in the portrayal of the human figure. Examples of this can be seen in the use of scale where Amarna artists portrayed Akhenaten’s queen, Nefertiti, at the same scale as her husband in family scenes, something not done before this time. It is also during this period that artists would begin taking inspiration from their contemporary setting as opposed to the conventionalized themes of previous dynasties (Malek 2003:
260-262). The conventional methods of artistic representation were discarded during this
Period and were instead replaced with an exaggeration of the Pharaoh’s figure used to further distinguish between statuses. The grid that artists used also changed, allowing them more space to represent the subjects. Art was now approached with a subjective outlook, very different from the objective outlook attributed to earlier artwork (Aldred
1961: 25-7).
The Amarna art style was part of a new, naturalistic period of Egyptian art, with new artistic freedoms and rules. Due to these new freedoms, the symmetry found in typical Egyptian art was relaxed (Malek 2003: 271). Visual unity in Amarna art was also created by using new methods of spatial organization. The use of sunken (incised) reliefs, in which the figures are carved into the surface instead of raised (excised) reliefs, in which the background was carved away from the figures, became prevalent during the
Amarna Period. These reliefs were easier for artists to create and during the Amarna 21
Period they were characterized by a careful modeling of detail which gave the relief more dimension (Malek 2003: 269). With the death of Akhenaten, Amarna was deserted, but the new artistic style associated with it was not so easily abandoned. The discovery of the tomb of Tutankhamen, c. 1341 BC – 1323 BC, Akhenaten’s successor, by Howard Carter in 1922 provided evidence that the Amarna style was still influencing artistic techniques after the death of Akhenaten (Aldred 1961: 30-31; Smith 1952: 100-101). Indeed, remnants of this style can be seen at Nefertari’s temple in Abu Simbel, where Ramesses represented his chief wife at the same scale as himself.
While the most dramatic changes in art from the Amarna Period would be reined in during the following dynasties, some of the influences can still be seen in animal depictions from the New Kingdom. Animal depictions continued to be used as exemplifying scenes of everyday life. These themes, which began as early as the Old
Kingdom, reached almost a point of abundance in the New Kingdom, with “fully zoomorphic forms” becoming an artistic staple by the reign of Amenhotep III, c. 1390–
53 BC (Malek 2003: 377). Like so many other elements of Egyptian art, fine biological details were also modified throughout the years in direct relation to changes in Dynastic rule. Scenes depicting the aspects of daily life eventually transformed into mostly ornamental scenes by the Late Period. The everyday life scenes from the later periods of
Egyptian history still depict animals as found in farming and hunting scenes, but had become completely divorced from the contemporary reality they appeared to represent
(Malek 2003: 368).
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Third Intermediate Period, Late Period Artistic Changes
The New Kingdom is remembered for the drastic stylistic changes that are associated with the Amarna Period. These stylistic changes would later be combined with the formal styles of Old Kingdom art during the Third Intermediate and Late Periods. As the New Kingdom declined, Egypt again fell under the rule of foreign kings. From the
Sais Dynasties to those of the Kushites, Nubians and Persians, Egypt was in a state of flux for most of the Third Intermediate and Late Periods. Kushite influence affected rules of proportion and during this time the human figure was drawn using a reduced grid size in which six new grid squares equaled five old grid squares (Malek 2003: 359). Artists also began using new methods to represent anatomical details, including a technique common to many Kushite reliefs, the usage of vertically incised lines to indicate bones and musculature in lieu of modeling (Malek 2003: 358). Thematic characteristics were adapted in the Third Intermediate Period to current religious and stylistic standards, including the use of hard stone statues which become characteristic in later periods. From the Third Intermediate Period onward, as Egypt was ruled by non-Egyptians, in order to avoid being challenged in asserting their right to rule, many of these foreign Pharaohs built temples following the architectural and artistic themes developed in the Old and
Middle Kingdoms (Wilkinson 2000: 27). During these Periods – into at least the reign of
Shebitku, c. 698-690 BC – blended realism and artistic stylization are the noticeable trends. However, the realistic elements associated with the Amarna style and the naturalistic approaches used under Kushite rule had completely ceased by Late Period
26th Dynasty and were instead replaced “by formally perfect but idealizing representations, showing a bland, slightly suppressed smile” (Malek 2003: 366). 23
The Late Period 25th and 26th Dynasties, c. 712 BC- 525 BC, were characterized by the integration of the archaizing trends which began showing up in art from the Third
Intermediate Period. These trends drew on Old and Middle Kingdoms themes and discarded New Kingdom styles. Consistent into the Late Period, art exhibited conventional themes and rounded modeling of figures (Baines and Malek 2000: 56;
Malek 1999: 369; Wilkinson 2000: 27). Late Period 25th Dynasty is notable for shifting the capital to Memphis as a result of the Kushite victory at the end of the Third
Intermediate Period which forced the abandonment of the ancient capital of Thebes.
While the early dynasties of the Late Period were marked by peaceful relations with
Kush, at the end of the 26th Dynasty, c. 525 BC, Egypt was invaded by the Persian king
Cambyses. Cambyses was followed by five more Persian kings, creating the 27th
Dynasty, but the reigns of these kings were marked by frequent revolts and finally, in c.
404 BC, the Persians were expelled from Egypt and Egypt was again ruled by natives for close to sixty years. The two succeeding dynasties were relatively short lived, and by the end of the Late Period, a time of peace and prosperity had arrived, led by Nectanebo I, c.
380- 362 BC, and his successor Nectanebo II, c. 360-343 BC (Robins 1997: 10, 23, 210).
During this time Kushite influence can still be seen in the representation of the human form, especially females, as “fuller and more curvaceous forms that anticipated the styles of the Ptolemaic Period”(Malek 2003: 269, 358).
Graeco-Roman Period Artistic Changes
The last dynasty of the Late Period was marked by the Persians again invading
Egypt, c. 343 BC. Their rule lasted almost twenty years until Alexander the Great moved into the country, to the intense welcome of the indigenous people. While Alexander 24
quickly moved on to his next conquest, he left in charge fellow Macedonian rulers who were openly accepted by the Egyptians. In c. 304 BC, the Ptolemaic Dynasty began when
Ptolemy I declared himself king of Egypt after bringing the body of Alexander into
Memphis (Robins 1997: 231). This dynasty lasted for two and half centuries, and the art of Egypt adhered closely to traditional New Kingdom styles. The main reason for this seems to be that the Ptolemys concentrated on other regions of the Mediterranean and therefore left the Egyptians to their own devices. During this Period, the capital again changed locations, to the new city of Alexandria, in which the official language was
Greek (Robins 1997: 231, 235). Changes in artistic style dated to the Ptolemaic Period show a connection between idealization and naturalism of the human form as artists begin using a variety of methods to create funerary art, including bald, close shaved egg shaped heads and differing hairstyles with closely cropped curly hair (Malek 2003: 385-
9).
The last of the Ptolemy line, Cleopatra VII, c. 51-30 BC, committed suicide at the end of the Hellenistic Period and Octavian, later Augustus, conquered Egypt during the September 31 BC battle of Actium. As a result of these events, in 27 BC Egypt became a part of the Roman Empire. Despite being under Roman rule, Egyptian civilization and Egyptian art remained very similar to the pharaonic style. Importantly, artistic style and subject matter were mostly unaffected, with some general Romanizing trends, until the advent of Christianity. These Christian themes eventually took root in
Alexandria in the mid-first century AD and spread through the rest of the country. It is important to note that mummification existed until the fourth century AD with modifications taking the form of encrusted portraits painted on wood boards that were set 25
into the wrapping at the head of the mummy case portraying the individual in the prime of life.
As a direct result of this continuity in both style and culture within Egypt, Rome regarded the country with suspicion, which translated into little Roman development in
Egypt during the years of Roman rule. In fact, the only new city founded by the Romans was Antinoopolis, in Middle Egypt on the Nile. Catacombs dating to the second century
AD have artistic elements from both Greek and Egyptian schools, including false sarcophagi decorated with animal themes, as well as depictions of Egyptian gods in reliefs. Overall, a mainly Egyptian style continued throughout Roman rule and into the
Byzantine era with few changes. The stylistic changes were mainly influenced by the influx of Christianity. The end of Egyptian Civilization came in c. 543 AD when
Justinian closed the last of the pagan temples (Smith 1952: 138-144).
In summary, art historians and archaeologists have identified significant changes in the style of ancient Egyptian art over the millennia encompassed by the Dynastic periods. Many of these changes were gradual, incremental, and seemingly evolutionary.
Certain epochs, however, were punctuated by more abrupt and charismatic innovations in style, of which the most salient is probably the Amarna style. Nevertheless, despite its slow evolution and occasional anomalies, ancient Egyptian Dynastic art is noteworthy for its aesthetic conservatism. One can argue, indeed, however subjectively, that the consistency in Dynastic Egyptian art style is far greater than its variation in time space.
That is, the change and variation, though identifiable and important, appears small against the background of overwhelming stylistic uniformity and constancy.
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CHAPTER 4
GEOGRAPHY AND GEOGRAPHIC VARIATION
As I am going to study geographic variation in ancient Egyptian art, I should say a few words about ancient Egyptian geography. Present-day Egypt borders the
Mediterranean Sea to the north and Sudan to the south. It lies between Libya in the west and Israel and the Red Sea to the east. Ancient Egypt was also located in the northeast corner of the African continent, though its borders and territories were often in a state of flux. The northern border of ancient Egypt was the Mediterranean Sea. Lower Egypt extended from the sea to the southern apex of the delta, where the Nile River branches into hundreds of distributaries. These distributaries lead into the Mediterranean Sea creating the Nile Delta region. The southern frontier of ancient Egypt was for the most part drawn at the first cataract of the northern flowing Nile, at Aswan (inland) and the island fort of Elephantine, but moved farther south during differing periods (Figure 7).
These changing borders, drawn south of Abu Simbel and north of Wadi Halfa, were included in Egyptian geography in New Kingdom texts, while the oases that run parallel to the Nile from northern Siwa to southern el-Kharga were under Egyptian rule during the majority of the Dynastic and Roman Periods (Malek 2003: 5-6).
Ancient Egypt was divided into two lands, the “red land” and the “black land.”
The former represented the barren desert, which protected Egypt along two of its borders
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and also supplied precious metals and gems. The ‘black land’ was the area on both sides of the fertile Nile along which the Egyptians, in both ancient and modern times, grew crops; the name derives from the black silt that is left after the annual inundation (British
Museum 2005). To be noted is the tremendous impact that geography had on the themes and style of ancient Egyptian art (Baines and Malek 2000: 14-15; Robins 1997: 13-14).
The royal cities changed on a dynastic basis. As a result, each of the major cities of ancient Egypt contributed to the formation of artistic style and technique. As mentioned earlier, the capital of Tell El-Amarna gave its name to the distinctive style associated with its period, while in response to the Hyksos invasion in the northern Delta region, a new art style referred to as “southern” emerged in the city of Thebes (see map,
Figure 8). “Southern” Theban art exhibited a combination of the established classical, formal technique blended with mannerism art (Aldred 1961: 8-9). For much of the Third
Intermediate Period, Upper and Lower Egypt operated independently of one another.
Abydos - in Upper Egypt - was known for perfection of style and technique, while
Abu Simbel – in Lower Nubia - is covered with enormous monuments to Ramesses II which optimize the effects of sunlight (Glanville 1957: 100-101). Pharaohs from the New
Kingdom, namely Thutmose I, Thutmose III, and Hatshepsut, constructed a number of enormous monuments at Deir el-Bahari – north of Thebes – which include Hatshepsut’s obelisks, the largest standing in the world. Egypt’s changing borders provided multiple options in regards to the location of royal cities. As Egypt is no stranger to looting and material borrowing within her own borders, there is an alarming number of artifacts that have no provenience. Also, monuments were moved in ancient times, or they may occur in secondary deposits—like the Narmer Palette (Figure 10). 28
Figure 8. Archaeological Map of Ancient Egypt (from the University of Chicago, Oriental Institute). 29
CHAPTER 5
QUANTITATIVE APPROACHES TO ART
There have been prior attempts to develop rigorous, quantitative approaches to the analysis of art. Of these, the most prominent in archaeology is symmetry analysis.
Designed to meet the archaeologist’s need to describe form and pattern, this type of analysis plays an important role in identifying, describing and quantifying variation in material culture. Dorothy Washburn (1977) developed this system to describe archaeological artifact designs. Based in part on the geometry of crystallography,
Washburn’s method of analysis involves identification and recording of the symmetrical patterns that incorporate transformations of translation, reflection, and rotation in artifact designs. Washburn created both a systematic procedure and a nomenclature to describe identified and recorded patterns. Washburn uses “symmetry analysis” to more accurately analyze the patterns found on ceramics, baskets, textiles, and so forth.
While Washburn’s analysis can be applied to “symmetrical” designs, much of the ancient artwork that has been found is not symmetrical. Egyptian art, for example, contains dramatic variations in scale – contractions or dilations – that render it non- symmetrical. For the same reason, most naturalistic art, including landscapes, portraits, and compositions that combine them, are not even approximately symmetrical
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Washburn’s own definition of the “symmetrical” operations states that the size is to remain constant” (Washburn quoted in Brown et al. 2005: 54), a constraint not obeyed by much of the art associated with early civilizations. While some mathematicians seem to include dilation or contraction mappings as a type of symmetry transformation, such scaling is not conventionally considered to be one of the canonical geometric transformations encompassed by symmetry (i.e., translation, reflection, rotation, and glide reflection). Obviously, techniques other than symmetry analysis must be used to analyze non-symmetrical patterns. Fractal analysis is one potential approach to analyzing non-symmetrical designs, which is why I chose to explore it in this research.
I must emphasize that patterns are not necessarily fractal merely because they are not symmetrical. More formally, a lack of symmetry is a necessary but not a sufficient condition for a pattern to be a fractal. Thus, fractals are only a subset of non-symmetrical patterns. Therefore, one must first show that a pattern is fractal before using fractal analysis to quantify the characteristics of the pattern.
Fractal geometry is the study of complex, irregular, and rough phenomena; it is the geometry of nonlinear processes (Brown 2001: 620). The father of fractal analysis,
Benoit Mandelbrot, recognized the fractality of cultural and natural phenomena, and coined the term fractal. He provided a technical definition of fractal patterns (Mandelbrot
1982; see Brown and Liebovitch [2010] for an extended explanation), but here I will limit the discussion to the most important qualities that fractals exhibit. Fractals are patterns that exhibit self-similarity, which means that they are composed of parts whose shapes are similar to the whole but smaller in scale. In addition, the number of component parts systematically increases rapidly as their scale shrinks (Brown 2001: 619; Brown et al. 31
2005: 40-1; Mandelbrot 1982: 14-15). In the simplest terms, this means a fractal is composed of many small pieces and a few large pieces, with the number of pieces decreasing rapidly as their size increases. In a fractal, the mathematical relationship between the number of constituent parts and their linear size is described by a power law
(Brown and Witschey 2003: 1624). Fractal analysis consists of determining whether these rules hold for a particular pattern, and if they do, statistically estimating the key parameter of the power law, its exponent, which allows one to calculate its fractal dimension. When this dimension is a fraction (rather than an integer) the pattern is definitely a fractal. Although fractals do exist that have integral dimensions, they rarely appear in empirical, random data sets such as those I am analyzing. I will be analyzing drawings of low-relief sculptures. It makes sense to treat these as patterns embedded in two dimensions. As such, they will necessarily have a fractional fractal dimension between 1 (the dimension of a line) and 2 (the dimension of a plane) (Taylor et al. 1999:
25-28.).
Power-laws are functions of the form f x xb , where the exponent, b, is the parameter to be measured. For the fractal or self-similarity dimension the following power-law function, in varying forms, is used:
1 a (1) sD
where a equals the number of self-similar pieces, s equals the linear scaling factors of the pieces to the whole, and D equals the dimension to be calculated. To solve for D, the 32
elements of the above equation can be rearranged (Mandelbrot 1983: 37 in Brown et al.
2005: 41):
log a D (2) log s
The fractal dimension, D, is not an integer for the majority of fractals. This dimension measures the set’s complexity and serves as the most important quantitative measure of the pattern’s qualities. In a fractal, as the number of “little pieces” increases in relation to the number of larger elements, the fractal dimension increases. As an example of this, one can imagine a line with a dimension between 1 and 2. The closer the line approaches 2, the more the line fills the plane and the more complex the curve becomes
(Brown et al. 2005: 41-42). In art, therefore, higher fractal dimensions tend to correspond to busier or more complicated compositions with a greater concentration of detail. I will be testing the proposition that the fractal dimensions of ancient Egyptian artworks varied through time or space.
A few examples of fractal measurement are in order to make these unfamiliar ideas concrete. For these examples, I will use the Cantor Set and Richard Taylor’s application of fractal analysis to Jackson Pollock’s Drip paintings. These will provide the clearest explanation, both visually and mathematically, for the fractal definition, while the drip paintings will help to clarify the analysis used for this project. The Cantor set is a primordial fractal created by Georg Cantor, a German mathematician working at the turn of the twentieth century. Cantor was conducting work on set theory, when he created what is now known as the Cantor set. Cantor started with a line segment representing the
33
interval [0, 1] (square brackets represent inclusion of endpoints). Cantor then removed the middle third of the line, leaving two equal length line segments [0, 1/3] and [2/3,1] behind (Figure 9); from these remaining segments, Cantor repeated this procedure ad infinitum providing an example of fractal dust:
Figure 9. Illustration of the first six iterations of the middle third Cantor Set (Peitgen et al. 1992: 172)
This iterative process, formed from the reduction in scale of the line segments, causes the Cantor set to be composed of self-similar parts, which causes the line segments to be scale invariant. To calculate the fractal dimension, D, of the Cantor set
Equation (2) is used. This equation relates the fractal dimension, D, to the size, s, and the number of pieces, a. The size, s, becomes 1/3 because each remaining line segment is one-third the length of the original, while the number of pieces, a, becomes 2 as the removal of the middle third of the line yields two remaining segments. After plugging in the numbers accordingly, the equation is:
34
log2 D 0.6309 (3) log1/ 3
The above equation indicates that not only is the Cantor Set a fractal, its fractal dimension is 0.6309.
The second example of a fractal is Jackson Pollock’s drip paintings, which were analyzed by Richard Taylor et al. (1999). This example will provide evidence of the fractality of art and of the usefulness of fractals in creating both a chronology and, in
Taylor’s case, a counterfeit detection system. Pollock’s famous “drip” paintings of the late 1940’s display a fractality. Taylor recreated Pollock’s drip painting technique, dripping paint from a swinging bucket onto enormous canvases in his barn. The recreation of Pollock’s continuous stream of paint, by dripping the paint from a bucket, displayed a chaotic process normally found in nature and in Pollock’s paintings. Taylor’s replication of Pollock’s technique displayed the same self-similarity that occurs in both
Pollock’s art and in nature.
Taylor et al. (1999) estimated the fractal dimension of various Pollock drip paintings using the box-counting method of fractal analysis. To do this, Taylor and his team scanned the Pollock painting with a computer generated mesh, the largest square being set to the size of the canvas while the smallest being set to the finest paint work. As the mesh square size became increasingly smaller, the number of squares that the image appeared in was measured. The paintings not only proved to be fractal in nature, but they exhibited increasing complexity over the years and therefore an increasing fractal dimension. This method not only proves the natural quality of Pollock’s paintings, but
35
also provides a means for routing out fakes (because the forgeries are not strictly fractal like the originals) and, determining chronology (because the dimensions changed systematically through time (Taylor et al. 1999: 25-28).
In sum, I will use fractal analysis (specifically, box counting) to determine whether ancient Pharaonic art was fractal and, if so, what the fractal dimensions of key examples were. Then I will explore whether their dimensions varied in a systematic way that can aid archaeological analysis.
36
CHAPTER 6
MATERIALS AND METHODS
To reiterate, my goals were to determine whether Egyptian bas-reliefs have fractal characteristics and, if so, whether the fractal dimension could be used to measure either chronological or geographic variation in art style. Therefore, I analyzed a variety of monuments from different locations in Upper and Lower Egypt, as well as from different periods of Egyptian history, from the Predynastic Period to the Late Period in order to capture a wide range of chronological and geographic variation. I analyzed exclusively low relief sculpture and endeavored, whenever possible, to analyze complete, well- preserved works for which the entire composition could be measured. I selected works that are exemplars of their style in order to try to capture the variation that has been recognized, documented, and analyzed qualitatively by art historians. I used the well- known box-counting method to evaluate the fractality of the patterns and estimate the fractal dimensions of them.
The purpose of box-counting is to estimate the statistical relationship between the number of elements of different sizes in the composition. This is done by superimposing a grid of squares on the design and then by counting how many of the squares are occupied by the figure as the size of the grid shrinks. So, the basic procedure of box- counting is that a grid of squares is overlaid on the image to be measured and the number
37
of boxes that include some portion of the figure are counted. The number of squares, N, required to cover the design will depend on the size of the squares, s, so N is a function of s, or Ns. The size of the grid is repeatedly reduced and the two changing variables, N and s, are recorded. As the grid shrinks, so does the size of the boxes, and therefore the number of occupied boxes increases. If the relation between the logarithm of versus the logarithm of s is linear on a graph, then the image is fractal because the linearity of the log-log relationship indicates that a power law relates the number of parts to their size. If the slope of the best fit line is b, then the fractal dimension, D, is D b (Brown et al. 2005: 55-6).
In order to use the box counting analysis to determine the fractal dimension of the selected works of ancient Egyptian monumental art, it was necessary to find drawn reproductions of the monuments. I used drawings rather than photographs – the obvious alternative – because photographs carry a vast amount of irrelevant information, such as glare or sheen from different textures of the stone or the carving, for example. Also, most monuments have suffered from some erosion or damage. I was not interested in measuring damage or erosion but rather the style and composition of the original works.
It would be difficult to edit photographic images to eliminate the irrelevant data.
Drawings, however, serve this purpose admirably. Through the drawing process, the artists have already extracted the vital symbolic information about the design and pattern of the carving and effectively eliminated the irrelevancies. Of course, drawings do show cracks, missing chunks, and erosion, but I selected well-preserved examples and used photographs to amend or correct drawings when necessary. Of course, Egyptologists have been drawing monuments since the dawn of the discipline, and so I was able to use 38
images from the well-known leaders in the field of Egyptology including N. de G.
Davies, Jean François Champollion, and Cyril Aldred.
Before analyzing the drawings of the monuments with the box counting programs,
I first scanned them in as black and white bitmaps at 600x600 dpi using a Brother MFC-
9840CDW scanner. Since the box-counting method counts the number of times a pixel in the image falls within a box, I had to erase all non-original lines and marks, indicating hachure, stippling, and other symbology used conventionally to indicate excision, damage, erosion, and so forth. To perform the image editing, I used the GIMP2 (GNU
Image Manipulation Program) software. GIMP2 is a free software program used to retouch and edit images. The program has tools to erase and paint, as well as threshold, invert, and options to turn images into grayscale – all of which were used on the selected relief drawings to get the clearest lines possible for analysis. GIMP2 is a widely used image manipulation program that has been employed in studies as far reaching as forensic analysis (Castiglione 2011: 6479-684), astronomical imaging (Rector et al.
2007), and biometric studies of eyelid shape (Rhee et al. 2012). Using the tools in
GIMP2, I completed all clearly marked broken lines with the “pencil” tool, but no other additions were made. I cropped the images closely so as not to erroneously include unrelated empty space in the analysis, which could have skewed the results. I also used
GIMP2 to manage file formats because the different fractal analysis programs I used all had varying format requirements. For example, Benoit will only measure white pixels on a black background for files in bitmap format. Therefore, I removed all existing indexed layers and I then inverted the images to white lines on a black background and confirmed the bitmap format. 39
I originally began analyzing the image bitmaps with two separate programs –
Benoit 1.3 and ImageJ. I later had the opportunity to use the Fractal3e analysis program as well. Although all three programs are widely used and appear to have been validated, their algorithms are all slightly different. I thought that using multiple programs would serve to cross-check and validate the results. Note that because the algorithms executed by the three programs are different, the results produced ought to vary at least slightly.
Nevertheless, I did expect them to be internally consistent for each program and for the results for each program to exhibit a logical and systematic relationship to the results given by the others. Below is an introduction to each software program and an explanation of the settings that were used in each for this research.
Produced by TruSoft International Inc, Benoit 1.3 is an often cited fractal analysis program that has been used in a wide variety of studies by researchers worldwide.
Researchers, scientists, mathematicians, and archaeologists have used Benoit 1.3 to study a diverse range of topics including metropolitan transportation networks (Burnett and
Pongou 2006: 241-247), molecular clouds (Datta 2003), mussel bed spatial patterns
(Crawford et al. 2006: 1033-1044), and remote sensing imagery (Zubrow 2007: 219-
235). Benoit offers the unusual option of rotating the image a certain number of degrees at each box size to find the best estimate of the box count at the grid size. It also offers the user the option of selecting different increments for reducing the grid size. Using the box counting command, I set the coefficient of box size decrease to 1.1 (the minimum) and the increment of grid rotation to 1.0 on a 0-90° scale (also the minimum), in order to gain the most precise results possible, even though these settings increased the processing time by several orders of magnitude. After the analysis was complete, I imported the raw 40
data file into Excel and calculated the logarithms of both the box count and the number of occupied boxes. Then I used Excel to graph the data on a scatter plot for each image and used least squares regression to fit a trend line to the double logarithmic data. The absolute value of the slope of the trend line is an empirical estimate of the fractal dimension. The results are discussed in detail by monument below in the Results chapter.
As mentioned earlier, I also used ImageJ software, which is a Java-based image manipulation program developed by the National Institutes of Health. The program, originally released in 1997, is widely used in biomedical science for studying microphotographs but has been utilized in many fields including, studying the spatial distribution of rural settlements in Guizhou Province (Zhou and Zhou 2011: 85-91), lithic use-wear analysis (Stevens et al. 2010: 2671-2678), and tomography imaging of South
African pottery (Jacobson et al. 2011: 240-243). ImageJ, like GIMP2, allows the user to edit and reformat bitmaps, but additionally allows for a variety of analyses, including the fractal box count. This function in ImageJ requires all images to be 8-bit binary and using the built in functions, I turned all of the image bitmaps binary prior to running them through the three analysis programs. As the images being analyzed are inverted, I selected the “black background” option and saved both the “Results” and the “Plot
Values” files once the analysis was complete. ImageJ records the “Plot Values” in logarithmic form and I imported these results into Excel, and then graphed the data on a scatter plot to show the log-log graph for each image, the results of which are discussed in-depth below.
Last, the third analysis program that I used, with the permission of its creator Dr.
Hiroyuki Sasaki, is the Fractal3e (Frac3) software that he originally created to evaluate 41
ecotypes of Zoysia japonica (Japanese lawn grass). Frac3 was created in 1991 and since then has been used in a variety of studies both inside and outside of Japan’s National
Institute of Livestock and Grassland Science, where Dr. Sasaki first used the software.
Physicists studying the fractal dimension of surface topographies (Guzman-Castaneda et al. 2012) and archaeologists studying geographical complexity in Romania (Andronache
2011: 175-185) have utilized the box counting function of Frac3 for their research. The software was created to be used with preprocessed images and allows the user to define the image area to be analyzed. As mentioned above, I closely cropped each image during the image cleaning step and, to ensure accuracy, I analyzed the exact image size and width in each program. As the images were all saved as inverted bitmaps, the “Except
Black” option was selected on the “Fractal Dimension” window and the fractal dimension and R(n) values were recorded. The relationship between box size and the number of times the boxes are intersected is also provided and, for each image, I imported this data into Excel, following the same process as with the results from the Benoit 1.3 and ImageJ programs, I calculated the logarithms of the data and graphed the results onto a log-log scatter plot.
42
CHAPTER 7
RESULTS
As previously mentioned, the artwork chosen for this research was selected from the major Dynastic periods of ancient Egyptian history, from complete monuments when possible. In total, twenty-seven reliefs were selected from the Predynastic Period, Old
Kingdom (OK), Middle Kingdom (MK), New Kingdom (NK) and Late Period. Below is a review of each image with the corresponding scatter plot graphs showing the fractal dimensions calculated from each of the three analysis programs. I have included the non- inverted images that I used during analysis, which reflect the removal of all non-original lines, damage marks, etc. Note that there are not many Predynastic or Middle Kingdom monuments to choose from, and thus the corpus of monuments I analyzed is dominated by Old Kingdom and New Kingdom pieces. My selection was non-random, but not arbitrary. The most important factor in the selection was the availability of good drawings of representative monuments. I did not select images specifically for their fractal qualities.
43
Figure 10. The Narmer Palette, Predynastic Period, 0/1st Dynasty: Above left, Obverse; right, Reverse (Promotora Española de Lingüística).
The Narmer Palette – discovered in the temple of Hierakonpolis (Kom el-Ahmar) by James Quibell and Frederick Green – is a raised relief siltstone palette, most likely used as a temple offering, showing the king on the Obverse (Figure 10, left) wearing the crown of Lower Egypt in procession above two serpopards – mythical serpent leopard hybrids – in confrontation and on the Reverse (Figure 10, right), wearing the crown of
Upper Egypt while about to strike a prisoner. The palette is commonly thought to be one of the earliest representations of Egyptian art that contains elements that would become central characteristics of Egyptian art – symmetry and two-dimensional representations.
The palette dates to c. 2990 BC and is currently housed in the Egyptian Museum in Cairo.
As both sides of the palette are preserved remarkably well, I scanned, cleaned and analyzed them as separate images, designated as Obverse and Reverse. After processing
44
the Obverse image of the Narmer Palette in GIMP2, I then analyzed the 821 kb bitmap with the three programs following the previously defined steps. I calculated the data and recorded the following fractal dimensions, D: Benoit 1.3 D = 1.50 (Figure 11), ImageJ D
= 1.47 (Figure 12), and Frac3 D = 1.52 (Figure 13).
6
5 y = -1.5024x + 4.9605 R² = 0.9943 4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 11. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Narmer Palette Obverse.
45
12
10 y = -1.4669x + 11.425 R² = 0.9956
8
6
4
2
Log(Number Boxes) of Log(Number 0 0 2 4 6
Log(Box Size(Pixels))
Figure 12. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes - Narmer Palette Obverse.
6
5 y = -1.5189x + 4.9807 4 R² = 0.9934
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 13. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes - Narmer Palette Obverse.
46
I then processed the reverse side of the Narmer Palette (Figure 10, right) in
GIMP2, and analyzed the 861 kb bitmap with the three analysis programs, resulting in the following fractal dimensions, D: Benoit 1.3 D = 1.50 (Figure 14), ImageJ D = 1.48
(Figure 15), and Frac3 D = 1.49 (Figure 16).
6
5
y = -1.5001x + 4.9002 4 R² = 0.9966
3
2 Log(Number Boxes) of Log(Number 1
0 0 0.5 1 1.5 2 2.5 Log(Box Size(Pixels))
Figure 14. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes - Narmer Palette Reverse.
47
12
10 y = -1.4747x + 11.286 R² = 0.9976
8
6
4
2
0 Log(Number Boxes) of Log(Number 0 2 4 6
Log(Box Size(Pixels))
Figure 15. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Narmer Palette Reverse.
6
5 y = -1.4937x + 4.8878 4 R² = 0.9964 3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 16. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Narmer Palette Reverse. 48
Figure 17. Menes Tablet, Predynastic Period, 1st Dynasty (Gardiner 1961: 405).
The Menes Tablet (Figure 17), discovered in 1897 by Jacques de Morgan in the recessed tomb at Naqada, is an ivory marker originally used to label the contents of the artifact that it was, in antiquity, connected to. Dated to the reign of Aha in Dynasty 1, c.
2955 BC, the tablet – most likely broken in antiquity - clearly shows the name of the king as well as a ship associated with the king’s seafaring. After processing with the image with GIMP2, I imported the 3,041 kb bitmap into each analysis program and calculated the following results: Benoit 1.3 D = 1.62 (Figure 18), ImageJ D = 1.56 (Figure 19), and
Frac3 D = 1.61 (Figure 20).
49
6 y = -1.6165x + 5.6053 5 R² = 0.9994 4
3
2
1
Log(Number Boxes) of Log(Number 0 0 1 2 3 Log(Box Size(Pixels))
Figure 18. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Menes Tablet.
14 y = -1.5623x + 12.789 12 R² = 0.9996
10 8 6 4 2 0 Log(Number Boxes) of Log(Number 0 2 4 6 Log(Box Size(Pixels))
Figure 19. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Menes Tablet.
50
6 y = -1.6073x + 5.6155 R² = 0.9993 5
4
3
2
1 Log(Number of Boxes Log(Number 0 0 1 2 3
Log(box Size(Pixels))
Figure 20. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Menes Tablet.
51
Figure 21. Mastaba of Meresankh III, Old Kingdom, 4th Dynasty – Main Room, East Wall, South of Entrance (Simpson 1974: 60).
In the 1970s William Kelly Simpson undertook the task of publishing the previously unpublished excavations of the Giza Necropolis excavations carried out by
George Reisner and the Joint Egyptian Expedition. Under the title of the Giza Mastaba
Series, which the University of Chicago has made available via the web, the series details the excavated mastabas tomb by tomb. The Mastaba of Queen Meresankh III, c. 2507
BC, cleared by Reisner in 1927 and numbered as Giza 7530-7540 (G7530-7540), is the first tomb discussed in detail in this series and is the entire subject of Volume I. In an attempt to explore the possible variation in fractal dimension by style or composition, I
52
chose to analyze two reliefs from this tomb. The first relief, from the east all of the south entrance of the Main Room (Figure 21), is notable for the business of the composition, with five full registers of activity in the description of ships and daily labors. After I processed the scanned line drawing in GIMP2, I recorded the following fractal dimension, D, from each analysis of the 1,053 kb bitmap: Benoit D = 1.53 (Figure 22),
ImageJ D = 1.49 (Figure 23), and Frac3 D = 1.54 (Figure 24).
6 y = -1.5328x + 5.1102
5 R² = 0.9926
4
3
2 Log(Number of Boxes) Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 22. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Meresankh III, Main Room, East Wall, South of Entrance.
53
12 y = -1.4899x + 11.79 R² = 0.9936 10
8
6
4
2
0 Log(Number Boxes) of Log(Number 0 2 4 6 Log(Box Size(Pixels))
Figure 23. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Meresankh III, Main Room, East Wall, South of Entrance.
6
5 y = -1.5436x + 5.1036 R² = 0.9909 4
3
2 Log(Number of Boxes) Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 24. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Meresankh III, Main Room, East Wall, South of Entrance.
54
Figure 25. Mastaba of Meresankh III, Old Kingdom, 4th Dynasty – South Door Jamb (Barta 2006: 260).
The Mastaba of Meresankh III, located in the Eastern Cemetery at Giza revealed a number of well-preserved reliefs of varying subjects and composition. This rock cut mastaba belonging to the wife of 4th Dynasty pharaoh Khafra, c. 2520-2494 BC, is one of the most elaborate in the Giza necropolis. The second relief I chose from this mastaba is still visually fractal; however, the contents are not as densely packed as those in the first selection, which is also from the Main Chamber. The above relief from the south door jamb of the Main Chamber focuses on a representation of Meresankh III holding a lotus to her nose as she stands beneath Anubis (Figure 25). I scanned the line drawing of this
55
relief and processed the image following the methods explained in Chapter 8. I recorded the following fractal dimension, D, from each analysis of the 819 kb bitmap: Benoit D =
1.44 (Figure 26), ImageJ D = 1.38 (Figure 27), and Frac3 D = 1.43 (Figure 28).
6
5
4 y = -1.4407x + 4.8221 R² = 0.9916 3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 26. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Meresankh III, South Door Jamb.
56
12 y = -1.3796x + 11.067 R² = 0.9933 10
8
6
4
2
0 Log(Number Log(Number of Boxes) 0 2 4 6 Log(Box Size(Pixels))
Figure 27. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Meresankh III, South Door Jamb.
6
y = -1.4302x + 4.8001
5 R² = 0.9921 4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 28. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Meresankh III, South Door Jamb. 57
Figure 29. Tomb Chapel of Werirenptah, Old Kingdom, 5th Dynasty (Robins 1997: 23).
Werirenptah, a high ranking official during the 5th Dynasty reign of Neferirkara c.
2446-2426 BC, is buried in the Saqqara necropolis tomb chapel most notable for the varied domestic and agricultural scenes found within. In Figure 29, just such a scene of domestic labors is shown from the east wall of the mastaba. Originally made up of more than six completely filled registers, much of which was lost over time, the relief demonstrates well-known characteristics of Egyptian art, including scaling, frontality and the use of overlapping figures to show depth. After all unoriginal damage lines were removed and the image was processed, the 20,002 kb bitmap was run through the three box counting programs resulting in the following fractal dimensions: Benoit D = 1.68
(Figure 30), ImageJ D = 1.60 (Figure 31), and Frac3 D = 1.67 (Figure 32).
58
7 y = -1.6815x + 6.4933 6 R² = 0.999
5
4
3
2
1
Log(Number Boxes) of Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 30. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Werirenptah.
16
14 y = -1.5996x + 14.788 R² = 0.9998 12
10
8
6
4
2 Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 31. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Werirenptah. 59
7
6 y = -1.6677x + 6.4875 R² = 0.9996 5
4
3
2
1
Log(Number Boxes) of Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 32. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Werirenptah.
60
Figure 33. Mastaba of Sekhemka, Old Kingdom, 5th Dynasty – East Wall, South Section (Simpson 1980: 112).
Volume IV of the Giza Mastaba Series details the mastaba of Sekhemka, Giza tomb 1029 (G1029), in the Western Cemetery of Giza. The Mastaba is dated to the just after the reign of Nyuserre Ini a 5th Dynasty pharaoh, c. 2416-2392 BC. As with the mastaba of Meresankh III, I selected two reliefs from this tomb to compare the variation in fractal dimension based upon style and composition. The above relief, Figure 33, from the south section of the east wall shows Sekhemka standing with his son, beside and above them are vertical registers of hieroglyphs. I processed and analyzed the 20,681 kb bitmap following the defined steps and recorded the following fractal dimension, D, from each program: Benoit D = 1.54 (Figure 34), ImageJ D = 1.48 (Figure 35), and Frac3 D =
1.56 (Figure 36).
61
7
6 y = -1.5367x + 6.1639 R² = 0.999
5
4
3
2
1 Log(Number of Boxes) Log(Number
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 34. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Sekhemka, East Wall, South Section.
14 y = -1.4828x + 14.053 R² = 0.9994 12
10
8
6
4
2
Log(Number Boxes) of Log(Number 0 0 1 2 3 4 5 Log(Box Size(Pixels))
Figure 35. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Sekhemka, East Wall, South Section. 62
7 y = -1.5562x + 6.1951 6 R² = 0.9983
5
4
3
2
1
Log(Number Boxes) of Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 36. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Sekhemka, East Wall, South Section.
Figure 37. Mastaba of Sekhemka, Old Kingdom, 5th Dynasty – South Wall (Simpson 1980: 113).
The second relief analyzed from the mastaba of Sekhemka is from the south wall of the tomb and shows the seated figure of Sekhemka as he receives a lotus blossom 63
from, presumably, his son (Figure 37). Below him are dancers and musicians and before him are registers of food offerings. I scanned and processed the line drawing of the relief and after analyzing the 22,209 kb bitmap with the three separate programs, I calculated the resulting data and recorded the following fractal dimensions: Benoit D = 1.56 ( Figure
38), ImageJ D = 1.45 (Figure 39), and Frac3 D = 1.59 (Figure 40).
7 y = -1.5634x + 6.2952 6 R² = 0.9976
5
4
3
2
1
Log(Number Boxes) of Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 38. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Sekhemka, South Wall.
64
y = -1.4459x + 14.244 14 R² = 0.9996 12
10
8
6
4
2 Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 39. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Sekhemka, South Wall.
7 y = -1.5892x + 6.3278 6 R² = 0.9978
5
4
3
2
1 Log(Number Boxes) of Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 40. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Sekhemka, South Wall. 65
Figure 41. Mastaba of Ptah-Hotep, Old Kingdom, 5th Dynasty (Aldred 2004: 87).
The limestone mastaba of vizier Ptah-Hotep, shared with his son Akhet-hotep, dates to the 5th Dynasty reign of Djedkare Izezi, c. 2388 – 2356 BC, and is located in the ancient necropolis at Saqqara. Originally discovered by Auguste Mariette in the 1850’s and further explored over the next forty years, the tomb is designated as D64. The mastaba is well known for the quality of the preserved reliefs, including the above image showing the sculptor of the 5th Dynasty tomb, Niankhptah, resting and drinking wine while boatmen engage in conflict (Figure 41). I scanned, processed and analyzed the line drawing bitmap, 4,402 kb, of this relief and calculated the following fractal dimensions:
Benoit D = 1.59 ( Figure 42), ImageJ D = 1.51 ( Figure 43), and Frac3 D = 1.61 (Figure
44).
66
7 y = -1.5864x + 5.7796 R² = 0.9984 6
5
4
3
2
1
Log(Number Boxes) of Log(Number 0 0 1 2 3 Log(Box Size(Pixels))
Figure 42. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Ptah-Hotep.
14 y = -1.5091x + 13.172 12 R² = 0.9998
10
8
6
4
2
Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 43. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Ptah-Hotep. 67
7 y = -1.6124x + 5.8139 6 R² = 0.9977
5
4
3
2
1 Log(Number Boxes) of Log(Number 0 0 1 2 3 Log(Box Size(Pixels))
Figure 44. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Ptah-Hotep.
Figure 45. Mastaba of Senedjemib Inti, Old Kingdom, 5th Dynasty – Room II, South Wall (Simpson and Manuelian 2000: 352).
68
Excavated by a joint team from Harvard University and the Boston Museum of
Fine Arts in the fall of 1912, the mastaba, Giza tomb numbered 2370 (G2370), of
Senedjemib Inti was constructed from limestone and made up of five rooms. Senedjemib
Inti, a vizier late in the reign of 5th Dynasty King Djedkare Izezi, c. 2388-2356 BC, oversaw major agricultural, financial and judicial matters which the mastaba reliefs detail through verbatim transcriptions of letters sent to Inti from the pharaoh. These letters, three in total, described some of the more important projects that Inti oversaw during his life and the reliefs depicting them on three walls in the anteroom were still well intact by the time Lepsius first visited the tomb in the 1840’s (Simpson and Manuelian 2000: 1-2,
23-4, 37, 89). On the south wall of the anteroom, one of these reliefs shows Inti being carried on a chair in procession while daily labors are carried out in the lower registers
(Figure 45). I processed the line drawing of the relief and calculated the following fractal dimensions for the 22,192 kb bitmap: Benoit D = 1.53 (Figure 46), ImageJ D = 1.40
(Figure 47) and Frac3 D = 1.57 (Figure 48).
69
7
6 y = -1.5309x + 6.238
R² = 0.9962 5
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 46. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Senedjemib Inti.
y = -1.3966x + 14.082 14 R² = 0.9996 12
10
8
6
4
2 Log(Number of Boxes) Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 47. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Senedjemib Inti. 70
7 y = -1.5649x + 6.2756 6 R² = 0.9967
5
4 3 2 1
Log(Number Boxes) of Log(Number 0 0 2 4 Log(Box Size(Pixels))
Figure 48. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Senedjemib Inti.
Figure 49. Mastaba of Qar, Old Kingdom, 6th Dynasty – Court C, Lower Half of North Wall (Simpson 1976: 106-9).
The mastaba of Qar, Giza tomb number 7101 (G7101), was cleared by George
Andrew Reisner in the 1924/5 season. The remaining structure of the mastaba of Qar is located in the Eastern cemetery at Giza and is almost completely below ground, the limestone superstructure having disappeared almost entirely by the time Reisner cleared 71
the tomb. A 6th Dynasty official, the titles in Qar’s tomb name him as an overseer for pharaoh Pepi I, c. 2289-2255 BC, in the towns that had built up around the pyramids during construction. The walls of the open Court C were mostly undamaged and the well preserved reliefs show Qar seated before food offerings while his funeral and ship procession travels to the embalming (Figure 49) (Simpson 1976: vii, 1, 5-6). The lower half of the north wall in Court C was separated onto four pages in the text and I scanned each page in individually. To align and merge the four separate images, I used Adobe
Photoshop CS6, and created the seamless image originally intended. I then followed the previously explained steps in cleaning, processing and analyzing, calculating the fractal dimensions for the 7,959 kb bitmap: Benoit D = 1.56 (Figure 50), ImageJ D = 1.46
(Figure 51), and Frac3 D = 1.60 (Figure 52).
7
6 y = -1.5642x + 6.0396
R² = 0.9944 5
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 50. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Qar.
72
14 y = -1.4615x + 13.761 R² = 0.9975 12
10
8
6
4
2 Log(Number of Boxes) Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 51. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Qar.
7
6 y = -1.5995x + 6.0585 R² = 0.9943 5
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 52. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Qar. 73
Figure 53. Mastaba of Idu, Old Kingdom, 6th Dynasty - West Wall, South of Stela Niche (Simpson 1976: 144).
In January of 1925, while crews were clearing out shafts that lay to the east of the mastaba of Qar (G7101, Figure 49), the mastaba of Idu, was discovered and crews immediately set about clearing out the burial chamber and sarcophagus. The mastaba of
Idu, Giza tomb 7102 (G7102), is similar to G7101 in that it is also mostly underground. 74
The tomb most likely belonged to the son or father of Qar and was also an official of the
6th Dynasty pharaoh Pepi I, c. 2289-2255 BC (Simpson 1976: 19, 26). The stela niche remained relatively well preserved at the time of clearing and shows Idu engaged in several tableaus of offerings (Figure 53). I followed the same procedures as with the previous images, scanning, processing and analyzing the line drawing bitmap, 1,150 kb. I recorded the following fractal dimension, D, for the relief: Benoit D = 1.56 (Figure 54),
ImageJ D = 1.51 (Figure 55), and Frac3 D = 1.57 (Figure 56).
6
5
4 y = -1.5571x + 5.2571 R² = 0.9906 3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 54. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Idu.
75
12 y = -1.5104x + 12.132
10 R² = 0.9903
8
6
4
2 Log(Number Boxes) of Log(Number
0 0 2 4 6 Log(Box Size(Pixels))
Figure 55. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Idu.
6
y = -1.5647x + 5.2527 5 R² = 0.9887
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 56. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Mastaba of Idu.
76
Figure 57. Temple of Mentuhotep II, Middle Kingdom, 11th Dynasty (Ullmann 2007: 16).
The first pharaoh to begin using the desert valley of Deir el-Bahari, on the west bank of the Nile across the river from Luxor, for his mortuary complex was the 11th
Dynasty pharaoh Mentuhotep II, c. 2061-1991 BC. Known for reuniting Egypt after the tumultuous years of the late Old Kingdom, Mentuhotep II decorated his mortuary temple at Deir el-Bahari with reliefs depicting various rituals of divinity, including the Bark
Journey, on behalf of Amun, represented on the southern outer wall of the sanctuary
(Figure 57) (Arnold 1997: 3, 74-5; Ullmann 2007: 7). I processed and analyzed the bitmap, 3,369 kb, and recorded the following fractal dimensions, D, based on their scatter plot graphs: Benoit D = 1.52 (Figure 58), ImageJ D = 1.45 (Figure 59), and Frac3 D =
1.55 (Figure 60).
77
6 y = -1.5172x + 5.5141
5 R² = 0.9976
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 58. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Temple of Mentuhotep II.
14 y = -1.4518x + 12.597 R² = 0.9986 12
10
8
6
4
2 Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 59. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Temple of Mentuhotep II.
78
6 y = -1.5525x + 5.5466
5 R² = 0.9958
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 60. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Temple of Mentuhotep II.
79
Figure 61. Tomb of Antefoker, Middle Kingdom, 12th Dynasty (Davies 1920: Plate VII).
Theban tomb 60 (TT60) belonged to Antefoker, a vizier to the 12th Dynasty pharaoh Senwosret I, c. 1971-1926 BC, is one of the best preserved Middle Kingdom tombs in the Theban necropolis and was intermittently cleared between 1907-1917 by
Weigall and Gardiner. Antefoker shares his tomb with his wife, Senet, who seems to be the primary figure in much of the tomb art, with the exception of the extended hunting scenes found on the north wall (Figure 61). I chose one of these reliefs to analyze from this tomb and processed the image. For the 16,337 kb bitmap, I recorded the following fractal dimensions: Benoit D = 1.52 (Figure 62), ImageJ D = 1.48 (Figure 63), and Frac3
D = 1.54 (Figure 64). 80
7
6 y = -1.5195x + 5.9902
R² = 0.9995 5
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels)
Figure 62. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Antefoker.
14 y = -1.4833x + 13.697 12 R² = 0.9996
10
8
6
4
2
Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 63. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Antefoker.
81
7 y = -1.5406x + 6.0253 R² = 0.9989 6
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 64. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Antefoker.
Figure 65. Amenemhat I Pyramid Temple, Middle Kingdom, 12th Dynasty (Arnold 1997: 77).
In the 12th Dynasty c. 1975 BC, Amenemhat I moved the capital city from Thebes back to Memphis and put a stop to the Theban tomb tradition of his predecessors. Instead, 82
he ordered his pyramid complex at el-Lisht to be built in the 6th Dynasty Memphis style, though due to internal conflicts artists had little in the way of ancient stylistic guides and instead created a unique style of their own. While Amenemhat’s pyramid temple is small, the building history is complex and material reuse can be seen throughout the complex, including a lintel relief showing the enthroned king observing the Sed-festival marking 30 years of rule and celebrating his continued reign (Figure 65) (Arnold 1997: 74-7). After preprocessing in GIMP2, the 4,579 kb bitmap was analyzed showing the following fractal dimensions: Benoit D = 1.51 (Figure 66), ImageJ D = 1.39 (Figure 67), and Frac3 D =
1.55 (Figure 68):
6 y = -1.5115x + 5.683
5 R² = 0.9946
4
3
2
1 Log(Number Boxes) of Log(Number 0 0 1 2 3 Log(Box Size(Pixels))
Figure 66. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Amenemhat I Pyramid Temple.
83
14 y = -1.3928x + 12.894 R² = 0.9981
12
10
8
6
4
Log(Number Log(Number of Boxes) 2
0 0 2 4 6 Log(Box Size(Pixels))
Figure 67. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Amenemhat I Pyramid Temple.
6 y = -1.5497x + 5.7119 R² = 0.9928 5
4
3
2
1
0 Log(Number Boxes) of Log(Number 0 1 2 3 Log(Box Size(Pixels))
Figure 68. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Amenemhat I Pyramid Temple.
84
Figure 69. Temple of Senwosret I, Middle Kingdom, 12th Dynasty (Ullmann 2007: 16).
A conspiracy ended the reign of the 12th Dynasty pharaoh Amenemhat I, c. 1991-
1962 BC and began the forty year reign of his son, Senwosret I. The longevity of
Senwosret’s reign and the proliferation of his temple building program guaranteed a standardization of style and technique among ancient Egyptian artists, which would extend well into the following Dynasties (Shaw 2000: 160-3). Senwosret I was the first of the Middle Kingdom pharaohs to have a wide-reaching building program and even though little remains of his temples and monuments, a partial relief from the southern portico of the temple of Senwosret I at Karnak corresponds clearly with a relief from the
11th Dynasty Temple of Mentuhotep II, c. 2061-1991 BC (Figure 57). The representation of the ritualistic bark journey of the king is clearly discernible in the remains Senwosret’s
Karnak temple (Figure 69) (Ullmann 2007: 7-8). I chose to analyze this relief due to its
85
similarity to the Mentuhotep relief analyzed from the 11th Dynasty. I processed the line drawing bitmap, 3,801 kb, and recorded the following fractal dimensions, D: Benoit D =
1.45 (Figure 70), ImageJ D = 1.34 (Figure 71), and Frac3 D = 1.46 (Figure 72).
y = -1.4518x + 5.4232 6 R² = 0.9934
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 70. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Temple of Senwosret I.
86
12 y = -1.3428x + 12.339 R² = 0.9962
10
8
6
4
Log(Number Boxes) of Log(Number 2
0 0 2 4 6 Log(Box Size(Pixels))
Figure 71. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Temple of Senwosret I.
6 y = -1.4638x + 5.439
5 R² = 0.993
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 72. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Temple of Senwosret I. 87
Figure 73. Pyramid Complex of Senwosret I, Middle Kingdom, 12th Dynasty (Robins 1997: 99).
As mentioned earlier, Senwosret I was the first Middle Kingdom pharaoh to have an extensive building program and his temples and monuments can be seen from Nubia
88
in the south to the Nile delta in the north. Senwosret I, as his father Amenemhat I had, built his pyramid complex at el-Lisht in the same Memphite style of the 5th and 6th
Dynasties (Robins 1997: 96-99; Wilkinson 2000: 133). Though the mortuary temple of
Senwosret’s pyramid complex suffered greatly from the poor materials used in Middle
Kingdom construction, remnants of the limestone inner enclosure wall remain showing at least one hundred separately, intricately carved falcon panels (Figure 73). I processed and analyzed the recreation line drawing of one of these panels and recorded the following fractal dimensions for the 3,701 kb bitmap: Benoit D = 1.57 (Figure 74), ImageJ D = 1.52
(Figure 75), and Frac3 D = 1.56 (Figure 76).
6 y = -1.5713x + 5.626
5 R² = 0.9989
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 74. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Pyramid Complex of Senwosret I.
89
14
12 y = -1.5206x + 12.872 R² = 0.9991 10
8
6
4
2
Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 75. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Pyramid Complex of Senwosret I.
6 y = -1.557x + 5.6322 R² = 0.9992
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 76. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Pyramid Complex of Senwosret I. 90
Figure 77. Temple of Amenemhat-Sobekhotep at Medmud, Middle Kingdom, 13th Dynasty: Temple Gates (Arnold 1997: 83).
By the end of the Middle Kingdom, during the 13th Dynasty, c. 1783-1640 BC, political unrest lead to non-royal pharaonic rulers, many of whom, in attempts to legitimize their rule and promote themselves, copied the architectural and artistic styles of those before them. In the third year of his reign, 13th Dynasty pharaoh Amenemhat-
Sobekhotep, c.1710 BC, built a temple at Medamud whose gates showed the same Sed- festival scene that is found in the 11th Dynasty Temple of Mentuhotep II c. 2061-1991
BC. I chose to analyze the line drawing of the Sed-festival doorframe (Figure 77), as it 91
was, much like the Bark Journey reliefs (Figures 57 and 69) a repeated theme found earlier in the Middle Kingdom (Figure 61). I processed and analyzed the image and recorded the following fractal dimensions, D, for the 11,262 kb bitmap: Benoit D = 1.62
(Figure 78), ImageJ D = 1.55 (Figure 79), and Frac3 D = 1.64 (Figure 80).
7
6 y = -1.6184x + 6.0236 R² = 0.999 5
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 78. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Temple of Amenemhat-Sobekhotep.
92
14
12 y = -1.5527x + 13.754
R² = 0.9994 10
8
6
4
Log(Number Boxes) of Log(Number 2
0 0 2 4 6 Log(Box Size(Pixels))
Figure 79. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Temple of Amenemhat-Sobekhotep.
7
6 y = -1.6348x + 6.0438
R² = 0.9986 5
4
3
2 Log(Number Log(Number of Boxes) 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 80. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Temple of Amenemhat-Sobekhotep. 93
Figure 81. Tomb of Rekhmire, New Kingdom, 18th Dynasty (Robins 1997: 28).
The political turmoil that marked the degradation in style and materials at the end of the Middle Kingdom and the entirety of the Second Intermediate Period ended in the
New Kingdom as early as the 18th Dynasty pharaohs Tuthmosis III, c. 1479-1425, and
Hatshepsut, c. 1473-1458 BC. During this time, the highest civil official in Thebes was the vizier Rekhmire, c. 1504-1480 BC, who built his tomb, Theban tomb 100 (TT100), in the western necropolis, much of which was still well preserved when Frederic Caillaud began exploring the massive tomb in 1819 (Robins 1997: 29; Bunson 1991: 226; Nova online 2012). Reliefs from TT100 show sculptors at work on the massive statues of a sphinx and an altar (Figure 81), activities that Rekhmire would have overseen in the course of his responsibilities as vizier and governor of Thebes. I analyzed the 7,485 kb image and recorded the following fractal dimensions: Benoit D = 1.54 (Figure 82),
ImageJ D = 1.45 (Figure 83), and Frac3 D = 1.57 (Figure 84).
94
7
6 y = -1.539x + 5.8728 R² = 0.9979
5
4
3
2
Log(Number of Boxes) Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 82. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Rekhmire.
14
12 y = -1.4504x + 13.34 R² = 0.9995
10
8
6
4 Log(Number of Boxes) Log(Number 2
0 0 2 4 6 Log(Box Size(Pixels))
Figure 83. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Rekhmire. 95
7
6 y = -1.566x + 5.9039
R² = 0.9975
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 84. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Rekhmire.
96
Figure 85. Tomb of Meryra, New Kingdom, 18th Dynasty (Amarna Period) (Davies 2004a[1903]: Plate XXII).
The ninth king of the 18th Dynasty, Amenhotep IV/Akenaten, c. 1353-1335 BC, ushered in a period of religious and artistic upheaval known as the Amarna Period, as discussed earlier. As this period in Egyptian history is known for the most dramatic changes both to the country and to its artistic style, I selected several works from the
Amarna tombs to analyze for possible variations in the fractal dimension associated with the new style. The first relief analyzed was found in the el-Amarna tomb of Meryra, c.
1343. During the reign of Akhenaten, Meryra served as both vizier and as the High Priest
97
of Aten, though he disappeared from the record less than six years after the death of the king. Meryra’s tomb, numbered Amarna tomb 4 (N4) by N. de G. Davies, was abandoned after the death of Akhenaten, and while the effigies and cartouches of the king and queen were destroyed in anger during antiquity, the rest of the reliefs in this tomb were left more or less intact (Bunson 1991: 11-2, 166; Davies 2004a[1903]:1-3, 7-8). On the east side of the south wall of Meryra’s tomb the attempts to remove Akhenaten from the known record can be seen in the destruction of the faces of the king and his wife from the relief showing the royal family making offerings to Aten (Figure 85). I processed and analyzed the line drawing of this relief, and after making the scatter plot graphs recorded the following fractal dimensions for the 25,705 kb bitmap: Benoit D = 1.55 (Figure 86),
ImageJ D = 1.40 (Figure 87), and Frac3 D = 1.58 (Figure 88).
7 y = -1.5457x + 6.3205 6 R² = 0.9961
5
4
3
2
1 Log(Number of Boxes) Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 86. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Meryra.
98
14 y = -1.3981x + 14.254
12 R² = 0.9995
10
8
6
4 Log(Number of Boxes) Log(Number 2
0 0 2 4 6 Log(Box Size(Pixels))
Figure 87. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Meryra.
7 y = -1.5836x + 6.3584 6 R² = 0.9958 5 4 3 2 1 Log(Number Boxes) or Log(Number 0 0 2 4
Log(Box Size(Pixels))
Figure 88. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Meryra.
99
Figure 89. Tomb of Panehesy, New Kingdom, 18th Dynasty (Amarna Period) (Davies 2004a[1905]: Plate XXIII).
Also from the Amarna Period, I chose a relief from the tomb of Panehesy,
Amarna tomb 6 (N6). This small tomb was found among the tombs to the north of el-
Amarna, and though some of the reliefs were destroyed after the death of Akhenaten, some of the reliefs survived and remained in the tomb during later Coptic reuse. The tomb of Panehesy, unlike any other tomb at el-Amarna, shows a personal tableau of
Panehesy himself, with his sister and daughters, sitting down to dine (Figure 89) (Bunson
1991: 201; Davies 2004a[1905]: 1-2, 9). I analyzed the 27,976 kb bitmap of the drawing of this relief and recorded the following fractal dimensions: Benoit D = 1.50 (Figure 90),
ImageJ D = 1.40 (Figure 91), and Frac3 D = 1.53 (Figure 92).
100
7
6 y = -1.5024x + 6.216 R² = 0.9972
5
4
3
2
1
Log(Number Boxes) of Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 90. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Panehesy.
14 y = -1.395x + 14.061 12
R² = 0.9991
10
8
6
4
Log(Number Log(Number of Boxes) 2
0 0 2 4 6 Log(Box Size(Pixels))
Figure 91. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Panehesy. 101
7 y = -1.5342x + 6.253 6 R² = 0.9975
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 92. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Panehesy.
102
Figure 93. Tomb of Meryra II, New Kingdom, 18th Dynasty (Amarna Period) (Davies 2004b[ 1905]: Plate XXXIII).
Also discovered in the cliffs of el-Amarna is the tomb of Meryra II, numbered
Amarna tomb 2 (N2), c. 1341 BC, the vizier who oversaw Queen Nefertiti’s affairs and household, who is most likely unrelated to the vizier Meryra (N4) discussed earlier. This small tomb was also abandoned when Akhetaten fell and suffers from the same destruction as the rest of the tombs in the el-Amarna area and was largely unfinished
(Bunson 1991: 166; Davies 2004b[1905]: 1-3, 33-38). Of the work that was completed,
103
an extensive relief on the east side of the south wall depicts Meryra II being rewarded by the king and queen (Figure 93). I analyzed the drawing of this relief and recorded the following fractal dimensions for the 28,876 kb bitmap: Benoit D = 1.64 (Figure 94),
ImageJ D = 1.50 (Figure 95), and Frac3 D = 1.67 (Figure 96).
7
6 y = -1.6345x + 6.5825 R² = 0.9975
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 94. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Meryra II.
104
16 14 y = -1.5009x + 14.895 R² = 0.9995
12 10 8 6 4
Log(Number Boxes) of Log(Number 2 0 0 2 4 6 Log(Box Size(Pixels))
Figure 95. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Meryra II.
7 y = -1.6668x + 6.6139 6
R² = 0.9968
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 96. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Meryra II.
105
Figure 97. Tomb of Apy, New Kingdom, 18th Dynasty (Amarna Period) (Davies 2004b[1906]: Plate XXXI).
The final Amarna period tomb chosen for analysis is that of the royal scribe Apy, c. 1340 BC, due to the incredibly well-preserved images of royal family. This tomb, though simple, rudely worked and incomplete, survived the retaliation that followed the end of Akhenaten’s reign, most likely by being filled in with sand before the atmosphere in el-Amarna became too violent (Davies 2004b[1905], v. IV: 19-20). Only the entrance and doorway were completed with reliefs, but these were preserved remarkably well.
From the entrance of Apy’s tomb, a relief showing the royal family making offerings to
Aten (Figure 97) is one of the best representations of the Amarna artistic style. I modified
106
and analyzed the 27,163 kb bitmap, recording the following fractal dimensions: Benoit D
= 1.59 (Figure 98), ImageJ D = 1.43 (Figure 99), and Frac3 D = 1.63 (Figure 100).
7
6 y = -1.5917x + 6.4731 R² = 0.9963 5
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 98. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Apy.
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16 y = -1.4336x + 14.584 14 R² = 0.9995
12 10 8 6 4
2 Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 99. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Apy.
7 y = -1.6275x + 6.507 6 R² = 0.9956
5
4
3
2
1 Log(Number of Boxes) Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 100. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Apy.
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Figure 101. Tomb of Neferhotep, New Kingdom, 18th/19th Dynasty (Davies 1973: Plate LI).
As the 18th Dynasty transitioned into the 19th Dynasty, Egypt started to recover from the reign of Akhenaten, and though the royal capital had moved back to Thebes, stylistic elements from the Amarna Period can still be seen in reliefs from this time, including those of the tomb of royal scribe Neferhotep, c. 1245 BC. The rock cut tomb of
Neferhotep, numbered as Theban tomb 49 (TT49), was first explored by Robert Hay in
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the 1830’s and further cleared between 1920-1930 by N. de G. Davies, is situated off of a central courtyard which also houses the entrance of three additional tombs, also of the
19th Dynasty. While the tomb itself is unimpressive and has seen wear by both time and reuse by local villagers, reliefs from both the inner and outer chambers were not too severely damaged. These reliefs give the only accounts of Neferhotep and his devotion to king Ay, c. 1279-1212 BC; notable is the relief found on the northeast pillar, south wall of the inner room wherein Neferhotep worships Amenhotep I and his mother, who had been deified after death (Figure 101) (Davies 1973: 1-7, 60-1). I processed and analyzed the 20,441 kb bitmap and recorded the following fractal dimensions: Benoit D = 1.67
(Figure 102), ImageJ D = 1.59 (Figure 103), and Frac3 D = 1.68 (Figure 104).
7
6
5 y = -1.6665x + 6.5321 4 R² = 0.9989
3
2
1 Log(Number Boxes) of Log(Number 0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 102. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Neferhotep.
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y = -1.5853x + 14.862 16 R² = 0.9997
14
12 10 8 6
4 Log(Number Boxes) of Log(Number 2 0 0 2 4 6 Log(Box Size(Pixels))
Figure 103. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Neferhotep.
7 y = -1.6838x + 6.5562
6 R² = 0.999
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 104. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Neferhotep. 111
Figure 105. Temple of Beit el-Wali, New Kingdom, 19th Dynasty (Champollion 1970a: Plate LXV).
One of ancient Egypt’s best known kings is the 19th Dynasty pharaoh Ramesses
II, c. 1212 - 1279 BC. Often referred to as Ramesses the Great, the king lived into his nineties, had dozens of wives and concubines, and fathered almost a hundred children throughout his sixty-plus year reign. During this time, Ramesses undertook large-scale restoration and building projects across Egypt, including his mortuary complex at Abydos and the rock cut temple of Amun at Beit el-Wali (Bunson 1991: 220-221; Wilkinson
2000: 217). The temple of Beit el-Wali is decorated with reliefs depicting the king
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triumphant over enemies, as can be seen from the fourth tableau on the right of the temple walls showing Ramesses fighting on foot while grabbing the hair of his Asiatic enemy (Figure 105). I processed and analyzed the 3,795 kb bitmap of the above relief, recording the following fractal dimensions: Benoit D = 1.46 (Figure 106), ImageJ D =
1.36 (Figure 107), and Frac3 D = 1.50 (Figure 108).
6
5
4 y = -1.4578x + 5.4299 R² = 0.9967 3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 106. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Temple of Beit el-Wali.
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12 y = -1.3608x + 12.345 10 R² = 0.9987
8
6
4
2 Log(Number Boxes) of Log(Number
0 0 2 4 6 Log(Box Size(Pixels))
Figure 107. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Temple of Beit el-Wali.
6 y = -1.4978x + 5.4572
5 R² = 0.994
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 Log(Box Size(Pixels))
Figure 108. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Temple of Beit el-Wali. 114
Figure 109. Tomb of Merenptah, New Kingdom, 19th Dynasty (Champollion 1970b: Plate CCL).
Ramesses the Great outlived all but seventeen of his sons and upon his death his fourteenth-born son, Merenptah, who was by this time well into his fifties, took control of
Egypt, c. 1212 BC. Merenptah’s reign lasted less than ten years and the king showed haste in the construction of his tomb, which is made up primarily of reused materials
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from surrounding temples. The tomb of Merenptah, numbered as Valley of the Kings tomb 8 (KV8), was excavated in 1896 by William Flinders Petrie, and it is from his excavations that we know the mortuary temple is made up from these reused materials
(Bunson 1991: 165; Shaw 2000: 302-5; Wilkinson 2000: 87-8). From this tomb two reliefs were chosen for the same reasons as previously mentioned in the tombs of
Meresankh III and Sekhemka, to see if the fractal dimension does, indeed change by relief style. The above relief showing the king being presented to Horus (Figure 109) was chosen due to the very dense nature of the composition. I analyzed this relief and recorded the following fractal dimensions for the 20,208 kb bitmap: Benoit D = 1.74
(Figure 110), ImageJ D = 1.67 (Figure 111), and Frac3 D = 1.74 (Figure 112).
7 y = -1.74x + 6.7052
6 R² = 0.9993
5
4
3
2 Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 110. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Merenptah.
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16 y = -1.6734x + 15.322 14 R² = 0.9996
12
10 8 6 4
2 Log(Number Boxes) of Log(Number 0 0 2 4 6 Log(Box Size(Pixels))
Figure 111. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Merenptah.
8
7 y = -1.7386x + 6.7059 R² = 0.9995
6
5
4
3
2
Log(Number Boxes) of Log(Number 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 112. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Tomb of Merenptah. 117
Figure 113. Temple of Merenptah, New Kingdom, 19th Dynasty (Champollion 1970c: Plate CCCI).
Merenptah was active in the military throughout his life and by the middle of his reign forces from Libya, a problem that had been brewing since the reign of Ramesses I, finally invaded along the western Nile border. Merenptah’s victories over these Libyan forces is recorded in the temple of Merenptah, built on the West Bank at Thebes, which includes a well preserved relief showing the king triumphantly leading prisoners of war
(Figure 113) (Bunson 1991: 165; Shaw 2000: 302-5; Wilkinson 2000: 187-8). I processed and analyzed the line drawing of this large relief and, for the 13,364 kb bitmap, recorded the following fractal dimensions: Benoit D = 1.67 (Figure 114), ImageJ D = 1.60 (Figure
115), and Frac3 D = 1.69 (Figure 116).
118
7
6 y = -1.6661x + 6.3796 R² = 0.999
5
4
3
2
1 Log(Number Boxes) of Log(Number 0 0 1 2 3 Log(Box Size(Pixels))
Figure 114. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Temple of Merenptah.
16 y = -1.6031x + 14.573 14 R² = 0.9998
12
10
8
6
4
Log(Number Boxes) of Log(Number 2
0 0 2 4 6 Log(Box Size(Pixels))
Figure 115. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Temple of Merenptah. 119
7
6
y = -1.6857x + 6.4051
5 R² = 0.999
4
3
2
Log(Number Log(Number of Boxes) 1
0 0 1 2 3 4 Log(Box Size(Pixels))
Figure 116. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Temple of Merenptah.
Figure 117. Great Triumphal Stela of Piye, Late Period, 25th Dynasty (Torok 1997: Figure 6).
In the dynasties following the long reign of Ramesses II, the kings of the 19th and
20th Dynasties would prove to have short lived reigns and by the end of the Third
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Intermediate Period, Egypt was under Nubian rule. It is from this Period that I chose the last relief to be analyzed. The 25th Dynasty Kushite king Piye, c. 747-716 BC systematically over took Egypt either by force or by diplomacy and by c. 747 BC Piye was recognized as the king of Egypt. Under the rule of the Nubian kings, Egypt was primarily left to its own devices, but distinct changes in artistic style are evident in Piye’s
“victory stela” in which the predefined rules of scaling by importance have changed. In the “lunette scene” from this Great Triumphal Stela, the figures are observed to be of the same scale, with only the hieroglyphs to identify the figures and their importance (Figure
117) (Shaw 2000: 337-8, 347, 353-4). Once again, following the steps laid out in the
Methods section, I scanned, processed and analyzed the black and white line drawing of this relief, recording the following fractal dimensions for the 541 kb bitmap: Benoit D =
1.46 (Figure 118), ImageJ D = 1.47 (Figure 119), and Frac3 D = 1.42 (Figure 120).
5 y = -1.4602x + 4.6953 R² = 0.9966
4.5
4 3.5 3 2.5 2 1.5
1 Log(Number Boxes) of Log(Number 0.5 0 0 1 2 3 Log(Box Size(Pixels))
Figure 118. Scatter plot of Benoit data showing the logarithms of the box sizes and number of occupied boxes – Great Triumphal Stela of Piye. 121
12 y = -1.4714x + 10.885
10 R² = 0.9968
8
6
4
2 Log(Number Boxes) of Log(Number
0 0 2 4 6 Log(Box Size(Pixels))
Figure 119. Scatter plot of ImageJ data showing the logarithms of the box sizes and number of occupied boxes – Great Triumphal Stela of Piye.
5 4.5 y = -1.4167x + 4.6704 R² = 0.9983 4 3.5 3 2.5 2 1.5
Log(Number Boxes) of Log(Number 1 0.5 0 0 1 2 3 Log(Box Size(Pixels))
Figure 120. Scatter plot of Frac3 data showing the logarithms of the box sizes and number of occupied boxes – Great Triumphal Stela of Piye. 122
Table 2, below, shows the results of each analysis listed chronologically, including geographic location. In the following chapter, I discuss and interpret the results of the analyses presented above.
Table 2. Analysis Results Listed Chronologically Period Dates, Benoit ImageJ FRAC3 Monument (Relief) Period, Dynasty BC Location D = D = D = Narmer Palette, Obverse Predynastic, 0-1st -3000 to -2650 Hierkanopolis 1.50236 1.4669 1.5189 Narmer Palette, Reverse Predynastic, 0-1st -3000 to -2650 Hierkanopolis 1.50014 1.4747 1.4937 Menes Tablet Predynastic, 1st -3000 to -2650 Naqada 1.6165 1.5623 1.6073 Mastaba of Meresankh III, Main Room Old Kingdom, 4th -2520 to 2494 Giza 1.44067 1.3797 1.4302 Mastaba of Meresankh III, Door Jamb Old Kingdom, 4th -2520 to 2494 Giza 1.53279 1.4899 1.5436 Tomb of Werirenptah Old Kingdom, 5th -2446 to -2426 Saqqara 1.68147 1.5996 1.6677 Mastaba of Sekhemka, East Wall Old Kingdom, 5th -2416 to -2392 Giza 1.53669 1.4828 1.5562 Mastaba of Sekhemka, South Wall Old Kingdom, 5th -2416 to -2392 Giza 1.56343 1.446 1.5892 Mastaba of Ptah-Hotep Old Kingdom, 5th -2388 to -2356 Saqqara 1.5864 1.5091 1.6124 Mastaba of Senedjemib-Inti Old Kingdom, 5th -2388 to -2356 Giza 1.53085 1.3967 1.5649 Mastaba of Qar Old Kingdom, 6th -2332 to -2283 Giza 1.56424 1.4615 1.5995 Mastaba of Idu Old Kingdom, 6th -2332 to -2283 Giza 1.55706 1.5104 1.5647 Temple of Mentuhotep II Middle Kingdom, 11th -2061 to -1991 Dier el-Bahari 1.51723 1.4518 1.5528 Tomb of Antefoker Middle Kingdom, 11th/12th -2060 to -1802 Beni Hassan 1.51949 1.4834 1.5406 Amenemhat I Pyramid Temple Middle Kingdom, 12th -1991 to -1962 el-Lisht 1.51151 1.3928 1.5497 Temple of Senwosret I Middle Kingdom, 12th -1971 to -1928 Karnak 1.45183 1.3428 1.4638 Pyramid Complex of Senwosret I Middle Kingdom, 12th -1971 to -1928 Karnak 1.57135 1.5206 1.557 Temple of Medamud Middle Kingdom, 13th -1783 to -1640 Medamud 1.61845 1.5527 1.6348 Tomb of Rekhmire New Kingdom, 18th -1504 to -1450 Thebes 1.53902 1.4504 1.5660 Tomb of Meryra New Kingdom, 18th -1353 to -1335 el Amarna 1.54568 1.3981 1.5836 Tomb of Panehesy New Kingdom, 18th -1353 to -1335 el Amarna 1.5024 1.3950 1.5342 Tomb of Meryra II New Kingdom, 18th -1353 to -1335 el Amarna 1.63454 1.5009 1.6668 Tomb of Apy New Kingdom, 18th -1350 to -1333 el Amarna 1.59168 1.4337 1.6275 Tomb of Neferhotep New Kingdom, 18th/19th -1279 to -1212 Thebes 1.66654 1.5853 1.6838 Temple of Beit el-Wali New Kingdom, 18th/19th -1279 to -1212 Beit Oualli 1.45778 1.3608 1.4987 Tomb of Merenptah I New Kingdom, 19th -1212 to -1202 Thebes 1.74002 1.6734 1.7386 Palace of Merenptah I New Kingdom, 19th -1212 to -1202 Thebes 1.66608 1.6031 1.6857 Great Triumphal Stela of Piye Late Period, 25th -747 to 716 Nubia 1.46019 1.4714 1.4167
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CHAPTER 8
DISCUSSION
The above results prove several points, the most essential of which is that ancient
Egyptian monumental art proves, in fact, to be fractal in nature. All the monuments analyzed exhibited strong fractal properties, as one would expect from simple observation of their compositions. For each monument analyzed in Chapter 7, note the strongly linear character of the relationship between the logarithms of box size and number of occupied boxes. The linearity of the relationship is summarized by the high values for the coefficients of determination (R²), which approach 1, the maximum that would indicate a perfect linear relationship. If these relationships were not linear, they would indicate that the images were not fractal, but as they are, I conclude that ancient Egyptian art from the
Dynastic periods is strongly fractal. While there may well be ancient Egyptian monuments or artworks that are not fractal, the standard and conventional compositions, which are well-represented by the pieces I analyzed, definitely have well-defined fractal characteristics reflected in the statistics presented above. Therefore, I am justified in attempting to use variation in the fractal dimension as a measure of chronological or regional variation.
All three fractal analysis programs produced internally consistent and generally similar results, suggesting that they were operating correctly and accurately characterized
124
the fractality of the patterns submitted to them. Benoit and Frac3 in particular produced results that were quite close, differing by only a few percent. It is noteworthy, however, that ImageJ produced estimates of the fractal dimension that were systematically lower than those of the other two programs. In the following graph (Figure 121), which shows the results in chronological order, the close correspondence between Benoit and Frac3, and the slightly lower values produced by ImageJ, are all apparent. The graph also highlights that ImageJ, despite its slightly lower estimates, consistently produced the same relative pattern of results: higher values from ImageJ correspond to higher values from the other programs and the reverse was also true. The close correlations among the results of the three programs can be seen statistically. I calculated the correlations among the three sets results (Table 3). The correlations are all highly significant.
Figure 121. Change in fractal dimension of ancient Egyptian art over the course of Dynastic Egypt - Benoit 1.3, ImageJ, and Frac3 combined results.
125
Table 3. Benoit 1.3. ImageJ, Fractal 3e Data Correlation Benoit Fractal D ImageJ Fractal D FRAC3 Fractal D Benoit Fractal D 1 ImageJ Fractal D 0.867803228 1 FRAC3 Fractal D 0.962641995 0.737282555 1
During the review process my advisor, Dr. Clifford Brown, was able to determine that the slightly divergent results produced by ImageJ were apparently the result of the number and range of the grid sizes implemented in the box-counting procedure. Forcing
ImageJ to use, for example, the same box sizes as Frac3 produced results much closer to those of the other two programs.
I used the Excel 2010 software to look for patterns in the results related to possible variation in the fractal dimension. I began with the question of a possible change in fractal dimension over time. It quickly became evident that the variation in fractal dimension was small and apparently random over time, with a variation no greater than
.40 between the lowest fractal dimension recorded and the highest fractal dimension recorded in each of the programs. This can be clearly seen in Figures 122, 123, and 124.
These results support the qualitative conclusions of art historians and Egyptologists – namely that the art of ancient Egypt remained remarkably consistent – almost static - throughout the whole of Dynastic Egypt. Thus, fractal analysis would not seem to be very helpful in quantifying stylistic change through time. To verify this pattern statistically, I calculated a separate single factor analysis of variance (ANOVA) on the results of each program by separating the data into three groups, Old Kingdom, Middle Kingdom, and
New Kingdom. Not surprisingly, the ANOVAs (Tables 4-6) show no significant difference among the mean fractal dimensions from each period. Having determined that
126
the variation was not evidently systematic across time, I then turned to the question of a possible geographic variation in the fractal dimension.
Figure 122. Change in fractal dimension of ancient Egyptian art over the course of Dynastic Egypt - Benoit 1.3.
127
Figure 123. Change in fractal dimension of ancient Egyptian art over the course of Dynastic Egypt - ImageJ.
Figure 124. Change in fractal dimension of ancient Egyptian art over the course of Dynastic Egypt - Fractal3e.
128
Table 4. Benoit ANOVA Singe Factor results Groups Count Sum Average Variance OK 12 18.6126 1.55105 0.00373 MK 6 9.18986 1.53164 0.00325 NK 10 15.8039 1.58039 0.00909
Source of Variation SS df MS F P-value F crit Between Groups 0.00975 2 0.00488 0.87671 0.42856 3.38519 Within Groups 0.13903 25 0.00556
Total 0.14878 27
Table 5. ImageJ ANOVA Singe Factor Results Groups Count Sum Average Variance OK 12 17.7796 1.48163 0.00378 MK 6 8.7441 1.45735 0.00622 NK 10 14.8721 1.48721 0.01053
Source of Variation SS df MS F P-value F crit Between Groups 0.00356 2 0.00178 0.26568 0.76882 3.38519 Within Groups 0.16751 25 0.0067
Total 0.17107 27
Table 6. Fractal3e ANOVA Singe Factor Results Groups Count Sum Average Variance OK 12 18.7481 1.56234 0.00384 MK 6 9.2987 1.54978 0.00295 NK 10 16.0016 1.60016 0.00982
Source of Variation SS df MS F P-value F crit Between Groups 0.01197 2 0.00599 1.02919 0.37194 3.38519 Within Groups 0.14542 25 0.00582
Total 0.15739 27
129
The basic geography of Egypt limits the possible geographic distribution of necropoles and temple complexes. Because most settlement (and monument architecture) was constrained to lie within the narrow north-south corridor of the Nile Valley, longitude is essentially irrelevant to the geographic analysis. Therefore, I analyzed fractal dimension in relation to latitude only. Several of the selected reliefs share latitude because they come from the same place or nearby locations. To get a clear view of the change in fractal dimension based on geographic location, I calculated the mean fractal dimension of each unique location to act as the y axis on a scatter plot, while the x axis represented the degree of latitude. I repeated this procedure for the results from each program, Benoit 1.3, ImageJ, and Fractal3e, in order to not confound the variation in the programs with the geographic variation. The results are illustrated below in Figures 125,
126, and 127 respectively. Visual inspection of each scatter plot suggests little in the way of obvious geographic patterning in the variation in fractal dimension. For each program, the dimensions show a geographic variation of no more than .20. The minimal change in fractal dimension recorded from across Egypt, from the Nile delta in Lower Egypt to the necropolis of Thebes in Middle Egypt extending even to the Temple of Beit el-Wali south of the First Cataract in Upper Egypt, further quantitatively confirms the geographic consistency of ancient Egyptian monumental art.
In fact the greatest variation recorded was compositional variation, which only supports the fractality of Egyptian art. This variation was recorded in reliefs that were more densely packed compositions, which produced higher fractal dimensions, such as the reliefs from the tomb of Merenptah in Thebes (Figures 109, 113). Relief compositions that were not as full, shall we say, that had fewer registers or figure repetitions, proved to 130
have lower fractal dimensions, such as the mastaba of Meresankh III at Giza (Figures 21,
25) and the Piye Triumphal Stela from Kush (Figure 117).
1.7
1.65
1.6
1.55 Benoit D = 1.5
1.45 Mean Fractal Dimension
1.4 30 28 26 24 22 20 18 Latitude ° North
Figure 125. Scatter plot showing the geographic variation of the fractal dimension – Benoit 1.3.
1.65
1.6
1.55
1.5 ImageJ D =
1.45 Mean Mean Fractal Dimension
1.4 30 28 26 24 22 20 18 Latitude ° North
Figure 126. Scatter plot showing the geographic variation of the fractal dimension – ImageJ.
131
1.7
1.65
1.6
1.55 Fractal3 D = 1.5
1.45 Mean Fractal Dimension
1.4 30 28 26 24 22 20 18 Latitude ° North
Figure 127. Scatter plot showing the geographic variation of the fractal dimension – Fractal3e.
In summary, the results of analyzing each of the twenty-eight Egyptian bas-reliefs with the three fractal analysis programs of Benoit 1.3, ImageJ, and Fractal3e both qualitatively and quantitatively support the general conclusions of art historians and
Egyptologists. The ANOVA tests confirm the small variance observed between each major Dynastic period in ancient Egyptian history. The low variation in fractal dimension across time and space confirms that the stylistic consistency that ancient Egyptian artists worked so hard to maintain, through painstakingly and repeatedly laying grids upon their work and using standardized measurements, was, in fact, successful in making the monuments of ancient Egypt the most remarkable and memorable in both the ancient and modern world.
The modest variation in my results for the different monuments seems clearly to be the result of the variability in the composition of different works, combined with differential preservation. Some of the works analyzed are fragmentary or have gaps due 132
to erosion or intentional defacement in antiquity. These lacunae could well account for some of the observed variation in the fractal dimensions measured.
133
CHAPTER 9
CONCLUSION
This study suggests that fractal analysis may be used as a quantitative tool in the study of ancient Egyptian monumental art. This art shows significant fractal qualities which can accurately be measured by box-counting. I used this method of analysis on twenty-eight separate reliefs dating from the Old Kingdom, Middle Kingdom, New
Kingdom and Late Period in ancient Egyptian history. I tried to answer three separate questions:
1. Is ancient Egyptian monumental art fractal?
2. If so, can a chronology be created based on the variation in fractal dimension?
3. If so, does the monumental art show regional variation?
Ancient Egyptian monumental relief art does, indeed, prove to be both visually and statistically fractal. This fractality, however, does not exhibit great variation over time or space. The fractal dimension of this artwork remains remarkably stable over time with variation largely attributed to style or composition. The data collected from the fractal analysis of the monumental art correlates with and provides a quantitative
134
measurement that supports the visual analysis a trained art historian or Egyptologist would make. While the results from the analysis do not lend themselves to evaluating variation in Egyptian art over time or space, the fractality of monumental relief art is not unimportant. An entire field of study is now open for Egyptian art and for the quantitative and statistical support in a field that is often based on subjective and qualitative analyses.
Fractal analysis has already proven its value in the fields of archaeology and art history, among many others, and has many applications to the field of Egyptology and the study of ancient Egyptian art. This is only the beginning.
135
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