Chapter 7 TOWARD VISUAL REASONING and DISCOVERY

Visual and Spatial Analysis. Eds. B. Kovalerchuk, J. Schwing, Springer 2004.

Chapter 7

TOWARD VISUAL REASONING AND DISCOVERY Lessons from the early history of mathematics

Boris Kovalerchuk Central Washington University, USA

Abstract: Currently computer visualization is moving from a pure illustration domain to visual reasoning, discovery, and decisions making. This trend is associated with new terms such as visual data mining, visual decision making, heterogeneous, iconic and diagrammatic reasoning. Beyond a new terminology, the trend itself is not new as the early history of mathematics clearly shows. In this chapter, we demonstrate that we can learn valuable lessons from the history of mathematics for visual reasoning and discovery.

Key words: Visualization, visual reasoning, visual discovery, history of mathematics.

1. INTRODUCTION

In Chapter 1, visuals were classified in three ways: (1) illustration, (2) reasoning, and (3) discovery. Illustration demonstrates the essence of entities involved and presents a solution statement without showing the underly- ing problem solving reasoning process. Reasoning sets up explanatory rele- vance of entities to each other and discovery finds relevant entities. These categories form the creativity scale shown in Figure 3 in Chapter 1 where illustration and discovery are the two extremes in this scale with many intermediate mixed cases. Reasoning occupies the middle of this scale. In Chapter 1, all three concepts have been illustrated with the Pythagoras Theo- rem: (1) visualization of the theorem statement, (2) visualization of the proof process for the theorem’s statement, and (3) visualization of the discovery process that identifies the theorem’s statement as a hypothesis. In visual decision making the listed categories have their counterparts: 154 Chapter 7

• visualization of a decision • explanation of a decision • visualization of the process of discovery of a decision. Computer visualization is moving from being pure illustration to reasoning, discovery, and decision making. New terms such as visual data mining, visual decision making, visual, heterogeneous, iconic and diagrammatic reasoning clearly indicate this trend. Beyond a new terminology, the trend itself is not new as the early history of mathematics clearly shows. In this chapter, we demonstrate that we can learn valuable lessons from the history of mathematics. The first one is that all three aspects had been implemented in the ancient times without the modern power of computer graphics: 1) Egyptians and Babylonian had a well developed illustration system for visualizing numbers; 2) Egyptians and Babylonian had a well developed reasoning system for solving arithmetic, geometric and algebraic tasks using visualized numbers called numerals; 3) Ancient Egyptians were able to discover and test visually a non- trivial math relation, known now as the number π. How can we learn lesson from this history? How can we accelerate the transition from illustration to decision-making and problem solving in new challenging tasks we face now using history lessons? At first, that history should be described in terms of visual illustration, reasoning and discovery. This will give an empirical base for answering posed questions. Traditionally texts on the history of mathematics have different focus. This chapter could be viewed as an attempt to create such an empirical base for a few specific subjects. The first lesson from this analysis is: inappropriate results of illustration stage hinder and harm the next stages of visual reasoning and decision making. Moreover, this can completely prevent visual reasoning and decision making, because these stages are based on visualization of entities provided in the illustration stage. The most obvious example of such a case is exhibited by Roman numerals. These numerals perfectly fulfill the illustration and demonstration role, but have very limited abilities to support visual reasoning for arithmetic (summation, subtraction, multiplication and division). Hindu-Arabic numerals fit reasoning tasks much better. The second lesson is that the most natural visualization that seems isomorphic to real world entities is not necessarily the best for reasoning and decision making. The Ptolemy Geocentric system was isomorphic to the observed rotation of the Sun around the Earth, but eventually it became clear that it does not provide advanced reasoning tools to compute the orbits of other planets. 7. Toward visual reasoning and discovery 155

The third lesson is that if we want to design (invent) visualization that will survive reasoning and decision making tests later we should be able to formulate future reasoning and decision making tasks explicitly at the time when design of visualization as illustration is started. The forth lesson is that if we design (invent) visualization without a clear vision of future reasoning and decision making (problem solving) tasks the chance is no better than a flip of a coin that the visualization will survive reasoning and decision making tests later. It seems that the history of mathematics points towards the conclusion that most initial visualizations of numerals were invented for illustration, description and recording purposes. Their usefulness for reasoning and problem solving was tested later. Those that survived, namely the Hindu-Arabic numerals, we use now. In essence, this history fits the idea of evolution with the survival of the fittest. In this chapter, we analyze the history of Egyptian and Babylonian numerals that support lessons learned presented above.

2. VISUALIZATION AS ILLUSTRATION: LESSONS FROM HIEROGLYPHIC NUMERALS

To provide an illustration it is sufficient to visualize concepts involved in the solution. Let us consider a simple arithmetic example, 3535+1717=5252. There is a justified computational procedure for getting this solution 5252, but the expression 3535+1717=5252 does not show the reasoning steps that lead us to the solution. In this example, concepts visualized are input, output, the summation operation and equality relation. Such visualization tasks were successfully solved in ancient Egypt, Greece, India and Mesopotamia by developing symbols for numbers and to some extent symbols for operations and relations too.

2.1. Egyptian numerals

Hieroglyphic numerals. Table 1 shows Egyptian hieroglyphic numerals and some of their ideographic meanings [Allen, 2001a, Williams, 2002c, Aleff, 2003, Bertin, 2003]. 156 Chapter 7

Table 1. Symbolic Egyptian Hieroglyphic numerals 1 10 102 103 104 105 106 107

1 2 3 4 5 6 7 vertical heal snare lotus bent burbot kneeling figure Sun stroke bone, coil of flower finger fish, with raised vault rope Tadpole arms, Heh-god Compare this with Roman and Hindu-Arabic numerals presented in Ta- ble 2. Romans had also other numerals: V for 5, L for 50, and D for 500.

Table 2. Roman and Hindu-Arabic numerals 1 10 102 103 104 105 106 107

1 10 100 1000 10000 100000 1000000 10000000

| X C M X C M

Table 3 shows some alternative design of hieroglyphs from sources listed above. Left and right forms were used on the left and right sides of the text to provide visual symmetric view of the wall. Several fonts have been developed for hieroglyphs. Table 3 uses Gardiner, Glyph basic, and Nahkt fonts [Bertin, 2003].

Table 3. Symmetric pairs and alternative glyphs Number Left form Right form

100 3

1,000 4 10,000 # 6 100,000 $ 7 7. Toward visual reasoning and discovery 157

Hieratic numerals. The Egyptian Hieratic numeral system is also 10 based. Hieratic symbols for 2 and 3 are repeated symbols for 1 ( | ), re- spectively (||, ||| ) and the symbol for 8 (=) is a repeated symbol for 4 (−). The same symbols || and ||| are used in Hieroglyphic and Roman systems for 2 and 3. Other unique symbols in the Hieratic system are [Fried- man, 2003b]:

5 = , 6= , 7= , 9= , and 10= 2.

This system also has unique symbols for 20,30,40,50,60,70,80,90 and 100. The symbol for 20 ( ′2 ) is directly based on symbols for 10 (2), and the symbol for 40 ( − ) is based on symbol for 4 ( − ). Symbols for 200, 300, 400 and 500 are based on the symbol for 100 ( ). They are drawn by adding one, two, three or four dots (.) above, 200 ( ),

300 ( ), 400 ( ), and 500 ( ) [Friedman, 2003b].

In Table 4, we show how numbers were composed using base glyphs presented in Table 1 using the idea of Hindu-Arabic decimal positional system that is a modern standard for number visualization. It is so common after several thousand years of use that we often miss the point that this is just one of the possible number visualization systems. It would be more transparent if we contrast it with a textual description of numbers. The textual description at most visualizes the sequence of sounds, e.g., the word “thousand” in phonetics based languages.

Table 4. Egyptian hieroglyphs and operations [font from Williams, 2002c] Modern symbols = 1=100 10 102 103 Egyptian hieroglyphs 1 2 3 4 3 Modern 3105=5+100+3*10 3 (backward notation) 5 0 100 3*10

Egyptian 3105 111113444 11111 3 444 158 Chapter 7

In Egyptian numeral system, there is a flexible sequence and every decimal has its own symbol. Thus, some hieroglyphic numerals are shorter than in modern notation, for instance: 1000110 = 237 The hieroglyphic system is less positional than the Hindu-Arabic system that we are using now. The change of the sequence of the components does not destroy the value of the number: 1000110 = 237 = 732 but for the Hindu-Arabic numeral 1000110 the backward sequence 0110001 is not equal to 1000110. The Roman system is an intermediate system between the Egyptian and the Hindu-Arabic systems. This system is more positional than the Egyptian system and less positional than the Hindu-Arabic system. At first glance, Tables 1 and 2 show that both Roman and Egyptian systems provide simpler visualization for numbers than the Hindu-Arabic system. However, only this system has survived as a reasoning and decision making tool for humans and was replaced by the Binary positional systems for computers only very re- cently. It is not clear which of these systems were developed first. It is most likely that all these systems were developed quite independently and then were tested for ease in solving mathematical tasks. It is possible that the Hindu-Arabic system is the oldest one. Egyptians actually used the flexibility of their system to present numbers in a variety of ways including several lines. Their numerals are truly two- dimensional (see Figure 1). Thus, Egyptians had a well developed illustration system for visualizing numbers. Figure 1 shows that Egyptians wrote a single number using 2-3 lines starting from larger digits. Alternatively, they used a single line, where larger digits are on the right because Egyptians wrote from right to left. The record for number 300003 shows also that Egyptians used a smaller size for smaller digits (all six ones “|” occupy the same space as a single glyph for 100,000). Writing for 1/25 in Figure 1 shows that they also used a format where digits are distributed in two columns. Writing for 3350 shows that Egyptians also used a “zipper” type of writing, where two of 2 symbols are moved down. Sometimes Egyptians used a larger “font” for larger digits. The symbol for 1000 is twice larger that the symbols for other digits in the Egyptian 1303 shown above. The same idea is implemented in writing 1010005, the digit for 1000000 is more than twice larger than digits for 10000 and 5. Thus, every number is represented as a complex icon/glyph/ numeral that is combined from simpler icons (numerals) for basic digits. 7. Toward visual reasoning and discovery 159

This combining has its own visual syntax. The presentation for 19607 does not follow the pattern that larger digits are also larger in size. It shows digits for 1000 smaller because there are nine of them. Thus, a more general rule is that a large number of equal digits is the main factor for drawing those digits smaller.

33 444 3 1113 4 333 R 2 2 2 22 21 2 2

120 1303+1/11 3350

R 6 6 6 2111

11111 211 7 111 6 111

300003 1010005 1/25

444 1111333 33 222 444 111333 222 444 5 1111113

19607 276 Figure 1. Free sequence of numeral components (based on [Friedman, 2003c; Williams, 2002c; Arcavi, 2003; Allen,2001a])

2.2. Babylonian numerals

The Babylonians inherited the Sumerian style of writing on clay tablets. Their arithmetic was positional and sexagesimal based on 60 with symbols for 1, 10, 60, 600, 3600, 36000, and 216000. We follow a simplified notation from [Allen, 2001b] where V is used for 1 and  for 10. In this notation

7341 = VV VV  V 2 because 734110 = 222160 = 2*60 + 2*60 + 21. Base 60 has many advantages. One of them is that other systems can be converted to this one (2, 3, 5, 10, 12, 15, 20, and 30 all divide 60) . 160 Chapter 7

The Babylonian system permitted some shortcuts. Without the shortcut, number 19 is represented as VVVVVVVVV. Table 5 shows a shortcut for this number that uses a subtraction idea (19 = 20-1). This way the symbol for 1 that is V is not repeated nine times [Allen, 2001b].

Table 5. Babylonian numeral 19 (based on [Allen, 2001b]) 19 is presented as 20-1, where 20 is   and 1 is V. The negation symbol is over the V symbol for 1. VVV 19 as 10+9, where symbol  stands for 10 and 9 small symbols V stand for 1. VVV  VVV

This brief description shows that the Babylonians had a well developed illustration system for visualizing numbers. It was not limited to integers; Babylonians also used fractions. Their abilities for reasoning with numbers included extracting square roots, solving linear systems, using Pythagorean triples such as (3, 4, 5): 32 + 42 = 52, and solving cubic equations using tables. Several of these actions can be qualified as visual reasoning too.

2.3. Results of arithmetic operations

People in ancient Egypt knew how to visually represent results of arithmetic for both integers and fractions (see Figures 2 and 3).

33211111+222221111111=33222222211 215 + 57 = 272 Figure 2. Visual adding integers

R R R R R R 2 2 2 2111 2111 2111 211 211 211 1/10 1/10+1/10=2/10 1/25 1/25+1/25=2/25 Figure 3. Visual adding fractions

Egyptians also knew how to add numbers visually (see Figures 4, 5) 7. Toward visual reasoning and discovery 161

Adding visually by columns, 1700 BC, Adding Adding using Egypt visually text (method column by does not exist column yet) 33 2 11 Two hundred 111 215 fifteen 222 1111 Fifty seven 22 111 57 two hundred 2222 11 272 33 222 seventy two Figure 4. Adding numbers using symbols vs. text Let us assume that we need to sum the numbers 215 and 57 written in words, two hundred fifteen plus fifty-seven. How can we do this using the words? A procedure to do this does not exist even after 4000 years of using numbers. The only method known now is converting verbal numbers to one of the symbolic (visual forms).

::: 2801 | ::: uu ::: 5602 = 2801*2 || ::: uuuu ||| :: 11204 = 5602*2 | 4 5 ::: 444 |||| 444 Sum 19607 = ||| ::: 2801 + 5602 + 11204 444 5 Figure 5. Visual summation (based on Arcavy, 2002)

162 Chapter 7

In the next section, we analyze the visual reasoning in Egyptian mathematics that is representative of the level of sophistication reached in the Ancient world in visual reasoning.

3. VISUAL REASONING: LESSONS FROM HIEROGLYPHIC ARITHMETIC

Reasoning includes procedures for getting results from arithmetic operations. A proof that the operation is correct came from summation and multiplication tables. In early Egypt, addition and subtraction were simple visual processes using the counting glyphs. To add two numbers, we collect all symbols of the same type and replace ten of them by one of the next higher order. For example, adding and subtracting 5 and 7 is shown in Table 6, where ten symbols | are substituted by a single symbol 2 that means 10.

Table 6. Visual arithmetic Addition Subtraction

7 ||||| || 7 ||||| || 5 ||||| 5 ||||| ||||| ||||| || result Result || compact result || 2

Addition and subtraction of 35 and 17 are shown in Table 7 where numbers are presented in backward sequence of decimal positions. These operations use the same visual techniques: grouping and substitution.

Table 7. Visual addition of 35 and 17 Number (in modern notation) Decimal position 0 | (10 ) 2 (101) || ||| 2 22 35 5 30

|| ||| || 2 17 7 10 || ||| || ||| || 2222 Sum 52 = (12 + 40) 12 40 Sum 52 with use “carry” || 22222 (after shifting 100 “|” to 101 position as 2) 2 50

Similarly, Table 8 shows the addition of 70 and 10. 7. Toward visual reasoning and discovery 163

Table 8. Summation (with opposite sequence of decimal positions) Number (in modern notation) | (100) 2 (101) || ||| 2 22 222 Sum 70 = 10 + 60 || ||| 10 6⋅10 Sum 70 = 10 + 60 2 22 2222 (after shifting 10 “|” to 101 position as “2”) 0 7⋅10

Multiplication and division were done by using visuals and a lookup multiplication table for only 2n, 4n, 8n and so on for every n. Numbers in the table were generated by repeated visual summation such as n + n = 2n then 2n + 2n = 4n and so on (see Table 7 for the number 25). For example, to multiply 25 by 11 the property 11 = 1 + 2 + 8 is used along with the two times multiplication table (Table 8): 25⋅11 = 25⋅(1+2+8) = 25 + 25⋅2 + 25⋅8 (see Table 10). Table 9. Two times multiplication table for number 25 1⋅25 | || ||| 2 2 25 2⋅25 || 22222 50=25+25 4⋅25 |||| 3 100=50+50 8⋅25 |||| |||| 33 200=100+100

Table 10. Multiplication 25⋅11 1⋅25 || ||| 2 2 25 2⋅25 22222 50 8⋅25 33 200 25⋅(1+2+8)=25⋅11 || ||| 22 22222 33 275=25+50+200 || ||| 22 22222 22222 22222 3 275 || ||| 22 22222 33 275 To get a feeling for the advantages of this visual process we just need to try to multiply 25 by 11 solely in a textual form. This simple task becomes very difficult to solve and even harder to prove that the result is correct. But the visual arithmetic computation process is not simple either if we try to record it completely. Below we show what happens when we add two numbers, 35 and 17, using standard arithmetic techniques. We use a spreadsheet visualization similar to the one used in MS Excel. In Table 11, number 35 occupies cells (1,3), (1,4) and number 17 occupies cells (2,3) and (2,4). The result should be located in cells (3,3) and (3,4). There are also cells (4,2) and (4,3) reserved to writing carries. We use symbols aij for a content of cell (i,j). In this notation, the algorithm for adding 35 and 17 consists of two steps:

Step 1: If a14 + a24 >9 then a34 := (a14 + a24) -10 & a43 := 1 else a34 := a14 + a24 & a43 := 0 164 Chapter 7

Step 2: If a13 + a23 + a43 >9 then a33 := (a13 + a23) -10 & a42 := 1 else a33 := a13 + a23 + a43 & a42 := 0

Table 11. Visual summation j=1 j=2 j=3 j=4 i=1 3 5 i=2 1 7 i=3 sum 5 2 i=4 carry 0 1

At first glance, this algorithm does not appear to be visual, but the placement of numbers is visual, similar to what users do everyday working with the Excel spreadsheet graphical user interface. People accomplish these steps easily visually, but it is hard to explain steps 1 and 2 without visuals, although it is almost a complete computer program.

4. VISUAL DISCOVERY: LESSONS FROM THE DISCOVERY OF π

Ancient Egyptians were able to discover and test visually a non-trivial mathematical relation between the diameter of the circle, D and its area, S. Now this relation is expressed using the number π, S=(π/4)D2. Egyptians discovered this relation for a specific diameter D=9 in the form S=(D-1)2= (9-1)2. It can be generalized to S=(D-D/9)2=(8D/9)2=(64/81)D2. We can compare the Egyptian coefficient 64/81=0.790123 with π/4= 0.785398 and notice that the formula discovered in Egypt is remarkably accurate. Below we discuss this discovery in more detail. Problem 50 from the Rhind papyrus (about 1550 BC) [Williams, 2002b] is about this relation between the diameter and the area of the circle:

“A circular field has diameter 9 khet. What is its area?”

One khet is about 50 meters. This papyrus is now in the British Museum. Its detailed description is presented in [Robins, Shute, 1998; Chance at al., 1927, 1929]. The papyrus was made by the scribe Ahmose and sometimes is called the Ahmose papyrus. The solution provided in the Rhind papyrus for problem 50 is [Friedman, 2003a] :

You are to subtract one ninth of it, namely 1: remainder 8. You are to multiply 8 two times: it becomes 64. This is its area in land, 6 "thousands-of-land" and 4 setat. 7. Toward visual reasoning and discovery 165

In modern terms, it means as we already mentioned:

If diameter D=9 then circle area S=(9-1)(9-1)=8⋅8=64.

In accordance with this formula, π number would be 3.160494 which is quite close to the correct value 3.141593. To get this value we just need to notice that S=(9-1)2=64=π(9/2)2. Thus, π=64/(9/2)2 = 3.160494. How was it possible to discover such an accurate result 3500 years ago? We follow Williams’ conjecture that it was done by means of visual discovery: “An alternate conjecture exhibiting the value of π is that the Egyptians easily observed that the area of a square 8 units on a side can be reformed to nearly yield a circle of diameter 9.” Figure 6 reconstructs how such a discovery could be made. It shows a variety of circles and squares with small circles inside. The square 8 by 8 has 64 small circles and the circle with diameter of nine small circles has 67 of those circles. These numbers 64 and 67 are similar with the difference less then 5% of each of them. The difference between any other pair of numbers in Figure 6 is greater, thus, it is discovered that the area of the square 8×8 closely interpolates the area of the circle with the diameter 9. This mental experiment could be conducted in ancient Egypt physically with small river rocks, apples, or seeds. Ancient Egyptians could easily collect hundreds of these items of almost the same size.

4 4 3

16 14 9 7

25 21 36

49 42 67 64

Figure 6. Visual experiment (adapted from [Williams, 2002a]) 166 Chapter 7

Now we need to analyze how after discovering the similarity of the number of rocks in some circle and square to discover the mathematical indicators to match the circle and the square areas. We already assumed that these indicators are the diameter (or radius) of the circle and the side of the square. It is a very realistic assumption. These indicators were already known to Egyptians as the main characteristics of a circle and a square. But we need to discover the relation between their squares, S2 =πR2 with an unknown coefficient π. In essence, this is a data mining task in modern terms. It could be tried in ancient times visually again by experimenting. For instance, rocks can be counted in the square that contained in the circle and in the square that con- tains the circle getting S1

A a b B h c g d D f e C where ABCD is a square and abcdefgh is an irregular octagon [Gnaedinger, 2001]. Problem 48 differs from problem 50 discussed above. Problem 48 is to justify of the result (reasoning, proof) of the already discovered statement

AreaCircle(9)=AreaSquare(8), (1)

7. Toward visual reasoning and discovery 167 where 9 is the length of the diameter of the circle and 8 is the length of the side of square. In contrast, problem 50 asks for discovery of the statement presented in (1). Figure 7 presents the inscribed octagon graphically. A a b B

h c

g d

D f e C Figure 7. Illustration for the Rhind problem 48 (based on [Wright, 2000])

This simple method provides a reasonable approximation of π. As now known, this approximation idea permits one to get π with any desirable accu- racy by using an n-gon with large n.

5. CONCLUSION

Examples in this chapter illustrate the power of visual discovery combined with mathematical computations and reasoning. Below we summarize the characteristics of ancient arithmetic based on the analysis in this chapter: 1. Ancient arithmetic was visual. 2. Ancient arithmetic involves explicit reasoning. 3. There are exact reasoning rules how to operate with visual entities to obtain the result of an operation. 4. Reasoning rules are specific for each arithmetic operation. 5. Rules of more complex operations are based on rules for simple operations (e.g., multiplication is based on addition and division is based on multiplication). The goal of each visual operation was well defined. Although formal models of the mathematical operations can be built, the actual Egyptian system is a mixture of visual intuitive procedures and formal manipulation with minimized double conversion between analytical and visual representation of the problem (see discussion of this subject in Chapter 1, Section 1.5). If we compare many modern visualization tools with the characteristics presented above, we see that we have not reached the level of sophistication known in ancient Egypt more than 3000 years ago. For instance, visual data mining does not go further then showing glyphs such as squares 168 Chapter 7 and the rectangles from different viewpoints, but visual guidance how to experiment using visual tools for pattern discovery are not mature yet. In Chapter 16, we present a new visual data mining technique based on Monotone Boolean functions that intends to fill this gap for the Boolean data type. The technique permits discovering patterns by analyzing structural in- terrelations between objects (cases) in the original visualization and by changing the visualization to its modifications that permits one to see a con- tinuous border between patterns if it exists. To build an advanced visual discovery system we need to start with a clearly stated goal, as was done in the examples analyzed in this chapter, such as the goal of discovering a formula to compute the area of the circle. In modern visual data mining, the goal is discovering patterns. At first glance, it looks that we also have a goal, but this goal is not that well stated as computing the area. The correctness of the area computation can be tested using well-stated and simple criteria. Like data mining, the goal in imagery conflation (see chapters 17-21) is to find matching features. However, there are no natural, well-stated formal criteria to test if the goal is reached, even if people are able to match features by visual inspection of two images using informal, tacit rules. There are two important questions: (1) How can we know if a task can be solved by visual means? (2) How do we select tasks to be solved by visual means? The answer to both questions based on our analysis of early history of mathematics is: The task should have a goal and a formalized criteria to judge that the goal is reached, as was the case for these mathematical tasks. For less formal tasks, visual reasoning is still possible, as chapters (BK) and DB indicate, but the conclusions may be much less conclusive and the methods may be less sophisticated. In Chapter 1 (Section 2.1) we discussed visualization, visual reasoning and visual discovery for the Pythagorean Theorem. The first two tasks were successfully solved visually many times (there are more than 300 different proofs of the theorem), but it is not the case for visual discovery of the theorem statement. It is difficult to formulate formal criteria for visual discovery. The goal can be formulated easily – to discover the theorem statement. However, we cannot assume that parameters and the types of relations (pol- ynomial or other) between them are known if it is true discovery. Thus, the task should be formalized. This can be done in many different ways and the solutions can be quite different. The early history of mathematics clearly shows the trend from illustration to visual reasoning and discovery. This chapter demonstrates that we can learn valuable lessons from this history. The main lessons are: • inappropriate results at the illustration stage harm the next stages of visual reasoning and decision making; 7. Toward visual reasoning and discovery 169

• to invent a visualization that will survive visual reasoning and decision making tests, reasoning and decision making tasks should be formulated explicitly when one designs visualization as illustration. Future work will use additional empirical information about the use of visual reasoning and visual discovery in ancient mathematics to analyze how to solve visually modern problems.

6. EXERCISES AND PROBLEMS 1. Compute visually 71 + 71 in the Egyptian hieroglyphic system using Ta- ble 12 for 70+70 as a prototype.

Table 12. Hieroglyphic arithmetic for 70 Number (in modern notation) Decimal position | 100 2 101 3 102 2222222 70 0 70 2222222 70 70 2222222

Sum 140 = 10*14 2222222 0 140 Sum 140 = 100 + 40 2222 3 (after shifting 10 2 to 102 position as 3) 0 40 100

2. Compute visually 145 + 145 in the Egyptian hieroglyph system using Ta- ble 12 for 140 + 140 as a prototype. Table 13. Hieroglyphic arithmetic for 140 Number (in modern notation) Decimal position | 100 2 101 3 102

140 2222 3 0 40 100 2222 3 140 40 100 2222 33 Sum 280 = 10*14 2222 0 80 200

3. Compute visually 37 + 74 + 280 in the Egyptian hieroglyph system using Table 14 for 35 + 74 + 280 as a prototype. 170 Chapter 7

Table 14. Hieroglyphic arithmetic for 35, 70 and 280 Number (in modern notation) Decimal position | 100 2 101 3 102 ||||| 222 35 5 3*10 2222222 70 7*10

280 2222 2222 33 0 8*10 200 22222 22222 ||||| 33 Sum 385 = 5 + 180 + 200 2222 2222 5 180 200 Sum 385 = 5 + 80 + 300 ||||| 2222 2222 333 (after shifting 10 2 to 102 position as 3)

4. Analyze efficiency of hieroglyphic visualization for arithmetic operations. Do you see any cases where the summation or multiplication using hieroglyphic numerals can be accomplished faster than using the Hindu-Arabic numerals? Tip: Start from the cases presented in exercises 1-3.

7. REFERENCES

Allen, D., (2001a) Counting and Arithmetic – basics, TAMU, 2001http://www.math.tamu.edu/~don. allen/history/egypt/node2.html Allen, D., (2001b) Babylonian Mathematics, TAMU, 2001, http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf Aleff, H., Ancient Creation Stories told by the Numbers, 2003, http://www.recoveredscience. com/const105hehgods.htm Arcavi, A., Using historical materials in the mathematics classroom,2002, http://www. arte- misillustration.com/assets/text/Rhind%20Papyrus.htm. Berlin, H., Foreign Font Archive, Hieroglyphic and Ancient Language Typefaces, 2003, http://user.dtcc.edu/~berlin/font/downloads/nahkt_tt.zip Chance, A., Bull,L. .Manning,H., Archibald, R., The Rhind Mathematical Papyrus, vol. 1, Oberlin, Ohaio, MAA, 1927; vol. 2, Oberlin, Ohaio, MAA, 1929. Friedman, B., (2003a)A Selection of Problems from the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus, 2003, http://www.chass.utoronto.ca/~ajones/cla203/ egyptmath2.pdf] Friedman, (2003b) B., Sample hieratic math answers, 2003, [http://www.mathorigins.com/ Image%20grid/BRUCE%20OLD_007.htm] Friedman, (2003c), B., CUBIT, 2003, http://www.mathorigins.com/image%20grid/ CU- BIT_002.htm Gnaedinger, F., Rhind Mathematical Papyrus, 2001, http://www.seshat.ch/home/rhind6.htm 7. Toward visual reasoning and discovery 171

Robins, G., Shute, C., The Rhind mathematical papyrus. An ancient Egyptian text, British Museum Publications, Ltd., London, 1998 Williams, S., (2002a) Egyptian geometry determining the value of π the Pythagorean theorem, 2002, http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt _geometry.html#ahmes10 Williams, S., (2002b) Rhind papyrus, 2002, http://www.math.buffalo.edu/ mad/Ancient- Africa/mad_ancient_egyptpapyrus.html#ahmes/rhind papyrus Williams, S., (2002c) Egyptian counting with hieroglyphs, the Mathematics Department, SUNY, Buffalo, 2002, http://www.math.buffalo.edu/mad/ Ancient-Africa/mad_ an- cient_egypt_arith.html Wright, J., Lecture Notes for Egyptian Geometry, 2000, http://www2.sunysuffolk.edu/ wrightj/MA28/Egypt/geometry2.pdf