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Escher's Tessellations and the 17 Wallpaper Groups Escher's Tessellations and the 17 Wallpaper Groups Kayla Echols, Andrew Baker PhD, Gwyn Bellamy PhD, Ulrich Kraehmer PhD University of Glasgow, Department of Mathematics, Glasgow, Scotland Spelman College, G-STEM, Math RaMP RESULTS THE TROUBLE WITH A LATTICE ABSTRACT Morandi definition: a lattice is a finitely generated abelian subgroup of Rn for some n. By the Fundamental Theorem of Finitely FUTURE RESEARCH Generated Abelian Groups, we know that every lattice is a free Abelian group and is isomorphic to Z for some integer n. Therefore, a Many chemist and physicist know and use the fact n lattice would have a basis as a Z-module. A 2-D lattice then has the form, 퐿 = 푚푙 + 푛푘|푚, 푛 ∈ 푍 of liner combinations of a set {l,k}. In 2-D there are 17 distinct symmetry groups, referred that there are only 17 possible wallpaper structures, to as wallpaper groups. In 3-D there are 230 distinct many do not know why. The research aims to give an However, if {1, 2} is a basis of a lattice, then we see a contradiction in Morandi’s definition. symmetry group. These groups are called space groups. uncomplicated description of how wallpaper patterns, Thus we must refer to Morandi’s claim, A basis for a lattice, L, is also a vector space basis for Rn, to define a Wallpaper pattern. Further research could include examining how these defined as periodic 2-dimensional tiling of a pattern 230 space groups are classified. over a plane where there is no overlapping or THE FIVE LATTICE TYPES gapping, are classified. Here we will use group Figure 1 Other future research includes analyzing how the 17 theory and linear algebra to note an inconsistency in wallpaper groups are classified using group Morandi’s definition of a lattice as well as observe Cohomology. how to classify wallpaper patterns by lattice type, group actions, and whether or not they obtain a split group extension. ACKNOWLEDGMENTS Parallelogram Hexagonal Rectangle Rhombus Square Special thank to the University of Glasgow Mathematics G = C ,C INTRODUCTION 0 1 2 G0=C4,D4 G0=C3,D3,S,D3,L, G0=D1,P,D2,P G0=D1,C,D2,C Department, Spelman College Mathematics Department, C6,D6 and Arcadia University Department of Global Studies. Maurits Cornelis Escher was a Dutch artist who Additionally, we would like to thank the National Science became obsessed with painting tessellations in the Foundation for funding this research through the Spelman Global Stem program (G-STEM, Award ID:DMS 1045557 early 20th century. Initially Escher was known for Figure 2 his nature drawings, linocuts, woodcuts and ) and the Math Research and Mentoring program(Math depictions of impossible structures. It was in 1925 RaMP, Award ID:IDHRD-0963629). when he drew a block print of interlocking lions. It became the first of many tessellations that Escher would produce. The artist became fixated on creating REFERENCES these 2-dimensional tiling. It was Escher’s brother who noticed that the tessellation sketches were very 1.Allenby, R. B. J. T. "5.12 Space Groups and Plane similar to crystalline structures. He then introduced Symmetry Groups." Rings, Fields, and Groups: An Escher to papers that laid out the crystallographic Introduction to Abstract Algebra. 2nd ed. pp.233-41. NON-SPLIT GROUP EXTENSIONS AND GLIDE REFLECTIONS like structure for the 17 possible wallpaper patterns. London: E. Arnold, 1983. By the end of the nineteenth century, these Figure 3 2. Dutch, Steven. "Three-Dimensional Space Groups." crystallographic groups, in both two and three University of Wisconsin- Green Bay, dimensions, were categorized by Fedorov, Schoenies, 3. Engle, Michael. "Part 3: Wallpaper Groups (2D) and Barlow, building on work by several others. In, Space Groups (3D)." Short Course on Symmetry and “The Classification of Wallpaper Patterns: From Crystallography (2011): Ann Harbor. Group Cohomology to Escher's Tessellations", 4. Fraleigh, John B. Chapter 3: Groups A First Course Morandi uses group theory, geometry, linear algebra, in Abstract Algebra. 7th ed. Reading, MA: and other mathematical ideals to classify the 17 Addison-Wesley Pub., 1967 wallpaper groups in such a way that undergraduate 5. Judson, Thomas W. Abstract Algebra: Theory and students and mathematical professionals can Applications. Boston, MA: PWS Pub., 1994 understand. This research gives an uncomplicated 6. Lay, David C. Linear Algebra and Its Applications. description of how wallpaper patterns are classified Figure 4 Boston: Pearson/Addison-Wesley, 2006. as well as point out interesting observations made 7. Morandi, Patrick J. "The Classification of from Morandi's work. Note that we define a Wallpaper Patterns :From Group Cohomology to wallpaper group as a set in R2 such that the Escher's Tessellations," New Mexico State University translational and rotational symmetry group of a is a Department of Mathematical Sciences, Las Cruces, discrete lattice. Escher's beautiful tessellations will New serve as visual representations of each wallpaper Mexico. group. 8. Sethuraman, B.A. A Gentle Introduction to Abstract Algebra (2012): p26. 17 WALLPAPER GROUPS METHODS This research was conducted by reading and • 1st symbol represents the lattice type; p for primitive and c analyzing “The Classification of Wallpaper Patterns: for centered (or rhombic). From Group Cohomology to Escher’s Tessellations” • The second symbol is the largest order of a rotation. by Patrick J. Morandi. • The third symbol is either a m, g, or 1.A m (resp.g) means there is a reflection line (resp. glide reflection line but not a reflection line) perpendicular to the x-axis while a 1 means there is no line of either type. • The fourth symbol is also either an m, g, or 1. In this case an m (resp. g) represents a reflection line (resp. glide reflection line) at an angle α with the x-axis, the angle depending on the largest order of rotation as follows: α = 180° for n = 1, 2; α = 60° for n = 3, 6; α = 45° for n = 4. .
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