Tessellations: Wallpapers, Escher & Soccer Balls Robert Campbell <ricampb@nsa.gov>
Tessellation Examples
What Is What is a Tessellation? A Tessellation (or tiling) is a pattern made by copies of one or more shapes, fitting together without gaps. A Tessellation can be extended indefinitely in any direction on the plane. What is a Symmetry? A Symmetry (possibly of a tessellation) is a way to turn, slide or flip it without changing it. What is a Soccer Ball? That s a silly question.
Tessellations
Other Tessellations Not Edge-to-Edge
Regular Polygons I Regular Polygons have sides that are all equal and angles that are all equal. Triangle (3-gon) A regular 3-gon is an equilateral triangle How many degrees are in each interior angle? Walking around the triangle we turn a full circle (360º) So in each of three corners we turn (360º/3) = 120º Each turn is an exterior angle of the triangle, and exterior + interior = 180º So, each interior angle is 180º - (360º/3) = 180º - 120º = 60º
Regular Polygons II Square (4-gon) A regular 4-gon is a square How many degrees are in each interior angle? Walking around the square we turn (360º/4) = 90º So, each interior angle is 180º - (360º/4) = 180º - 90º = 90º Other Regular Polygons Pentagon (5-gon): 180º - (360º/5) = 180º - 72º = 108º Hexagon (6-gon): 180º - (360º/6) = 180º - 60º = 120º 7-gon: 180º - (360º/7) = 180º - 51 3/7º = 128 4/7º Octagon (8-gon): 180º - (360º/8) = 180º - 45º = 135º 9-gon: 180º - (360º/9) = 180º - 40º = 140º Decagon (10-gon): 180º - (360º/10) = 180º - 36º = 144º 11-gon: 180º - (360º/11) = 180º - 32 8/11º = 147 3/11º Dodecagon (12-gon): 180º - (360º/12) = 180º - 30º = 150º
Regular Tessellations I Regular Tessellations cover the plane with equal sized copies of a regular polygon, matching edge to edge. Need 360 around each vertex Try the triangle: How many degrees in each interior angle? 60 So put (360 /60 ) = 6 triangles around each vertex
Regular Tessellations II Square Each interior angle is 90 Four copies of 90 makes 360 So put four squares at each vertex Pentagon Each angle is 108 [180 - (360 /5) = 108 ] Four is too many [4(108 ) = 432 > 360 ] Three is too few [3(108 ) = 324 < 360 ] So, no regular tessellation with pentagons
Exercise: Regular Tessellations What Regular Tessellations Exist? Edge-to-Edge A single choice of regular polygon, of a single size
Regular Tessellations III Hexagon Each angle is 120 [180 - (360 /6) = 120 ] Three copies of 120 makes 360 So put three hexagons at each vertex
Archimedean Tessellations I Archimedean Tessellations (also called Semi-Regular Tessellations) are edge-toedge, made up of regular polygons, and all vertices have the same sequence of polygons around them. Question: What sort of vertex types (sequences of polygons around a vertex) will work?
Vertex Types I Question: Which sets of regular polygons fit exactly around a vertex? Example: 3 Triangles and 2 Squares (60º + 60º + 60º) + (90º + 90º) = 360º Two possible arrangements: (3.3.3.4.4) and (3.3.4.3.4) Example: 2 Triangles and 2 Hexagons (60º + 60º) + (120º + 120º) = 360º Two possible arrangements: (3.3.6.6) and (3.6.3.6)
Vertex Types II Question: Which sets of regular polygons fit exactly around a vertex? Close, but not quite: Pentagon, Hexagon & Octagon 108º + 120º + 135º = 363º 360º
Exercise: Vertex Types Find as many sets as you can of regular polygons which fit perfectly around a vertex (whose angles sum to 360 ) Recall: The interior angles of: Triangle (3-gon): 60º Square (4-gon): 90º Pentagon (5-gon): 108º Hexagon (6-gon): 120º 7-gon: 128 4/7º Octagon (8-gon): 135º 9-gon: 140º Decagon (10-gon): 144º 11-gon: 147 3/11º Dodecagon (12-gon): 150º
Vertex Types III The sets which add to 360º exactly are: 3.3.3.3.3.3 3.3.3.3.6 3.3.3.4.4 (and 3.3.4.3.4) 3.3.4.12 (and 3.4.3.12) 3.3.6.6 (and 3.6.3.6) 3.4.4.6 (and 3.4.6.4) 3.7.42 3.9.18 3.8.24 3.10.15 3.12.12 4.4.4.4 4.5.20 4.6.12 4.8.8 5.5.10 6.6.6
Archimedean Tessellations II Example: (3.3.3.4.4) Non-Example: (3.3.6.6) Doesn t work as a pure (3.3.6.6) tessellation But it does work as a 2-uniform tessellation with vertex types (3.3.6.6) and (3.6.3.6)
Archimedean Tessellations III Non-Example: (5.5.10) Lay down a 10-gon Every face of the 10-gon must glue to a 5-gon Every outer face of a 5-gon faces a 10-gon The outer vertex of each 5-gon has (impossible) type (5.10.10) of (108 +144 +144 ) = 396 > 360
Exercise: Archimedean Tessellations Build tessellations of vertex form: (3.4.6.4) (3.3.4.3.4)
(3.4.6.4) Solutions: Archimedean Tessellations (3.3.4.3.4)
Tessellating Triangles What triangles tessellate? Glue two triangles together to form a quadrilateral By rotating Or by flipping Now tile with copies of this quadrilateral
Other Tessellations What non-regular Polygons Tessellate (edge-to-edge)? How about quadrilaterals? Squares? Rectangles? Parallelograms? Trapezoids? Other? A D C C B B C B D A A D D A B C
Tessellating Pentagons How about pentagons? Not all But some
Open Problem: Tessellating Pentagons Find all types of pentagons which tessellate the whole plane.
Heesch s Problem
Open Problem: Heesch for more than five layers Find a tile with which you can make six concentric layers, but no more. Also for seven layers Also for eight layers etc?
More Information Wikipedia [http://en.wikipedia.org] {Frieze Group, Wallpaper Group, Tessellation, Platonic Solid} Books: Introduction to Tessellations, Seymour & Britton The Tessellations File, de Cordova Tilings and Patterns, Grunbaum & Shephard Geometric Symmetry in Patterns and Tilings, Horne Transformation Geometry, G. Martin Kali (Free) [http://geometrygames.org/kali/]