Tessellations: Wallpapers, Escher & Soccer Balls
Robert Campbell
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-to- edge, 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 3.10.15 3.3.3.3.6 3.12.12 3.3.3.4.4 (and 3.3.4.3.4) 4.4.4.4 3.3.4.12 (and 3.4.3.12) 4.5.20 3.3.6.6 (and 3.6.3.6) 4.6.12 3.4.4.6 (and 3.4.6.4) 4.8.8 3.7.42 5.5.10 3.9.18 6.6.6 3.8.24 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) Solutions: Archimedean Tessellations
(3.4.6.4)
(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? D C A Trapezoids? B B C B C A D A Other? D D C A B 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/]