Tessellations: Wallpapers, Escher & Soccer Balls. Robert Campbell

Transcription

1 Tessellations: Wallpapers, Escher & Soccer Balls Robert Campbell

2 Tessellation Examples

3 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.

4 Tessellations

5 Other Tessellations Not Edge-to-Edge

6 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

7 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 /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 /11º = 147 3/11º Dodecagon (12-gon): 180º - (360º/12) = 180º - 30º = 150º

8 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

9 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

10 Exercise: Regular Tessellations What Regular Tessellations Exist? Edge-to-Edge A single choice of regular polygon, of a single size

11 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

12 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?

13 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: ( ) and ( ) Example: 2 Triangles and 2 Hexagons (60º + 60º) + (120º + 120º) = 360º Two possible arrangements: ( ) and ( )

14 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

15 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

16 Vertex Types III The sets which add to 360º exactly are: (and ) (and ) (and ) (and )

17 Archimedean Tessellations II Example: ( ) Non-Example: ( ) Doesn t work as a pure ( ) tessellation But it does work as a 2-uniform tessellation with vertex types ( ) and ( )

18 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 ( ) of ( ) = 396 > 360

19 Exercise: Archimedean Tessellations Build tessellations of vertex form: ( ) ( )

20 ( ) Solutions: Archimedean Tessellations ( )

21 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

22 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

23 Tessellating Pentagons How about pentagons? Not all But some

24 Open Problem: Tessellating Pentagons Find all types of pentagons which tessellate the whole plane.

25 Heesch s Problem

26 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?

27 More Information Wikipedia [ {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) [