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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-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
3.3.3.3.6
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)
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?
Trapezoids?
Other? A
B
C
D
A
B
C
D
A
B
C
D A
B
C
D
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/]
