Chapter 20: TilingsLesson Plan

Tilings

Tilings with Regular Polygons

Tilings with Irregular Polygons

Using Translations

Using Translations Plus Half-Turns

Nonperiodic Tilings

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Mathematical Literacy in Today’s World, 7th

ed.

For All Practical Purposes

© 2006, W.H. Freeman and Company

Chapter 20: Tilings Tilings

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Tilings Tilings are repeated shapes

(or tiles) that cover a flat surface without gaps or overaps. Examples: Our ancestors

used stones to cover the floors and walls of their houses and selected colors and shapes to create pleasing designs that can be very intricate.

Tilings can be works of art and are often used in religious buildings, as well as other dwellings. The picture shows an Arab

mosaic.

Tiling – A tiling (tessellation) is a covering of the entire plane with nonoverlapping figures.

Chapter 20: Tilings Tilings with Regular Polygons

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Regular Polygons (Regular tilings use regular polygons.) An enclosed plane figure whose sides are all the same length and

whose inside angles are all equal is called a regular polygon. Examples of regular polygons:

A triangle has three equal sides and three equal interior angles. A square has four equal sides and four equal interior angles. Other polygons: A pentagon has five sides, a hexagon has six sides,

and a polygon with n sides is an n-gon.

Each interior angle of a five-sided polygon (pentagon) is 72°. Angles are measured in degrees.

Exterior angle – An angle formed by one side and the extension of an adjacent side.

Interior angle – The angle inside a polygon formed by two adjacent sides. The interior angles of any

regular polygon add up to 360°.

Chapter 20: Tilings Tilings with Regular Polygons

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Tilings with Regular Polygons The tiles used are shaped as regular polygons. All the same kinds of polygons are used in one particular tiling. Each tile is arranged in the same order, meeting at each vertex.

Monohedral tiling – Tiling that uses only one size and shape of tile. Regular tiling – Monohedral tiling that uses regular polygons. Edge-to-edge tiling – The edge of a tile

coincides entirely with the edge of a bordering tile.

A tiling by right triangles that is edge-to edge For our case, we mostly refer to this type of edge-to edge tiling (edges may be curvy).

A tiling that is not edge-to edge The horizontal edges of two adjoining squares do not exactly coincide.

Chapter 20: Tilings Tilings with Regular Polygons

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Only Three Regular Tilings The only regular tilings are the ones with equilateral triangle, with

squares and with regular hexagons (top three on the right, below).

Semiregular Tiling A systematic tiling that uses a mix

of regular polygons with different numbers of sides but in which all vertex figures are alike – the same polygons in the same order.

Vertex figure is the pattern of polygons surrounding a vertex in a tiling — pentagons do not work!

Chapter 20: Tilings Tilings with Irregular Polygons

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Edge-to-Edge Tilings with Irregular Polygons Tilings that use irregular polygons that may have some sides

longer than others or some interior angles larger than others but still fit together edge-to-edge.

In monohedral tilings with irregular polygons, all of the irregular polygons used in the tiling have the same size and shape.

Tiling with triangles can be used where any triangle can tile the plane.

The diagram shows a tiling using scalene triangles (triangles with all sides of different lengths and all interior angles of different sizes).

Two scalene triangles can fit together and form a parallelogram (a quadrilateral whose opposite sides are parallel).

Tiling with Triangles

Two scalene triangles can form a parallelogram.

Chapter 20: Tilings Tilings with Irregular Polygons

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Edge-to-Edge Tilings with Irregular Polygons Tiling with quadrilaterals (a polygon with four sides)

Any quadrilateral, even one that is not convex, can tile. How to tile quadrilaterals with opposite sides that are not parallel:

Fit together two copies of the quadrilateral, forming a hexagon whose opposite sides are parallel.

Such hexagons fit next to each other to form a tiling.

Convex quadrilaterals tiling the plane

Nonconvex quadrilaterals tiling the plane

Chapter 20: Tilings Tilings with Irregular Polygons

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Edge-to-Edge Tilings with Irregular Polygons

Tiling with pentagons (five sides) Five classes of convex

pentagons (five sides) can tile the plane.

Tiling with hexagons (six sides) Exactly three classes of

convex hexagons can tile the plane.

Convex polygons with seven or more sides cannot tile.

The diagram shows the three types of convex hexagon tile.

Chapter 20: Tilings Using Translations

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Translation A rigid motion that moves everything

a certain distance in one direction. Simplest case is when the tile is just

translated in two directions: Copies are laid edge-to-edge in rows. Each tile must fit exactly into the ones

next to it, including its neighbors above and below.

M.C. Escher (1898–1972), a Dutch artist, devoted much of his career of making prints to creating tilings with tiles in the shapes of living beings (a practice forbidden by Muslims). Example of print interlocking birds: Escher No. 128 (Bird).

Chapter 20: Tilings Using Translations

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Translation Criterion A tile can tile the plane by translation if either:

1. There are four consecutive points A, B, C, and D on the boundary such that:(a) The boundary part from A to B is congruent by

translation to the boundary part from D to C, and (b) The boundary part from B to C is congruent by

translation to the boundary part form A to D.

2. There are six consecutive points A, B, C, D, E, and F on the boundary such that the boundary parts AB, BC, and CD are congruent by translation, respectively, to the boundary parts ED, FE, and AF.

The individual tiles shown are from two Escher prints.

The points are marked to show they fulfill the criteria for tiling by translations.

Chapter 20: Tilings Using Translations

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Similarly, change one of the other two sides and copy that change on the side opposite it.

Revise as necessary, always making the same change to opposite sides.

How to Create Tiling Using Translation Starting from any irregular polygon that can be tiled.

Example: Tiling starting from a parallelogram Make a change to the boundary on one side, then copy that

change to the opposite side.

Chapter 20: Tilings Using Translations Plus Half-Turns

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Centrosymmetric Symmetric by 180° rotation around its

center. Conway Criterion

A criterion for determining whether a shape can tile by means of translations and half-turns.

Conway Criterion A tile can tile the plane by translations and half-turns if there are

six consecutive points on the boundary (some of which may coincide, but at least three of which are distinct)—call them A, B, C, D, E, and F—such that: The boundary part from A to B is congruent by translation to

the boundary part from E to D, and Each of the boundary parts BC, CD, EF, and FA is

centrosymmetric.

Chapter 20: Tilings Nonperiodic Tiling

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Nonperiodic Tiling A tiling in which there is no regular repetition of the pattern by

translation. In the example below, there is no direction (horizontal, vertical,

or diagonal) in which we can move the entire tiling and have it coincide exactly with itself. Because the sum of the offsets (1/2 + 1/3 + 1/4 + . . . + 1/n) never

adds up to exactly a whole number.

Chapter 20: Tilings Nonperiodic Tiling

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Penrose Tiles Sir Roger Penrose discovered the

“Penrose pieces” in 1973. His tiles “darts and kites” obtained from a single rhombus. Using only these two pieces, he created nonperiodic tilings; however, the same two pieces cannot be used to create periodic tilings. In the past it was believed that if you

can construct nonperiodic tilings with one or more tiles, you can construct periodic tilings from the same tiles.

This was disproven in 1964 with a set of tiles with 20,000 different shapes, and later done with as few as 100 shapes. This is why it was so amazing for Penrose to do it with only 2 tiles!

Penrose tiling by kites and darts colored with five colors.