Transcript A tiling

Chapter 20: Tilings
Lesson Plan
 Tilings
 Tilings with Regular Polygons
 Tilings with Irregular Polygons
 Using Translations
 Using Translations Plus Half-Turns
 Nonperiodic Tilings
© 2006, W.H. Freeman and Company
For All Practical
Purposes
Mathematical Literacy in
Today’s World, 7th ed.
Chapter 20: Tilings
Tilings
 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
 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.
 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°.
Each interior angle of a five-sided
polygon (pentagon) is 72°. Angles
are measured in degrees.
Chapter 20: Tilings
Tilings with Regular Polygons
 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
 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).
Vertex figure is the pattern of polygons surrounding
a vertex in a tiling — pentagons do not work!
 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.
Chapter 20: Tilings
Tilings with Irregular Polygons
 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.


Tiling with Triangles
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).
Two scalene triangles
can form a parallelogram.
Chapter 20: Tilings
Tilings with Irregular Polygons
 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.
Nonconvex quadrilaterals
tiling the plane
Convex quadrilaterals
tiling the plane
Chapter 20: Tilings
Tilings with Irregular Polygons
 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
 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
 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
 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.
 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.
Chapter 20: Tilings
Using Translations Plus Half-Turns
 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
 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
 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.