-'A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose
who investigated these sets in the 1970s. The aperiodicity of the
Penrose prototiles implies that a shifted copy of a Penrose tiling will
never match the original. A Penrose tiling may be constructed so as to
exhibit both reflection symmetry and fivefold rotational symmetry.'
-'A Penrose tiling has many remarkable properties, most notably:
- It is non-periodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original.
- It is self-similar,
so the same patterns occur at larger and larger scales. Thus, the
tiling can be obtained through "inflation" (or "deflation") and any
finite patch from the tiling occurs infinitely many times.
- It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.'
- 'Penrose tilings are simple examples of aperiodic tilings of the plane. A tiling is a covering of the plane by tiles with no overlaps or gaps; the tiles normally have a finite number of shapes, called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only tiles congruent to these prototiles. The most familiar tilings (e.g., by squares or triangles) are periodic: a perfect copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction. Such a translation is called a period
of the tiling; more informally, this means that a finite region of the
tiling repeats itself in periodic intervals. If a tiling has no periods
it is said to be non-periodic. A set of prototiles is said to be aperiodic
if it tiles the plane, but every such tiling is non-periodic; tilings
by aperiodic sets of prototiles are called aperiodic tilings'
-'Penrose's first tiling uses pentagons and three other shapes: a
five-pointed "star" (a pentagram), a "boat" (roughly 3/5 of a star) and a
"diamond" (a thin rhombus).
To ensure that all tilings are non-periodic, there are matching rules
that specify how tiles may meet each other, and there are three
different types of matching rule for the pentagonal tiles. It is common
to indicate the three different types of pentagonal tiles using three
different colors.'
-Penrose's second tiling uses quadrilaterals called the "kite" and
"dart", which may be combined to make a rhombus. However, the matching
rules prohibit such a combination.
Both the kite and dart are composed of two triangles, called
Robinson triangles, after 1975 notes by Robinson.
- The kite is a quadrilateral whose four interior angles are
72, 72, 72, and 144 degrees. The kite may be bisected along its axis of
symmetry to form a pair of acute Robinson triangles (with angles of 36,
72 and 72 degrees).
- The dart is a non-convex quadrilateral whose four interior
angles are 36, 72, 36, and 216 degrees. The dart may be bisected along
its axis of symmetry to form a pair of obtuse Robinson triangles (with
angles of 36, 36 and 108 degrees), which are smaller than the acute
triangles.
The matching rules can be described in several ways. One approach is
to color the vertices (with two colors, e.g., black and white) and
require that adjacent tiles have matching vertices.
Another is to use a pattern of circular arcs (as shown above left in
green and red) to constrain the placement of tiles: when two tiles share
an edge in a tiling, the patterns must match at these edges.
These rules often force the placement of certain tiles: for example, the concave
vertex of any dart is necessarily filled by two kites. The
corresponding figure (center of the top row in the lower image on the
left) is called an "ace" by Conway; although it looks like an enlarged
kite, it does not tile in the same way.
Similarly the concave vertex formed when two kites meet along a short
edge is necessarily filled by two darts (bottom right). In fact, there
are only seven possible ways for the tiles to meet at a vertex; two of
these figures – namely, the "star" (top left) and the "sun" (top right) –
have 5-fold dihedral symmetry (by rotations and reflections), while the remainder have a single axis of reflection (vertical in the image).
All of these vertex figures, apart from the ace and the sun, force the placement of additional tiles.
-The third tiling uses a pair of rhombuses (often referred to as "rhombs" in this context) with equal sides but different angles.
Ordinary rhombus-shaped tiles can be used to tile the plane
periodically, so restrictions must be made on how tiles can be
assembled: no two tiles may form a parallelogram, as this would allow a
periodic tiling, but this constraint is not sufficient to force
aperiodicity, as figure 1 above shows.
There are two kinds of tile, both of which can be decomposed into Robinson triangles.
- The thin rhomb t has four corners with angles of 36, 144, 36, and 144 degrees. The t rhomb may be bisected along its short diagonal to form a pair of acute Robinson triangles.
- The thick rhomb T has angles of 72, 108, 72, and 108 degrees. The T
rhomb may be bisected along its long diagonal to form a pair of obtuse
Robinson triangles; in contrast to the P2 tiling, these are larger than
the acute triangles.
The matching rules distinguish sides of the tiles, and entail that
tiles may be juxtaposed in certain particular ways but not in others.
Two ways to describe these matching rules are shown in the image on the
right. In one form, tiles must be assembled such that the curves on the
faces match in color and position across an edge. In the other, tiles
must be assembled such that the bumps on their edges fit together.
There are 54 cyclically ordered combinations of such angles that add
up to 360 degrees at a vertex, but the rules of the tiling allow only
seven of these combinations to appear (although one of these arises in
two ways).
-Starting with a collection of tiles from a given tiling (which might be a
single tile, a tiling of the plane, or any other collection), deflation
proceeds with a sequence of steps called generations. In one generation
of deflation, each tile is replaced with two or more new tiles that are
scaled-down versions of tiles used in the original tiling. The
substitution rules guarantee that the new tiles will be arranged in
accordance with the matching rules.
Repeated generations of deflation produce a tiling of the original axiom shape with smaller and smaller tiles.
