Aperiodic Tilings
Alexandre Karassev
Tilings
• A tiling (or tessellation) is a cover of the plane
(or space) by nonoverlaping regions
Tilings in nature
Escher’s tilings
http://www.mcescher.com/
3D Tilings and Crystals
3D Tilings and Crystals
Na Cl
BN
Cu
Tilings by regular polygons
Tiles
• A tile is a polygonal region of the plane (not
necessarily convex)
• Two tiles are called
– identical (congruent) if one can be transformed to the
other by shift and rotation
– of the same type (or similar), if one is a rescaling of the
other
Identical tiles
Tiles of the same type
Example: two different tilings by squares
Matching rules
• Matching rules specify a way of joining
individual tiles (e.g. edge to edge matching)
• Matching rules can be enforced in a number
of ways, including:
– vertex labeling or coloring
– edge labeling or coloring
– edge modifications
Examples
Edge modification
Vertex coloring
Homework:
Draw the resulting tiling
Homework
• Any triangle can tile the plane
• Any quadrilateral (even non-convex)
can tile the plane
• Which pentagons can tile the plane?
Find at least one
• Find a convex tile that can tile the plane in exactly
one way
• Can a regular tetrahedron tile the space?
What about other regular polyhedra?
• What about a non-regular tetrahedron?
Periodic and non-periodic tilings
• A tiling is called periodic if it can be shifted to
perfectly align with itself in at least two nonparallel directions
• A tiling is called non-periodic if it cannot be
shifted to perfectly align with itself
• Do non-periodic tilings exist?
Trivial example
Trivial example
Is there a non-periodic tiling of the plane consisiting
of identical tiles?
Less Trivial example
Yes: cut squares “randomly”
Homework: Make the cutting process more algorithmic to create a non-periodic tiling
Homework
• Find other examples of non-periodic tilings by
copies of a single triangle
• Can non-periodic tilings be created using
copies of a single square? What about
rectangles?
Source of more interesting examples:
substitution tilings
• A partial tiling of the plane consisting of finitely
many tiles is called a patch
• Let S be a finite set of distinct tiles and S’ is a set of
bigger (inflated) tiles, similar to those from S under
the same rescaling
• Suppose that each tile in S’ can be cut into a finite
number of tiles that belong to S
• Let P be a patch consisting of tiles from S
• Rescale (inflate) P and then cut each tile in P to
produce bigger patch that still uses tiles from S
Example: armchair tiling
Source: Wikipedia
Why is the armchair tiling
non-periodic?
Theorem
If, in a substitution tiling, every next generation
of tiles can be composed back into larger tiles in
a unique way, the resulting tiling of the plane is
non-periodic
Another example
Source: Wikipedia
Conway’s pinwheel tiling
(explicitly described by Charles Radin in 1994)
John Conway
Why is the pinwheel tiling
non-periodic?
Theorem
In the pinwheel tiling, every triangle appears
rotated in infinitely many ways
(reason: the angle arctan (1/2) is not a rational
multiple of pi)
Nevertheless…
• The armchair and two Conway’s triangles can
also tile plane periodically
• Are there finite sets of tiles that can tile
plane only non-periodically?
• Such finite sets of tiles are called aperiodic
and the resulting tilings are called aperiodic
tilings
Wang’s Conjecture and Discovery of
Aperiodic Tilings
• Conjecture (Wang, 1961): if a set of tiles can
tile the plane, then they can always be
arranged to do so periodically
• Berger (1966): conjecture is false, and thus
aperiodic tiles exist (first set contained 20,426
tiles)
Smaller sets of aperiodic tiles
• Raphael Robinson, 1971: 6 tiles
• Roger Penrose, 1973 : discovery of sets
containing 2 tiles
• More small sets where also found by Robert
Ammann
• Unsolved Problem: does there exist one
aperiodic tile?
Penrose Tiles and Tilings
• Pentagons, “diamond”,
“boat”, “star”
• Two rhombuses
• “Kite” and “dart”
Sir Roger Penrose
Kite, dart, and golden triangle
Golden ratio:
36o
ϕ
72o
1
Kite, dart, and golden triangle
Golden ratio:
36o
ϕ
72o
1
Kite, dart, and golden triangle
Golden ratio:
36o
ϕ
72o
1
Kite, dart, and golden triangle
Golden ratio:
36o
ϕ
72o
1
Kite and dart: matching rules
Prohibited configuration:
Possible vertex configurations
Source: Wikipedia
“Star” tiling
Kite and dart are aperiodic
Theorem
Any tiling of the plane by kites and darts that
follows matching rules is aperiodic
Why can we tile the whole plane?
• Extension theorem
Let S be a finite set of tiles and let Dn denote the
disc of radius n centered at the origin. Suppose
that for any n there exists a patch Sn conisting of
tiles from S such that Sn covers Dn. Then tiles from
S can tile the whole plane.
• Note: patches Sn do not have to be extensions of
each other, and moreover, do not have to be
related in any other way!
Substitution rule for kite and dart
Applying it to kite and dart
• We need to show that • This can be done
kites and darts can tile through the process of
arbitrary large regions
substitution and
of the plane
deflation/inflation
Source: Wikipedia
A patch of a Penrose tiling
Application of aperiodic tilings:
quasicrystals
In 1984 Dan Shechtman announced the discovery of new
type of crystal-like structure
Quasicrystals
• In 1984 Dan Shechtman announced the discovery of a
material which produced a sharp diffraction pattern with a
fivefold symmetry
• This type of rotational symmetry is prohibited by
crystallographic restrictions for usual (periodic) crystals, and
thus the new material must be “aperiodic crystal”
• Previously (in 1975) Robert Ammann had extended the
Penrose construction to a three-dimensional icosahedral
equivalent
• Since Schehtman’s discovery, hundreds of different types of
quasicrystals were found, including naturally occurring ones
• Schehtman received Nobel prize in Chemistry in 2011
Thank you!
• Questions
faculty.nipissingu.ca/alexandk
[email protected]