1. Aperiodic Tilings
Alexandre Karassev

2. Tilings
A tiling (or tessellation) is a cover of the plane (or space) by nonoverlaping regions

3. Tilings in nature

4. Escher’s tilings

5. 3D Tilings and Crystals

6. 3D Tilings and Crystals
Na Cl
B N Cu

7. Tilings by regular polygons

8. 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
Tiles of the same type
Identical tiles

9. Example: two different tilings by squares

10. 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

11. Examples Edge modification Vertex coloring Homework:
Draw the resulting tiling

12. 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?

13. Periodic and non-periodic tilings
A tiling is called periodic if it can be shifted to perfectly align with itself in at least two non-parallel directions
A tiling is called non-periodic if it cannot be shifted to perfectly align with itself
Do non-periodic tilings exist?

14. Trivial example

15. Trivial example
Is there a non-periodic tiling of the plane consisiting of identical tiles?

16. Less Trivial example Yes: cut squares “randomly”
Homework: Make the cutting process more algorithmic to create a non-periodic tiling

17. 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?

18. 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

19. Example: armchair tiling
Source: Wikipedia

20. 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

21. Another example
Source: Wikipedia

22. Conway’s pinwheel tiling (explicitly described by Charles Radin in 1994)
John Conway

23. 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)

24. 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

25. 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)

26. 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?

27. Penrose Tiles and Tilings
Pentagons, “diamond”, “boat”, “star”
Two rhombuses
“Kite” and “dart”
Sir Roger Penrose

28. Kite, dart, and golden triangle
Golden ratio:
36o ϕ
72o 1

29. Kite, dart, and golden triangle
Golden ratio:
36o ϕ
72o 1

30. Kite, dart, and golden triangle
Golden ratio:
36o ϕ
72o 1

31. Kite, dart, and golden triangle
Golden ratio:
36o ϕ
72o 1

32. Kite and dart: matching rules
Prohibited configuration:

33. Possible vertex configurations
Source: Wikipedia

34. “Star” tiling

35. Kite and dart are aperiodic
Theorem Any tiling of the plane by kites and darts that follows matching rules is aperiodic

36. 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!

37. Substitution rule for kite and dart

38. Applying it to kite and dart
We need to show that kites and darts can tile arbitrary large regions of the plane
This can be done through the process of substitution and deflation/inflation
Source: Wikipedia

39. A patch of a Penrose tiling

40. Application of aperiodic tilings: quasicrystals
In 1984 Dan Shechtman announced the discovery of new type of crystal-like structure

41. 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

42. Thank you! Questions faculty.nipissingu.ca/alexandk