Penrose Tilings

Thomas Fernique

Moscow, Spring 2011 Penrose tilings Matching rules Some properties Pentagrids

1 Penrose tilings

2 Matching rules

3 Some properties

4 Pentagrids Penrose tilings Matching rules Some properties Pentagrids

1 Penrose tilings

2 Matching rules

3 Some properties

4 Pentagrids Penrose tilings Matching rules Some properties Pentagrids The trouble with Kepler tilings Penrose tilings Matching rules Some properties Pentagrids The trouble with Kepler tilings Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Regular pentagons almost tile a bigger pentagon. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Each pentagon can in turn be tiled by smaller pentagons. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Holes can be filled by diamonds. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Consider such a diamond with its neighborhood. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Consider such a diamond with its neighborhood. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Tile pentagons by smaller pentagons and fill diamond holes. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

The rest can be tiled with a star, a boat and a pentagon. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Consider the star and the boat, with their neighborhood. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Consider the star and the boat, with their neighborhood. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Consider the star and the boat, with their neighborhood. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Tile pentagons by smaller pentagons and fill diamond holes. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

The rest can be tiled with the same tiles (and neighborhood) Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

tiling of the plane by pentagons, diamonds, boats and stars. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

Method: inflate, divide and fill diamond-shaped holes – ad infinitum. Penrose tilings Matching rules Some properties Pentagrids Tilings by pentagons, diamonds, boats and stars

This yields a hierarchical tiling of the plane (hence non-periodic). Penrose tilings Matching rules Some properties Pentagrids Tilings by kites and darts Penrose tilings Matching rules Some properties Pentagrids Tilings by kites and darts Penrose tilings Matching rules Some properties Pentagrids Tilings by kites and darts Penrose tilings Matching rules Some properties Pentagrids Tilings by kites and darts Penrose tilings Matching rules Some properties Pentagrids Tilings by kites and darts Penrose tilings Matching rules Some properties Pentagrids Mutual local derivability

Equivalence relation on tilings:

Definition (MLD tilings) Two tilings are said to be mutually locally derivable (MLD) if the one can be obtained from the other by a local map, and vice versa.

Example: the two previous tilings are MLD. Penrose tilings Matching rules Some properties Pentagrids Tilings by thin and fat rhombi Penrose tilings Matching rules Some properties Pentagrids Tilings by thin and fat rhombi Penrose tilings Matching rules Some properties Pentagrids Tilings by thin and fat rhombi Penrose tilings Matching rules Some properties Pentagrids Tilings by thin and fat rhombi Penrose tilings Matching rules Some properties Pentagrids Tilings by thin and fat rhombi Penrose tilings Matching rules Some properties Pentagrids

1 Penrose tilings

2 Matching rules

3 Some properties

4 Pentagrids Penrose tilings Matching rules Some properties Pentagrids The trouble with Penrose tilings

The previous tilesets also admit periodic tilings. Penrose tilings Matching rules Some properties Pentagrids The trouble with Penrose tilings

The previous tilesets also admit periodic tilings. Penrose tilings Matching rules Some properties Pentagrids The trouble with Penrose tilings

The previous tilesets also admit periodic tilings. Penrose tilings Matching rules Some properties Pentagrids Three aperiodic tilesets (Penrose, 1974)

Penrose’s trick: notch edges to enforce the hierarchical structure.

(tiles up to rotation) Penrose tilings Matching rules Some properties Pentagrids Three aperiodic tilesets (Penrose, 1974)

Penrose’s trick: notch edges to enforce the hierarchical structure.

(tiles up to rotation) Penrose tilings Matching rules Some properties Pentagrids Three aperiodic tilesets (Penrose, 1974)

Penrose’s trick: notch edges to enforce the hierarchical structure.

(tiles up to rotation) Penrose tilings Matching rules Some properties Pentagrids Penrose tilings (formal)

Definition (Equivalence) Two tilesets are equivalent if any tiling by one tileset is mutually locally derivable from a tiling by the other tileset.

Exercise: prove the equivalence of Penrose tilesets.

Definition (Penrose tilings) A Penrose tiling is a tiling mutually locally derivable from a tiling of the whole plane by one of the three previous Penrose tilesets. Penrose tilings Matching rules Some properties Pentagrids Robinson triangles

Notched rhombi are equivalent to colored rhombi. Penrose tilings Matching rules Some properties Pentagrids Robinson triangles

Notched rhombi are equivalent to colored rhombi. Penrose tilings Matching rules Some properties Pentagrids Robinson triangles

Colored rhombi are equivalent to Robinson triangles.

(tiles up to rotations and reflections) Penrose tilings Matching rules Some properties Pentagrids Robinson macro-triangles

Robinson macro-triangle: homothetic unions of Robinson triangles. Up to this homothety, triangles and macro-triangles are equivalent. Penrose tilings Matching rules Some properties Pentagrids Inflate and divide

Pattern + inflate/divide ad infinitum tiling of the plane (K˝onig). Penrose tilings Matching rules Some properties Pentagrids Inflate and divide

Pattern + inflate/divide ad infinitum tiling of the plane (K˝onig). Penrose tilings Matching rules Some properties Pentagrids Inflate and divide

Pattern + inflate/divide ad infinitum tiling of the plane (K˝onig). Penrose tilings Matching rules Some properties Pentagrids Inflate and divide

Pattern + inflate/divide ad infinitum tiling of the plane (K˝onig). Penrose tilings Matching rules Some properties Pentagrids Inflate and divide

Pattern + inflate/divide ad infinitum tiling of the plane (K˝onig). Penrose tilings Matching rules Some properties Pentagrids Inflate and divide

Pattern + inflate/divide ad infinitum tiling of the plane (K˝onig). Penrose tilings Matching rules Some properties Pentagrids Group and deflate

Conversely, fix a tiling of the plane. Consider a thin triangle (if any). Penrose tilings Matching rules Some properties Pentagrids Group and deflate ?

What can be its red neighbor? Penrose tilings Matching rules Some properties Pentagrids Group and deflate

!

A thin triangle would yield an uncompletable vertex. Penrose tilings Matching rules Some properties Pentagrids Group and deflate

This is thus a fat triangle. Penrose tilings Matching rules Some properties Pentagrids Group and deflate

We can group both to form a thin macro-triangle. Penrose tilings Matching rules Some properties Pentagrids Group and deflate

Consider a remaining fat triangle (if any). Penrose tilings Matching rules Some properties Pentagrids Group and deflate

Its red neighbor is fat (otherwise it would be already grouped). Penrose tilings Matching rules Some properties Pentagrids Group and deflate

?

What can be its blue neighbor? Penrose tilings Matching rules Some properties Pentagrids Group and deflate !

A fat triangle would yield an uncompletable vertex. grouped into a thin macro-triangle

Penrose tilings Matching rules Some properties Pentagrids Group and deflate

This is thus a thin triangle . Penrose tilings Matching rules Some properties Pentagrids Group and deflate

This is thus a thin triangle grouped into a thin macro-triangle. Penrose tilings Matching rules Some properties Pentagrids Group and deflate

We can group the three triangles to form a fat macro-triangle. Penrose tilings Matching rules Some properties Pentagrids Group and deflate

Hence, any tiling by Robinson triangles. . . Penrose tilings Matching rules Some properties Pentagrids Group and deflate

. . . can be uniquely seen as a tiling by Robinson macro-triangles. Penrose tilings Matching rules Some properties Pentagrids Group and deflate

Macro-triangles can be consistently replaced by triangles. . . Penrose tilings Matching rules Some properties Pentagrids Group and deflate

. . . and by deflating we get a new Penrose tiling. Penrose tilings Matching rules Some properties Pentagrids Group and deflate

Group/deflate ad infinitum aperiodicity of Robinson triangles. Penrose tilings Matching rules Some properties Pentagrids

1 Penrose tilings

2 Matching rules

3 Some properties

4 Pentagrids Penrose tilings Matching rules Some properties Pentagrids Uncountability

Proposition Penrose tilesets admit uncountably many tilings of the plane.

Proof: track a tile in the tiling hierarchy infinite tile sequence; tiles of isometric tilings sequences with the same tail ends; infinite tile sequence tiling; countably many sequences with the same tail end; uncountably many infinite tile sequences. Proposition Penrose tilings are quasiperiodic.

Proof: first check for r = Diam(tile), then group/deflate.

Penrose tilings Matching rules Some properties Pentagrids Quasiperiodicity

Pattern of size r (in a tiling): tiles lying in a closed ball of radius r.

Definition (Quasiperiodic tiling) A tiling is said to be quasiperiodic if, for any r > 0, there is R > 0 such that any pattern of size r appears in any pattern of size R.

Quasiperiodic ≡ bounded gap ≡ repetitive ≡ uniformly recurrent.

Any periodic tiling is also quasiperiodic (take R = r + period). Penrose tilings Matching rules Some properties Pentagrids Quasiperiodicity

Pattern of size r (in a tiling): tiles lying in a closed ball of radius r.

Definition (Quasiperiodic tiling) A tiling is said to be quasiperiodic if, for any r > 0, there is R > 0 such that any pattern of size r appears in any pattern of size R.

Quasiperiodic ≡ bounded gap ≡ repetitive ≡ uniformly recurrent.

Any periodic tiling is also quasiperiodic (take R = r + period).

Proposition Penrose tilings are quasiperiodic.

Proof: first check for r = Diam(tile), then group/deflate. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

π Tile angles multiple of 5 only 5- or 10-fold symmetries (or 2-fold). Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

Up to isometry, three 5-fold elementary patterns. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

Inflate/divide two 5-fold Penrose tilings of the plane. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

Inflate/divide two 5-fold Penrose tilings of the plane. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

Inflate/divide two 5-fold Penrose tilings of the plane. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

They are different even up to decorations. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

They are different even up to decorations. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

Conversely, a symmetry center must live in the whole hierarchy. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

Conversely, a symmetry center must live in the whole hierarchy. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

Conversely, a symmetry center must live in the whole hierarchy. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

There is thus only two 5-fold Penrose tilings of the plane. Penrose tilings Matching rules Some properties Pentagrids Rotational symmetry

The uncountably many others have only local 5-fold symmetry. Penrose tilings Matching rules Some properties Pentagrids Remind quasicrystals

Point-holes at vertices of a Penrose tiling 5-fold diffractogram. Penrose tilings Matching rules Some properties Pentagrids Remind quasicrystals

Point-holes at vertices of a Penrose tiling 5-fold diffractogram. Penrose tilings Matching rules Some properties Pentagrids Remind quasicrystals

Point-holes at vertices of a Penrose tiling 5-fold diffractogram. Penrose tilings Matching rules Some properties Pentagrids Remind quasicrystals

Point-holes at vertices of a Penrose tiling 5-fold diffractogram. Penrose tilings Matching rules Some properties Pentagrids Vertex atlas

In Penrose tilings, Robinson triangles fit in 8 ways around a vertex. Penrose tilings Matching rules Some properties Pentagrids Vertex atlas

Up to decorations, this yields a vertex atlas of size 7. Penrose tilings Matching rules Some properties Pentagrids Vertex atlas

Proposition: any tiling with this vertex atlas is a Penrose tiling. Penrose tilings Matching rules Some properties Pentagrids Vertex atlas

A similar vertex atlas for tilings by thin and fat rhombi. Penrose tilings Matching rules Some properties Pentagrids Vertex atlas

A similar vertex atlas for tilings by kites and darts. Penrose tilings Matching rules Some properties Pentagrids Ammann bars

Draw these two particular billiard trajectories in each triangle. Penrose tilings Matching rules Some properties Pentagrids Ammann bars

1

1

4

Some trigonometry ratio characterizing the trajectories. Penrose tilings Matching rules Some properties Pentagrids Ammann bars

This draws on Penrose tilings five bundles of parallel lines. Penrose tilings Matching rules Some properties Pentagrids Ammann bars

This draws on Penrose tilings five bundles of parallel lines. Penrose tilings Matching rules Some properties Pentagrids

1 Penrose tilings

2 Matching rules

3 Some properties

4 Pentagrids Penrose tilings Matching rules Some properties Pentagrids Pentagrid (De Bruijn, 1981)

6

5

S j 4

3

2

-3-4 0-1-2 1

−j 2iπ/5 Shift sj ∈ R grid {z ∈ C | Re(zζ )+sj = 0}, where ζ = e . Penrose tilings Matching rules Some properties Pentagrids Pentagrid (De Bruijn, 1981)

6

5

S j 4

3

2

-3-4 0-1-2 1

−j Strip numbering: Kj (z) = dRe(zζ ) + sj e. Penrose tilings Matching rules Some properties Pentagrids Pentagrid (De Bruijn, 1981)

Pentagrid: grids 0,..., 4, with s0 + ... + s4 ∈ Z and no triple point. Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

1)f(z z1

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

zf()2 )f(z z2 1 z1

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

zf()2 )f(z z2 1 z1 zf()3 z3

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

zf()2 z )f(z 2 z 1 1 zf() z 3 3 z 4 zf()4

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

zf()2 z )f(z 2 z 1 1 zf() z 3 3 z 4 zf()4 z 5 zf() z6 5 zf()6

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From pentagrids to rhombus tilings

X j f (z) := Kj (z)ζ maps each mesh onto a rhombus vertex. 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids Indices

3 2 3 2 2 3 2 3 2 3 2 3 4 2 2 4 1 3 4 3 3 3 3 3 3 3 3 3 2 2 2 2 2 3 2 4 4 2 4 2 4 1 2 2 1 2 2 2 1 2 1 3 1 1 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 4 4 4 4 3 3 3 3 2 1 3 3 1 3 3 4 3 1 1 3 3 3 3 2 3 2 3 2 2 2 2 2 2 2 2 2 4 1 1 1 3 3 3 3 3 3 3 2 3 2 2 2 2 2 2 2 3 4 4 3 4

X i(z) := Kj (z) maps each mesh into {1, 2, 3, 4}. (check) 0≤j≤4 Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

?

Can the rhombi of such tilings be consistently notched? Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

2 3

2 3 3 3 2 2 3 3 2 2 4 1

4 1

Lemma: each rhombus can be endowed by indices in only two ways. Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

3 2 3 2 2 3 2

4 1 3 4 3 3 3 3 3 2 2 2

2 4 2 4 2 1 2 1 3 1 3 3 3 3 2 2 2 2 4 4 4 3 2 1 3 3 1 3 3

3 3 3 2 2 2 2 2 2 4 1 1

3 3 2 3 3 2 2 2

4 3 4

Lemma: each rhombus can be endowed by indices in only two ways. Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

2 3

2 3 3 3 2 2 3 3 2 2 4 1

4 1

Double-arrow notchings from 3 to 4 and from 1 to 2: consistent. Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

3 2 3 2 2 3 2

4 1 3 4 3 3 3 3 3 2 2 2

2 4 2 4 2 1 2 1 3 1 3 3 3 3 2 2 2 2 4 4 4 3 2 1 3 3 1 3 3

3 3 3 2 2 2 2 2 2 4 1 1

3 3 2 3 3 2 2 2

4 3 4

Double-arrow notchings from 3 to 4 and from 1 to 2: consistent. Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

2 3

2 3 3 3 2 2 3 3 2 2 4 1

4 1

Notchings of remaining edges are forced. Is it consistent? Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

2 3

2 3 3 3 2 2 3 3 2 2 4 1

4 1

Remark: single-arrow notchings go from acute to obtuse angles. Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

Lemma: along an edge between 2 and 3, acute/obtuse angles match. Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

This yields the consistency of the single-arrow notchings. Penrose tilings Matching rules Some properties Pentagrids From rhombus tilings to Penrose tilings

3 2 3 2 2 3 2

4 1 3 4 3 3 3 3 3 2 2 2

2 4 2 4 2 1 2 1 3 1 3 3 3 3 2 2 2 2 4 4 4 3 2 1 3 3 1 3 3

3 3 3 2 2 2 2 2 2 4 1 1

3 3 2 3 3 2 2 2

4 3 4

This yields the consistency of the single-arrow notchings. Theorem (de Bruijn, 1981) The Penrose tilings are exactly the tilings derived from pentagrids.

Proof: Fix a Penrose tiling φ = φ(0); (n) inflate/divide n times Penrose tiling φ ; (n) find ~sn such that φ~sn and φ agree on B(0, 1); (0) n n group/deflate n times φg (~sn) and φ agree on B(0, τ ); n 5 (0) g (~sn) ∈ [0, 1) accumulation point t φ = φ = φt .

Penrose tilings Matching rules Some properties Pentagrids From Penrose tilings to pentagrids

Let φ~s be the rhombus tiling derived from the pentagrid of shift ~s.

Lemma: group/deflate φ~s yields φg(~s), where g(~s)j = sj−1 + sj+1. Penrose tilings Matching rules Some properties Pentagrids From Penrose tilings to pentagrids

Let φ~s be the rhombus tiling derived from the pentagrid of shift ~s.

Lemma: group/deflate φ~s yields φg(~s), where g(~s)j = sj−1 + sj+1.

Theorem (de Bruijn, 1981) The Penrose tilings are exactly the tilings derived from pentagrids.

Proof: Fix a Penrose tiling φ = φ(0); (n) inflate/divide n times Penrose tiling φ ; (n) find ~sn such that φ~sn and φ agree on B(0, 1); (0) n n group/deflate n times φg (~sn) and φ agree on B(0, τ ); n 5 (0) g (~sn) ∈ [0, 1) accumulation point t φ = φ = φt . Penrose tilings Matching rules Some properties Pentagrids Application

Remind: there are uncountably many Penrose tilings; exactly two of them have a global five-fold symmetry.

Proof:

uncountably many shifts s0 + ... + s4 ∈ Z; ~ 2 2 0 center of symmetry iff sj = 5 or sj = − 5 . Bibliography

Some references for this lecture:

Roger Penrose, Pentaplexity: a class of non-periodic tilings of the plane, Eureka 39 (1978). Nicolaas Govert de Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane, Indag. Math. 43 (1981). Marjorie Senechal, Quasicrystals and Geometry, Cambridge University Press, 1995. Chap. 6.

These slides and the above references can be found there:

http://www.lif.univ-mrs.fr/~fernique/qc/