The corona limit of Penrose tilings is a regulardecagonShigeki Akiyama1 and Katsunobu Imai2
1

Institute of Mathematics & Center for Integrated Research in Fundamental Scienceand Technology, University of Tsukuba
Tennodai, Tsukuba, Ibaraki, 305-8571 Japanakiyama@math.tsukuba.ac.jp
2
Graduate School of Engineering, Hiroshima University,
Higashi-Hiroshima 739-8527 Japan
imai@hiroshima-u.ac.jp

Abstract. We define and study the corona limit of a tiling, by investigating the signal propagations on cellular automata (CA) on tilings
employing the simple growth CA. In particular, the corona limit of Penrose tilings is the regular decagon.

1

Introduction

Since the discovery of quasi-crystals, quasi-periodic tilings like Penrose tiling
attracted a lot of attention as their possible mathematical models. Spectral studyof tiling dynamical system and Schrödinger operator on quasi-periodic structure
are developed in order to analyze their long-range order and quantum mechanicalmotion of the particle on quasi-periodic structure [8].
Cellular automata working on quasi-periodic tilings [10, 2, 13] are also studied. In particular, the intensive studies of the Game of Life[3] on Penrose tilings
by Owens and Stepney are paid attentions [11, 12]. As the result, recently, acellular automata simulator for reaction-diffusion media which also working ona Penrose tiling is released [7] and the first glider pattern on Penrose tilings isalso found [5].
We are interested in the difference of signal propagation of cellular automata
working on between the normal periodic cells and quasi-periodic tilings. In thiscontext, Chidyagwai and Reiter showed that the broken symmetry of quasiperiodic tilings, while still retaining a highly organized structure, could be used
to simulate the complex growth of snow crystals [2]. They could produce global
n-fold symmetry models where regular hexagonal grids could only produce 6-foldsymmetry models.
In this paper, we introduce corona limit which naturally visualizes thegrowth pattern of signal propagation. We show that the corona limit of a Penrose
tilings is a regular decagon. The speed of convergence depends on the version oftilings and adjacency condition of it.

2Corona limit

A tiling T is a covering of R2 by finitely many polygonal tiles and their images byisometry (translation, rotation, and flip) which overlap only at their boundaries.
Two tiles A, B are adjacent (resp. edge adjacent) if they share a point (resp. anedge). For brevity, we also say that A and A itself are adjacent (edge adjacent)as well. A patch P is a finite set of tiles in T . The 1-st corona P (1) of P isa patch consisting of all tiles which is adjacent to a tile of P. The n-th corona
P (n) is defined as a corona of the (n − 1)-th corona P (n−1) for n = 2, 3, . . . . Anedge corona P [0] and n-th edge corona P [n] are defined in the same manner but
by edge adjacency. If a sequence of shrunk patches
1 (n)
P
n

n = 1, 2, . . .

converges to a non-empty compact set K under Hausdorff metric, we say that K
is a corona limit. Here for two non empty compact sets A and B, the Hausdorff
metric is defined asinf{ε > 0 | A ⊂ B[ε] and B ⊂ A[ε]}with X[ε] := {y ∈ R2 | ∃x ∈ X ky − xk ≤ ε}. To see the geometric meaning
better, the limit is rephrased as limn n1 P (n) − c by fixing a point c in the initialpatch P. Then the coronas grow around the center c, and we renormalize themby the factor 1/n to obtain the corona limit. Since the tiling has finitely many
shapes, the diameter and inradius (maximal radius of the inscribed ball) of then-th corona are bounded from below and above by positive constant multiples ofn. Thus the corona limit contains the origin as an inner point, and is bounded.
Interestingly, one can show that if a corona limit K exists, then it does notdepend on the choice of the initial patch P. To see this, we claim that thecorona limit of P and that of P (n) are identical, and for any two patches P,Q,
there exist a positive integer m that Q ⊂ P (m) and P ⊂ Q(m) . Therefore wesay that K is the corona limit of T . We can similarly define the edge corona
limit. For periodic tilings, the corona limit is usually easy to obtain by simple
induction. For the standard square tiling the corona limit is a square, and theedge corona limit is also a square but rotated π/4. It often becomes a hexagon
for Archimedean tilings.

3

Penrose Tilings

A Penrose tiling is a tiling generated by a set of tiles with matching conditions.
They tiles the plane but only in non periodic way. Among many versions ofPenrose tilings, there are two types of tilings generated by two quadrilateral
proto-tiles : two rhombus tiles (a fat and a thin), and a kite and a dart. To avoid
periodic arrangements, several matching conditions are known to be added onthese tiles [4]. Ammann bars are one of such matching condtions [6]. Each tile inFig. 1 has Ammann bar line segments and Fig. 2 illustrates a possible rhombus

tiling and its substitution rules. Let ϕ be the golden ratio. To get a tiling froma patch, iterate the magnification by ϕ and substitution for each tile by the rulein Fig. 2. The line segments of each tile must be continued straight across theboundary. They form parallel lines of five different slopes and the gap length ofeach parallel lines is one of L and S. It is known that there exist infinitely manytranslationaly inequivalent quasi-periodic tilings of the plane with these tiles.
rhombuskite and dart

1/4
ϕ
2

ϕ
2
ϕ−1
2

ϕ
2π
5

fat

ϕ

1/4
ϕ−1
2

ϕ

ϕ−1
2

ϕ

π
5

thin

1
1

1/4

ϕ−1
2
1/4

Fig. 1. Rhombus, and Kite and Dart tiles with the line segments of Ammnann bars.

Fig. 2. A rhombus tiling and its substitution rules.

4

Growth Cellular Automaton on Penrose Tilings

We define a cellular automaton A as follows: choose a two-dimensional tiling T .
Each tile τ has a state c(τ ) ∈ Q and neighborhood tiles defined by N (τ ). We
call all assignments of state to the tiles C : T → Q as configuration. The local
function of A is defined by the assignment of states in N (τ ) and returns thenext state of τ . Thus its simultaneous application to each tile in C defines theglobal evolution A : C → C. When the tiling is the regular square grid, we denotethe vertex adjacent neighborhood by Moore neighborhood (NM ) and the edge
adjacent neighborhood by von Neumann neighborhood (NN ).
In order to define a CA on a Penrose tiling, we need to extend the above
neighboring relation. Because it is not a lattice tiling, congruent tiles may havedifferent neighborhoods. Generalized von Neumann and Moore neighborhoods
are illustrated in Fig. 3 and Fig. 4, respectively [11].

Rhombus, von Neumann

Kite and dart, von Neumann

Fig. 3. The generalized von Neumann neighborhoods on Penrose tilings [11]Rhombus, Moore

Kite and dart, Moore

Fig. 4. The generalized Moore neighborhoods on Penrose tilings [11]From this section, we focus on a simple cellular automaton, growth cellular
automaton.
Definition 1. Growth cellular automaton is a cellular automaton A whose state
set is Q = {0, 1}. Its local function is defined such that if the state of a focus tile
is 0 and at least one of its neighborhood tiles’ state is 1 then its state changes to
1. Any tile of state 1 never changes its state.
Let C0 be a configuration such that a single cell’s state is 1 and the others
are 0. Let Pn be the shape formed by state 1 cells by n-step evolutions of A from
C0 as its initial configuration.
It is clear that in the case of square Moore neighborhood, Pn is a square whichside length is 2n + 1 and in the case of square von Neumann neighborhood, Pn
is a π/4-rotated square which diagonal length is 2n + 1.

We show the simulated results of the Penrose tilings. Fig. 5 illustrates theresults after 10-step and 30-step evolutions (The scale of the figures of 10 and 20
steps are different). Because the |NM | is larger than |NN | on average, the size of
Pn is larger in the case of Moore neighborhoods than the case of von Neumannneighborhood. Each Pn seems to converge to a regular decagonal shape. Eventhe case of rhombus Moore neighborhood, it eventually converges to a regulardecagonal shape (See Section /refsec:speed).

Kite Dart Neumann Kite Dart Moore

Rhomb Neumann

Rhomb Moore

t=10

t=30

Fig. 5. The evolution of growth CA A on Penrose tilings (t = 10, 30 steps).

In the next section we show its proof in the case of rhombus von Neumannneighborhood.

5

Rhombus von Neumann neighborhood caseIn this section, we prove the experimental observation of the previous section in
the case of rhombus von Neumann neighborhood.
Choose a patch of the star shape in any rhombus Penrose tiling T as inFig. 6. It has five symmetric crossing Ammann bars. We denote the central
point of the star by O. We call the five symmetric Ammann bars across thestar as a0 , b0 , c0 , d0 , e0 . We denote by ε the distance between O and one of theAmmann bars. We denote by Xi for each Ammann bar parallel to X0 whereX ∈ {a, b, c, d, e} and for all integers i. We also denote X−i by X̄i . The lengthof the gap between two Ammann bars Xi and Xi+1 (denoted by Xi Yj ) is L or
S. We denote by Xi Yj the cross point of Xi and Yj .
Proposition 1. Let ∂Di be connected lines formed by the points: ai bi , bi ci , ci di ,¯ i , ēi ai , ai bi . There exists a constant k, for any i(> k),di ei , ei āi , āi b̄i , b̄i c̄i , c̄d¯i , dē∂Di forms a decagon (Di ) and Di ⊂ Di+1 .

b-1 b0
b1
a-1 a0 a1
c0

e0

d0

Fig. 6. Amman bars and their index.

Proof. We denote by X̂i the distance between O and Xi . For any two Ammannbars Xi and Yi which angle is π/5, if X̂i /Ŷi > cos(π/5) then ∂Di forms a decagon.
Because X̂i = X0 , Xi ± ε and X0 , Xi is the i-th addition of L or S, the valueX̂i /Ŷi converges to 1 as i grows. So there exists a constant k, for any i(> k),X̂i /Ŷi > cos(π/5).
We call Di (where i satisfies that each Di forms a decagon) as a sequence ofuniaxial Ammann bar decagons (Fig. 7).
Proposition 2. Decagon Di converges to the shape of regular decagon as as igoes to infinity.
Proof. Because the ratio of the numbers of L and S for parallel Ammann bars
converges to ϕ (10.6.8 in [6], [9]), i.e., for any large i, the distance between O
and Xi converges to the same length. Thus Di converges to a regular decagon
as i goes to infinity.
Definition 2. If the most part of a tile τ is in the gap between two Ammannbars Xi and Xi+j then we call τ is a gap tile of Xi and Xi+j . We denote byG(Xi , Xi+j ) all gap tiles of Xi and Xi+j .
Definition 3. G(Xi , Xi+1 ) is said to be filled when all tiles in G(Xi , Xi+1 ) except fat tiles of which diagonal line is perpendicular to the bars have state 1. We
do not concern the states of these fat tiles.
Fig. 8 shows a filled S-gap and a filled L-gap.
Proposition 3. For any tile τ ∈ G(Xi , Xi+1 ), the following properties hold:1. τ intersects only one of two Ammann bars Xi and Xi+1 ;2. If τ intersects Xi (Xi+1 ) then there exists an edge adjacent tile in
G(Xi , Xi+1 ) which intersects the bar Xi+1 (Xi ) and τ has no adjacent tilein G(Xi+1 , Xi+2 ) (G(Xi−1 , Xi ));a-2 a-1

a0 a1 a2
D8
D7

b-1 b0
b1

D6
D5
D4
D3

c0

D2

d0

e0

Fig. 7. A sequence of uniaxial Ammann bar decagons Di (i ≥ 3)

Fig. 8. S and L filled gaps.

L1

LS1

S1

L2

LS2

S2

Illegal pair

L3

Fig. 9. Possible pairs according to Ammann bars.

3. If τ intersects Xi (Xi+1 ), then τ has an edge adjacent tile in G(Xi−1 , Xi )(G(Xi+1 , Xi+2 )).
Proof. There are 16 possible combinations of two tiles. Removing rotationallysymmetric pairs, there are only eight pairs in Fig. 9. Moreover one of them is
an illegal combination. The pairs L1 (S1, S2) is the component of two pairs ofparallel Ammann bars with L-gaps (S-gaps), respectively. The pairs L2 and L3
are the component of one pair of parallel Ammann bars with L-gaps and they are
symmetric with each other. The pairs LS1 and LS2 are the component of parallel
Ammann bars with an L-gap and an S-gap and they are also symmetric witheach other. Blue (Green) line segments are used for L-gaps (S-gaps), respectively.
To exhaust the possible placements along an S-gap, we start from a pair S1.
Although S1 contains line segments for two distinct S-gaps, they are symmetric.
So it is enough to consider one pair of them. Fig. 10 shows the first possible
extensions. The right figure in Fig. 10 is the only possibility.

1

2

3

illegalFig. 10. Possible extensions from a pair S1. Fig. 11. The 2nd step of possible extensions from a pair S1

The second step, there are three possibilities in Fig. 11. S1 appears again in
Fig. 11-3. Fig. 11-1 and Fig. 11-2 have a fat tile of the same angle along the Sgap. But the latter is an illegal placement because there is no feasible placementin the next step (Fig. 12-2). Thus we can continue the next step only in thecase of 1. The final step, the only possible placement is Fig. 12-1 and S1 appears
again.
Fig. 14 is the possible extensions to the opposite direction from a pair S1.
Because any thin tile cannot be placed to the opposite side of S1 as in Fig. 13,
fat tiles forced to be placed next to S1. Fig. 14-1 and 2 have already appearedin Fig. 11-1 and 2. S1 appears again in Fig. 14-3. Thus there is no different
placement except their symmetric cases appeared in the both side of S1.
The downside (upperside) of S2 appears in Fig. 10 right (Fig. 11), respectively. The upper side of LS1 (and the symmetric version, LS2) appears in Fig.5
right. The downside of LS1 (and the symmetric version, LS2) only accepts a thintile to form the shape S1. Thus there is no more S-gap patterns connected from
S1, S2, LS1, and LS2.

12 illegal

Fig. 12. The final step of possible extensions from a pair S1.

1

illegal2

3

Fig. 13. Possible and illegal placement of a Fig. 14. The possible extensions to thetile to the opposite direction from a pair S1. opposite direction from a pair S1.

Because most of the patterns forming an S-gap are S1, S2, LS1, and LS2, theother patterns appeared in an S-gap must be generated by the above process. So
all possible placements except their symmetric cases along an S-gap are included.
In the same way, we consider the possible placements along an L-gap, we
start from a pair L1. Although L1 contains line segments for two distinct Lgaps, they are symmetric. So it is enough to consider one pair of them. Fig. 15-1
shows the first possible extensions. 1b is the only feasible placement and theonly placements in the next step are Fig. 15-2.

1 illegal
1a illegal

1b2a2 illegal

3 illegal

42bFig. 15. The possible extension from a pairFig. 16. Possible extension from a pair
L1.
L1 (2).

We consider the next step of Fig. 15-2a. The only feasible placement is Fig. 164 and a thin tile in the next step (Fig. 17). Thus we do not need to consider thecase of Fig. 15-2b separately because the angle of the thin tile in Fig. 17 is thesame.

1 illegal

2 illegal

3

Fig. 17. The possible extension from a pairFig. 18. Possible extension from a pair
L1 (3).
L1 (4).

The next step, only the case Fig. 18-3 is possible and there are two options
Fig. 19 after the extension. L1 appears again in Fig. 19-1. The final extension
in Fig. 20 shows there is only one possibility. The opposite side of L1 appears in
Fig. 20-3.

12

1 illegal

2 illegal

3

Fig. 19. The possible extension from a pairFig. 20. Possible extension from a pair
L1 (5).
L1 (6).

Then we consider the opposite side of L1. Fig. 21 is the possible extensions to
the opposite direction from a pair L1. In Fig. 21-1, it appears the opposite sideof L1 again and Fig. 21-2 (3) has the same shape as Fig. 15-2a (2b), respectively.
Thus all possible patterns appeared from L1 are shown in the previous figures.

1

2

3

Fig. 21. Possible extension to the opposite direction from a pair L1.

The upper side of L2 (and its symmetric version) appears in Fig. 15-2. The
downside of L2 (and its symmetric version) appears in Fig. 18-3. The downsideof LS1 (and the symmetric version, LS2) appears in Fig. 15-2a. The upper sideof LS1 (and the symmetric version, LS2) only accepts a thin tile and succeeding
placements are the same as that in Fig. 17. Thus there is no more L-gap patterns
connected from L1, L2, L3, LS1, and LS2.
To sum up, all possible placement of tiles along S-gaps and L-gaps are included in the above figures. Thus it is clear that the proposition agree with theall patterns in the figures.
Thus the following proposition holds:
Proposition 4. Let A be a von Neumann neighborhood growth cellular automaton on a rhombus Penrose tiling. All tiles of its initial configuration are state 0
except a filled gap of two Ammann bars Xi and Xi+1 . After two steps executionof A, G(Xi+1 , Xi+2 ) and G(Xi−1 , Xi ) are filled and all gaps outside of Xi+2 or
Xi−1 are remained to be unfilled.
Definition 4. Let Di be a sequence of uniaxial Ammann bar decagons. For an
integer k, we call Dk is filled when all tiles in Dk has state 1 except fat tiles towhich diagonal line is perpendicular and crossing with each line in ∂Dk . We do
not concern the states of these fat tiles.
Proposition 5. Let A be a von Neumann neighborhood growth cellular automaton on a rhombus Penrose tiling and Di be a sequence of uniaxial Ammann bar
decagons. Suppose all tile has state 0 in its initial configuration except a filleddecagon Dk for an integer k. Then after two-step executions of A, Dk+1 is filled
and the outside of Dk+1 is remained to be unfilled.
Proof. By Prop. 4, the outside of Dk+1 is remained to be unfilled. We need toshow that all tiles in G(Xk , Xk+1 ) ∩ Dk+1 are filled after two-step executions. It
is enough to show that the closest tile (in Dk+1 ) to each vertex of Dk+1 can bereached in two-step executions from its associated vertex of Dk . Two crossing

points of two pairs of two neighboring parallel Ammann bars of which angle is
4π/5 are the candidates of vertices of Dk and Dk+1 . In the pictures of proof of
Prop. 3, black circles and white circles are all such candidates of points. It iseasily check that they all have two-step distance.
Theorem 1. The edge corona limit of rhombus Penrose tilings is a regulardecagon.
Proof. Let A be a von Neumann neighborhood growth cellular automaton on arhombus Penrose tiling and Di be a sequence of uniaxial Ammann bar decagons.
Even if the given initial configuration is not connected, executing enough stepsof A, it is possible to change the shape of state-1 cells as follows:
1. the shape formed by the state 1 tiles is connected;
2. the shape contains a star and at least one of Ammann bar decagon Di forsome i,i.e., there exists integers i and j(> i) such that Di is filled and tiles outside of Dj
are state 0. We denote this initial patch by P [0] . By Prop. 5, Dn+i is filled andtiles outside of Dn+j are state 0 after 2n-step executions, i.e., Dn+i ⊂ P [2n] ⊂Dn+j holds. Because j − i is a constant, P [n] /n converges to the regular decagon
limn→∞ D[n/2] /n as n tends to infinity.

6

The difference of growth speeds

In the previous section, we show the wavefront of a growth cellular automaton ona rhombus Penrose tilings eventually forms a regular decagon and the speed of
the wavefront passing through L- or S-gap is l/2 per step, l ∈ {S, L}. Although
the signal propagation of the Moore neighborhood case is more fluctuated, thewavefront also forms a regular decagon and its speed is l, l ∈ {S, L} by theexistence of thin tiles along an S-gap. In the case of kite and dart tilings, it isalso possible to show the similar result in the same way.
Each growth speed across a group of Ammnann bars are shown in Table 1.
They actually agree with the difference of growth illustrated in Fig. 5.
Table 1. Growth speeds.

von Neumann
Moore

rhombusL/2, S/2
L, S

kite and dart
L/3, S/2
L/2, SIn Section 4, we pointed out that the convergence speed to a decagonal shapeis very slow in the case of rhombus Moore neighborhood. To inspect the behavior,
we modify the local function of the growth cellular automaton as follows:Definition 5. A cellular automaton A+ whose state set is Q = {0, 1, 2, 3, . . . }( |Q| is the maximum number of tiles sharing a vertex). Its local function isdefined such that if the state of a focus tile is 0 then its state is changed to thesum of all non-zero cells in N . Any tile with state ≥ 1 never changes its state.
Fig. 22 illustrates the evolutions of A+ in the case of square and rhombus
Penrose tilings. A light blue (state 1) tile receives a signal from a neighboringtile and a darker tile receives two or more signals at a time from two or moreneighboring tiles. Intuitively a light blue line segment shows that the existence ofthe fastest signal propagation at the sites. In the square cases, the line segments
are the diagonal lines of a square formed by the ‘light speed’ of the cellular space.
In the rhombus cases, there are many light blue line segments of short length.
This mainly due to the fluctuation of signal propagations caused by the existenceof alternations of L and S gaps. But even the tiling is the same, the situationis quite different between the von Neumann and the Moore neighborhood cases.
In the Moore neighborhood case, there are many self similar light blue line
segments and complicated signal collisions occur. This is the reason for the slow
convergence to a regular decagonal shape in the viewpoint of signal propagationof cellular automata.
As the result, employing rhombus tiling and Moore neighborhood might bemore suitable for the cellular automata simulation of, for example, chemical
reactions than the other cases. Because it anyway achieves the most homogeneous
local signal propagation in the above options.

0
1
2
3 or more

Square, Neumann

Square, Moore

rhombus, Neumannrhombus, Moore

Fig. 22. The evolutions of A+

7

Conclusion

In this paper, we showed that the corona limit of Penrose tilings is a regulardecagon, thanks to the existence of Ammann bars. This might suggest somerelation between corona limits and windows of the cut and project scheme which
generate the tilings. However we know little on this connection. This idea may
fail if the cut and project scheme is non Archimedean. For the case of the chairtiling (Fig.10.1.5 in [6]), its corona limit is easily shown to be an octagon. The
associated cut and project scheme is realized with 2-adic internal space [1] whose
window is not a four dimensional hypercube.
It is an intriguing question to understand what decides the shape of thecorona limit.
Acknowledgments. Shigeki Akiyama is supported by the Japanese Society
for the Promotion of Science (JSPS), Grant in aid 21540012. Katsunobu Imai
thanks Takahiro Hatsuda from SOLPAC Co., Ltd. for the helpful discussions
and gratefully acknowledges the support of JSPS KAKENHI Grant Number
26330016.

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