Some observations on a substitution
rule with singular vertex configrations
高知大学・数学コース

林 浩子 (Hiroko Hayashi)
小松 和志 (Kazushi Komatsu)
Department of Mathematics,
Kochi University

1. Preliminary
We recall several basic deﬁnitions.
A patch of a tiling is a ﬁnite subset of the tiles in a tiling. A vertex
conﬁguration of a vertex in a tiling is the patch consisting of all tiles whichcontains the vertex.
Let S = {S1 , S2 , · · · , Sl } be a ﬁnite set of polygons Sj . When each tile Tin a tiling T is congruent to some Si ∈ S, S is called a prototile set of T .
A set of matching rules for a prototile set S is a ﬁnite set of patches that
may appear in the tilings admitted by S.
Fix λ(> 1). For S = {S1 , S2 , · · · , Sl }, any Sk (∈ S) is decomposed into
λ−1 scale-down copies λ−1 S = {λ−1 S1 , λ−1 S2 , · · · , λ−1 Sl } of S. This decomposition is called a substitution rule of S if such a decomposition is possible.
Let Φ denote a substitution rule of S, and Φ[Sk ] denote the decompositionof Sk ∈ S. Similarly, let σ(Sk ) denote the decomposition of λSk into S. A
patch in a tiling corresponding to σ n (Sk ) is called a supertile with level n ofSk .
A substitution is primitive if the substitution rule is a linear map thatcan be represented by a primitive matrix. An n × n matrix A is said to beprimitive if its entries are nonnegative integers and if there exists a positive
integer k such that all the entries of Ak are positive.
A substitution is said to force the border if there exists a positive integern such that any two level n supertiles of the same type have the same patternof neighboring tiles.
1The predecessor P for a tiling T under a substitution σ is a tiling suchthat σ(P ) = T . That is the tiling that substitutes to T. A substitution tiling
T is a tiling that has a inﬁnite sequence of predecessor tilings Pn such that
σ n (Pn ) = T . A singular vertex conﬁguration is one in a substitution tilingwhich does not appear in supertiles.
The prototiles of Penrose tilings are four types of triangles with arrows
on the edges (two triangles a, b in Figure 1 and their mirror images). The
substitution rule of prototiles is given as in Figure 2 and their mirror images([1]).

Figure 1: Two prototiles witharrows of Penrose tilings

Figure 2: The substitution rule of Penrose tilings

The prototiles of Danzer tiling are six types of triangles with arrows
on the edges (three triangles a, b, c in Figure 3 and their reﬂections). The
substitution rule of prototiles is given as in Figure 4 and their mirror images([5]).

Figure 3: Three prototiles witharrows of Danzer tiling (θ = π/7)

Figure 4: The substitution rule ofDanzer tiling

One of the purpose of this note is to analyze construction of tilings withrotational symmetry obtained by substitution rules (See [4]). We can construct nonperiodic tilings with a given prototile set by the up-down gen2

eration using substitution rule (cf.[1]). The whole plane cannot always becovered by the up-down generation. In particular, Penrose tiling and Danzer
tiling with rotational symmetry cannot be constructed only by the up-downgeneration procedure. It is necessary to extend the tilings to the whole planeby arranging unbounded conﬁgurations obtained by the up-down generation
procedure.
We have the following questions:(1) For which n, can tilings with n-fold symmetry be constructed only by
the up-down generation procedure ?
(2) To construct tilings, how can we arrange unbounded conﬁgurations
obtained by the up-down generation procedure ?
It seems that theses questions are open.
2. Some observations
2.1. Question (1)If we do not need primitivity, we have the following answer for Question(1).
”If there is a substitution with n-fold tiling, we can make a substitutionthat can generate an n-fold tiling from up-down generation.”
Substitutions with n-fold tiling are constructed in [2] and [5]. In order topresent this answer, we use Penrose tiling as an example. In Figure 5 above
four triangle tiles are the original Penrose tiles. We add two tiles of star
shape. These are two shapes of vertex conﬁgurations at the center of 5-foldsymmetry. We see that this substitution is primitive.
By the similar way, we can make a substitution from the substitutionof Danzer tiling. This new substitution is not primitive, because the vertex
conﬁguration at the center of 7-fold symmetry never appear in any supertileof triangle tiles (cf.[3]).
So we have the following new question:(1)’ For which n, can tilings with n-fold symmetry be constructed onlyby the up-down generation procedure of some primitive substitution?

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Figure 5: Prototile

Figure 6: Substitution

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2.1. Question (2)As is known, the substitution of Penrose tilings forces the border. So
the arrangement of unbouded conﬁgurations is unique in the case of Penrosetilings.
We ﬁnd some singular vertex conﬁgurations in Danzer tilings(There is no
singular vertex conﬁguration in the substitution of Penrose tilings). (i), (ii)of Figure 7 is singular vertex conﬁgurations. These do not exist in the list of39 vertex conﬁgurations which appear in supertiles (cf.[3]).

Figure 7: Vertex conﬁgurations (i), (ii)
When we subdivides vertex conﬁguration (i) (resp. (ii)) and rescaling,
we have patch (I) (resp. (II)) in Figure 8 (resp. 9) which includes (i) (resp.
(ii)). By repeating subdivision and rescaling, we get a tiling with vertex
conﬁguration (i) (resp. (ii)). Vertex conﬁguration (i) (resp. (ii)) appears only
in one place, and other vertex conﬁgurations are in 39 vertex conﬁgurations.
In addition, this tiling is invariant under the substitution.
Note that the above results remains true even if some of a’s and reﬂectionsof a in vertex conﬁgurations (i), (ii) are changed as you like.
We observe the tiling with only one vertex conﬁguration (i). Then we
can see that there are diﬀerent neighboring tiles in supertiles of prototile atype. Hence the substitution of Danzer tilings doesn’t force the border. The
arrangement of unbouded conﬁgurations is not unique in the case of Danzer
tilings,
So we have the following new question:(2)’ Nonexistence of singular vertex conﬁgurations is the necessary andsuﬃcient condition for that a substitution forces the border.
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Figure 9: The Patch (II)Figure 8: The Patch (I)References[1] N. G. de Bruijn, Updown generation of Penrose patterns, Indagationes
Mathematicae, New Series 1 (1990), 201–220.
[2] E. O. Harriss, Non-periodic rhomb substitution tilings that admit ordern rotational symmetry, Discrete Comput. Geom. 34 (2005), 523–536.
[3] H. Hayashi, Y. Kawachi, K. Komatsu, A. Konda, M. Kurozoe, F. Nakano,N. Odawara, R. Onda, A. Sugio and M. Yamauchi, Notes on vertex atlasof planar Danzer tiling, Preprint.
[4] K. Kato, K. Komatsu, F. Nakano, K. Nomakuchi and M. Yamauchi, Remarks on 2-dimensional quasiperiodic tilings with rotational symmetries,Hiroshima Math. J., 38 (2008), 385-395.
[5] K. -P. Nischke and L. Danzer, A Construction of Inﬂation Rules Basedon n-Fold Symmetry, Discrete Comput. Geom. 15 (1996), 221–236.

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