#### Local rules for pentagonal quasi-crystals

Discrete Comput Geom
Local Rules for Pentagonal Quasi-Crystals
Le Tu Q u o c Thang 0
0 Department of Mathematics, SUNY at Buffalo , Buffalo, NY 14214 , USA
The existence of different kinds of local rules is established for many sets of pentagonal quasi-crystal tilings. For each t c ~ there is a set ~ of pentagonal filings of the same local isomorphism class; the case t = 0 corresponds to the Penrose filings. It is proved that the set ~t admits a local rule which does not involve any colorings (or markings, decorations) if and only if t = m + nr. In other words, this set of tilings is totally characterized by patches of some finite radius, or r-maps. When t = ( m + nv/5)/q the set ~t admits a local rule which involves colorings. For the set of Penrose tilings the construction here leads exactly to the Penrose matching rules. Local rules for the case t = 71 are presented.
Introduction
The aim of this paper is to find local rules (or matching rules) which force tilings to
belong to a specific set of tilings. The tilings considered in this paper are the sets o f
pentagonal quasi-crystal tilings, also sometimes called quasi-periodic tilings with
fivefold symmetry, or generalized Penrose two-dimensional tilings. These tilings have
received a great deal of attention among mathematicians and physicists (see [dB1],
[KP], and [IS]). The m e t h o d used in this paper can be applied to other classes of
quasi-periodic tilings of two or higher dimensions. The infinitely many sets of
pentagonal quasi-crystals are parametrized by t ~ R; we denote them by gtt. T h e
exact definition of these tilings are given in Section 1. They have many nice
properties, and they are quasi-periodic by perhaps all the known definitions of
quasi-periodicity. F o r each t the set ~ consists o f tilings of the same local
isomorphism class: every finite part of a tiling in ~t can be found in any other tiling
in ffTt.All these tilings are obtained by the strip projection m e t h o d [dB1], [KP].
/
"/jyJ
;
"'/
(a)
)
/
(b)
The set of tilings J 0 was discovered by Penrose; it is perhaps the best known
among nonperiodic tilings. As an example we give here the local rule which forces
tilings to belong to ~0 (see [P] and [dB1]). Let us consider two rhombs whose acute
angles are ~-/5 and 27r/5, and whose edges are of the same length. The edges are
equipped with arrows as in Fig. l(a). A tiling of a two-dimensional plane by these
"arrowed" rhombs is said to satisfy the Penrose local rule if every edge of this tiling
has a definite arrow, that is, the arrows of an edge coming from two rhombs incident
to this edge are coincident.
Suppose T is a tiling of the plane by copies of these rhombs, without any arrows.
We say that T satisfies the Penrose local rule if there is a way to put arrows on
edges of rhombs of T such that the arrowed tiling satisfies the Penrose local rule. A
fundamental result of De Bruijn states that the set J0 is the set of all tilings
satisfying the Penrose local rule.
The main question of this paper is whether there are similar local rules for other
sets gtt of pentagonal quasi-crystals. This question is also studied in [KP] and [IS].
Note that a "local rule," in some sense, contains information in a local finite
radius. It is far from trivial to decide when a local rule forces tilings, say, to be
nonperiodic, or quasi-periodic, or to belong to a specific set consisting of tilings of
the same isomorphism class. The question whether a set of (usually quasi-periodic in
some sense) tilings admits a local rule also has importance for physics. It seems that
only such sets of tilings can serve as model for real quasi-crystals such as those
discovered in 1984 (see discussions in [K], [Lev], and [LPS2]). The similar question of
finding a set of prototiles such that every tiling by these prototiles must be aperiodic,
or, more difficult, quasi-periodic in some sense but not periodic, seems interesting
and has been investigated by many authors.
There are two types of local rules discussed in this paper; their definition are
given in Section 1. The main difference between them is the first one does not
involve any coloring (marking, decorations) while the second one does.
O u r chief results can be summarized as follows. First we prove that if t = (m +
n v ~ ) / q , then the set ffTtadmits a local rule of type 2. Furthermore, this local rule
consists of patches containing only two neighboring tiles. The case t = 0 corresponds
to the Penrose tilings, and the construction of local rules here leads exactly to the
Penrose local rule. Hence De Bruijn's result can be regarded as a special case of our
results. We prove that if t = m + n~" where ~- = (1 + ~ - ) / 2 , then this local rule can
be realized by a local rule of type 1; hence ~ admits a local rule of type 1. It follows
from a result of Ingersent and Steinhardt [IS] that t = m + n~" is also a necessary
condition for the existence of local rules of type 1. Hence we have a criterion for the
existence of local rules of type 1. As an illustration, we describe the local rule (of
type 2) for the case t = ~1-. Actually, we have to refine the method so that we could
get a simple local rule. This local rule is the Kleman-Pavlovitch local rule [KP]
enhanced with some condition on the vertex which is very similar to the A m m a n n
and Socolar local rules for eight-fold and twelve-fold tilings.
The existence of local rules of type 2 has been established for sets of quasi-crystals
based on quadratic irrationality of any dimensions [LPS1], [LPS2]. The eight-fold
two-dimensional and icosahedral three-dimensional tilings are special cases of these
results.
We would like to emphasize the fact that many sets of tilings that do not admit
local rules of the first type do admit local rules of the second type. An example is the
set of tilings with eight-fold symmetry (see [B], [dB2], [Lel], and [Sol]), or any set ~-t
with t = ( m + n v / 5 ) / q where t is not of the form m + n,c. For those sets o f tilings,
the problem of coloring a tiling to get a colored tiling satisfying the local rule of type
2 is not a local problem: one cannot decide how to color a tile by inspecting a
bounded region around this tile. As observed by Senechal [Se] there is a tiling in ffTt
which can be colored in two different ways. This is very different from the set of
penrose tilings. For this class, we can decide how to arrow the edges of a tile by
inspection around this tile within the radius 2; and every tiling can be arrowed in a
unique way.
Here is a very rough sketch of the proof of the existence of a type 2 local rule. By
the cut method, there is a periodic tiling ~'t of R 4 whose tiles are prisms with the
base parallel to a two-dimensional plane E; and every tiling in ff~t is obtained as a
slice of : t by a two-dimensional plane parallel to E. In other words, a tiling in ffTtis
the projection of the tiles of : t which meet a fixed 2-plane parallel to E. Suppose T
is a tiling such that every pair of neighboring tiles of T is a translate of a pair of
neighboring tiles of some tiling in ffTt(i.e., T satisfies some special local rule). Then
we can lift T onto @t: we can choose a collection of tiles of ~'t which project onto
tiles of T. The lift has an important property: for every pair of neighboring members
in this collection, there is a plane parallel to E which meets the interior of both of
them. In some cases this is enough to prove that:
(-k) There is a plane parallel to E meeting all the members of this collection.
This means T is defined by a slice of ~t, i.e., T belongs to ~t.
Technically the proof of (~-) is rather complicated. First we have to prove that our
local rule is a weak local rule in the sense of Levitov [Lev]. Actually this was
established in a general setting in [LPS2]; we give a proof for our case in the
Appendix. Then we have to study the boundary of the tiles in @t and in general we
have to replace @t by its "refinement" so that the boundary matches some
conditions. This step corresponds to the coloring of the local rules. The method used here
can be easily generalized for other sets of quasi-periodic tilings obtained by the
projection method. There is no specific property of the case investigated here that is
used for the proof, except that in this case, due to some symmetry, we can reduce
the number of verifications. The general result of the existence of type 2 local rules
(local rules involving coloring) is formulated in Section 7, Theorem 7.1.
To establish the existence of a type 1 local rule we do need some specific property
of the case under investigation. Actually we prove that if t = m + n~-, there is some
finite radius such that the coloring of a tile is uniquely defined by the configuration
of the tiling inside the disk of this radius around the tile. It also follows from the
proof that the radius is linearly dependent on n.
The paper is organized as follows. In Section 1 we introduce definitions and
preliminary facts. In Section 2 some facts about the cut method are recalled. We
follow the paper [ODK]. In Section 3 we give the proof of the existence of a local
rule of type 2 for the case t = 0; the proof is readily generalized in Section 4 for the
case t = (m + nv/5)/q. We also prove that the local rule obtained for t = 0 is
equivalent to the Penrose local rule. Section 5 is devoted to the case t = rn + n~';
the existence of a local rule of type 1 is proved. In Section 6 we prove a technical
result used in Sections 3 and 4. Section 7 contains the example t = 89 some
generalizations, and concluding remarks. In the Appendix we prove a generalization
of a result of Levitov concerning the weak local rule.
1. Definitions and Preliminary Facts
For technical convenience some of our definitions (tilings, prototiles, etc.) are more
special than is generally the case. In particular, we use translational congruence
instead of the usual congruence which involves rotations and reflections (our results
can be easily reformulated in terms of the usual congruence).
1.1. On Tilings and Local Rules
Two subsets of R k are called congruent if the second is a translate of the first. We
always distinguish between two congruent polyhedra.
A tiling of R k is a family of k-dimensional polyhedra, called the tiles of this tiling,
which covers R k without overlaps (that is, the interiors of two different tiles have
empty intersection). In this paper, except in the case of the oblique periodic tilings
which appear later, all tilings are assumed to be face-to-face type, i.e., the
intersection of every two tiles is a c o m m o n facet of lower dimension, if not empty. A vertex,
edge, facet, etc., of a tiling is, respectively, any vertex, edge, facet, etc., of one of its
tiles. For a given tiling the translation classes of tiles are called the prototiles of this
tiling. All the tilings encountered in this paper are tilings of some Euclidean space
with a fixed origin 0. We use the following definition of convergence of tilings
(compare [Ral] and [Ro]).
Definition. A sequence of tilings T1, T2. . . . . of Ek converges to a tiling T if, for
every r > 0, there is a natural number N such that, for i > N, the tiling T/ coincides
with T inside the disk Ur with center at 0 and radius r.
We define the closure ~ of a set X/ of tilings as the set of all the limits of
sequences of tilings belonging to X/. A set of tilings is closed if it is coincident with
its closure.
A n r-map o f a tiling T at a vertex v is the collection of all the tiles lying inside the
ball of radius r centered at v. More generally, an r-map is any r-map of any tiling at
any vertex. Two r-maps are congruent if the second is a translate of the first.
Definition. A local rule of type 1 of radius r is a finite set of r-maps. A tiling T
satisfies a local rule ~r of type 1 of radius r if every r-map of T is congruent to one
from ~r
By "a local rule of type 1" we mean a local rule of type 1 of some radius.
1.2. Coloring
A colored polyhedron is a pair (P, j ) w h e r e P is a polyhedron and j is an arbitrary
element, called the color of this polyhedron. Two colored polyhedra are congruent if
their colors are the s a m e and the second is a translate of the first. A colored tiling
(resp. a colored r-map) is a tiling (resp. r-map) whose tiles are colored polyhedra.
Two colored r-maps are congruent if the second is a translate of the first and colors
of the corresponding tiles are coincident. As in the previous section, we can define
colored r-maps of colored tilings, limits of sequences of colored tilings, and closures
of sets of colored tilings.
The following definition of a local rule of type 2 is introduced only for the
two-dimensional case.
A n edge-configuration is a collection of two colored polygons having a full
common edge. Two edge-configurations are congruent if the second is a translate of
the first and the corresponding colors are the same. For a colored tiling of ~2 the
edge-configuration of an edge is the pair of colored tiles incident to this edge.
Definition. A local rule of type 2 is a finite set of edge-configurations. A colored
tiling satisfies a local rule of type 2 ~ ' if the edge-configuration of every its edge is
Remarks.
congruent to one from ~ ' . A noncolored tiling satisfies this local rule if it can be
colored to become a colored tiling satisfying this local rule.
Definition. A set of noncolored tilings admits a local rule of any type if it is the set
of all tilings satisfying this local rule.
It is easy to see that if a set of tilings admits a local rule of any type, then this set
is closed.
1. The coloring makes the number of prototiles become larger. For example the
two Penrose tiles in Fig. l(b) are not congruent.
2. The Penrose local rule is a local rule of type 2. All the proofs for the absence
of local rules (maybe under some restrictions) in [B], [L], [Lel], and [IS] are
only for local rules of type 1.
3. In [Lel], [Le2], and [LPS2] local rules of type 1 are called simply "local rule";
local rules of type 2 are special cases of "local rules with coloring" there.
Danzer [D2] calls local rules of type 2 "strictly local matching rules." Our
definition of a local rule of type 1 corresponds to the r-rule of Levitov [Lev].
1.3. The Superspace R 5
In the Euclidean space R 5 with origin 0 we fix a standard base 60, 81, o~2,E3, 84. Let
Z 5 be the lattice of integer points, and let Qv~- be the set of all numbers of the form
a + bye-, where a and b are rational numbers. If x is a point in ( Q v ~ ) 5, then
x = y + zv~- where y and z are rational points in R 5, i.e., points with rational
coordinates. The point y - z~/5 is called the conjugate of x. If X and Y are subsets
o f R 5 put X + Y = { x + y l x ~ X , y ~ Y } , a n d - X = { - x l x ~ X } .
A subspace (as a vector space) of R 5 is called a homogeneous plane and its
translates are called planes; an n-plane is a plane of dimension n. A plane F is
called rational if F is homogeneous and spanned by vectors with rational
coordinates.
Consider the action of cyclic group Z 5 = ( g i g s = 1) on R 5 by cyclic permutation
of the base: g ( 8 i) = (8i+ 1) (mod5). The space R 5 decomposes into three invariant
subspaces E, F,, and A. Here A is the one-dimensional subspace spanned by
6 = (80 + 81 + 82 + e 3 + 84)/5, E is the 2-plane spanned by two vectors with
coordinates ( 4 , ~ - 1 , - V3- - 1 , - ~ / 5 - 1, x/5 - 1) and (v~ - 1,4,~/5
1, - v ~ - 1, - v~- - 1), F.. is the 2-plane spanned by the conjugates of these two
vectors. The element g acts on E as rotation by 27r/5, on E as rotation by 47r/5, and
on A as the identity.
Putting E J-= E + A, s = E 9 E 9 A = E 9 E Let p , p , p be,
respectively, the projectors of R 5 on E , E , E x , A. We define ei = ~ 8 i ) , ei = p (o~ , and
ei = g'J(si). The projection o f every o f 8i on A is 5. Hence Pa(~) is a multiple of 6
for every ~ ~ Z 5.
F o r a real number t let Et = ~ + t6. D e n o t e the 4-plane E + E't by Rt4. T h e n
ff~5 = (-Jt~R R4. For example, R~ is the rational 4-plane E 9 E; it is the set of all
points whose coordinates sum to 0. l e t A = R g C3 7/5 be the set o f all integer points
lying in ~4; it is the lattice generated by e 0 - el, 8l - ~2, ea - 63, e3 - 64.
The following is easy to check.
Proposition 1.1.
of S~.
p ( ~ ) = p(~), where ~, 71 ~ 7/5, i f a n d only i f ~ - ~l is a multiple
Definition. Let x = p ( ~ ) where ~ = E 4i=oni~i, ni ~ 7/, is an integer point. The
index(x) is the remainder o f the sum Ea=oni modulo 5 (see also [dB1] and [KP]).
It follows from Proposition 1.1 that the index is well defined; it is defined only for
points in p(7/5). For x, y ~ p(7/5) we have index(x + y ) =- index(x) + index(y)
(rood 5).
The projection p(7/5) on E is a dense 7/-module generated by five vectors
e l , . . . , e5 which point to the vertices of a regular pentagon (see Fig. 2). The set
p (7/5) is not dense in E l but is contained and dense in the union of parallel and
equidistant 2-planes E + k 6, where k ~ 7/.
The following is also easy to check.
Proposition 1.2.
(a) I f pa( ~ ) = m 6, then m ~ 7/ and m - index(p( s~)) (rood 5).
(b) I f Pa(~) = O, where ~ ~ 7/5, then ~ belongs to A (recall that A = R~ N 7/5).
(c) I f two rational p o i n t s ~, ~' have the s a m e projection on E , then ~ = ~'.
(d) I r a rational k-plane F contains E, then it contains R 4 = E 9 E.
1.4.
The Strip Projection M e t h o d
If Vl, u 2. . . . . uk are vectors of ~5 let Pol(v 1. . . . . vk) be the polyhedron:
Vk)=l"i~~k~iiv=i~'i1E[O~l]l
e
O2
P~
T h e set y = Pol(e0, Opl,E2, ~3, ~ is the unit hypercube o f the lattice 7/5. T h e set
Pol(6i, e:) (for 0 _ i < j _< 4) is called a 2-facet of 3' and its translates ~: + Pol(ei, s:),
where sr ~ 7/5, are called the 2-facets of the lattice 7/5.
T h e set o f all pairs (i, j ) with 0 < i < j _< 4 is denoted by M. For convenience a
r h o m b in this p a p e r always refers to a translate of one of ten P1 = Pol(ei, ej) with
I = (i, j ) ~ M. U p to rotation there are only two r h o m b s f r o m the ten Pi, they are
shown in Fig. l(a).
F o r every a ~ E let S~ be the strip defined by shifting the unit hypercube y
along t h e 2 - p l a n e E + a : S ~ = E + a + 7, a E E
Definition. a ~ E is regular if the b o u n d a r y of the strip S~ does not contain any
integer point. Otherwise a is singular.
T h e following facts are f u n d a m e n t a l in the strip projection method. F o r regular a
the union o f all the 2-facets of the lattice 7/5 lying in the strip S~ is a
two-dimensional continuous surface. This surface contains all the vertices of the lattice 7/5
falling inside S~ and has an obvious polyhedral structure. By projecting this
polyhedral structure along E + o n t o E we get a tiling T~ of E. N o t e that there are no
overlaps: the restriction of p to this surface is one-to-one. T h e prototiles are the ten
r h o m b s P1, I ~ M. This m e t h o d of obtaining the tilings T~ is the strip projection
method, applied to our setting (see [dB1], [GR], and [ODK]).
D e n o t e b y S r ( r e s p . 5~t) the set of all tilings T~ for regular a ~ E (resp. a ~ E't)
and their translates.
Definition. A tiling o f the closure J o f J is called a pentagonal quasi-crystal.
O f course if a ~ E't is regular, then any translate of T~ is in 4 , but other tilings
in ~ exist as well. W e describe such tilings in Section 2.5.
Two i m p o r t a n t properties o f pentagonal quasi-crystals are [dB1], [KP], [ODK]:
Proposition 1.3.
(a) Local isomorphism: for every two tilings in ~tt, for every r > O, every r-map of
the first tiling is congruent to an r-map o f the second.
(b) Symmetry: if T is a tiling of ~tt, then the rotation o f T by 27r/5 also belongs
tor
R e m a r k . S o m e authors consider only regular cases, that is, the sets JTt and 3 - but
not their closures. However, as argued above, the question of finding a local rule for
~a~t,5r is not relevant, because these sets are not closed. In each set of ~tt,3-there are
two different tilings coincident in a half-plane.
2. The Cut Method
2.1. The Cut Method in the Superspace
A set X in ~5 is called a prism if X = p ( X ) + p (X). If X and Y are prisms, then
their intersection is also a prism, and we have the nice formula
X
Y = [ p ( X ) p ( Y ) ] + [p ( X ) A p (Y)].
(
1
)
This makes it easy to study the intersection of two prisms.
Definition: Suppose X is a prism such that p z ( X ) is a polyhedron. We define the
parallel boundary of X as
p ( X ) + O(pJ-(X)),
where 0(p 9 ( X ) ) is the boundary of p (X).
The parallel boundary of a prism is a part of its boundary.
For I = (i0, i l) ~ M recall that P1 is the rhomb Pol(ei0, eil). Let
PI = - Pol(e/~ , e:~jl , eye),
where (i0, il, J0, Jl, J2) is a permutation of (
0, 1, 2, 3, 4
). Put C1 = Pt + P~ ; it is a
prism.
Consider the family @ consisting of all prisms of the form C 1 + ~ for I ~ M and
~ 7/5. It is proved in [ODK] that this family covers ~s without any overlaps, i.e., it
is a tiling of Es. The tiling ~' is not of face-to-face type and is invariant under
translations by vectors of 7/5. The parallel boundary B of G, by definition, is the
union of the parallel boundaries of all the tiles of C. It is a cellular complex of
dimension 4. If a ~ E is such that E + a does not meet B, then all the
intersections of E + a with tiels of @ form a cover of E + a without overlaps and
define a tiling of E + a. Projecting along E we get a tiling of E, called the tiling
defined by ~ and a. If E + a meets B, then all the intersections of E + a with G
covers E + a with overlaps. The fundamental result of [ODK] can be stated as
follows.
Theorem 2.1. The 2-plane E + a does not meet B if and only if a is regular and in
this case the tiling defined by and a is coincident with the tiling T~ obtained by the
strip projection method.
It follows that the set I r of all singular points is p (B), Ir = p i (B). We have the
following description of Ir. For I = (i, j ) ~ M let H I be the 2-plane spanned by eg
and e ~ . Then H t + p (7/5) is a dense family of parallel 2-planes in E .
Proposition 2.2 (see [ODK]). The set I r o f all singular points is the union of ten
families o f parallel planes H I + p (zs), I ~ M:
I r = p (B) = I,.J ( / / i + p (77s)).
I ~ M
Of course I r is a dense set in E x but it has measure 0.
2.2. Index o f Rhombs
W e have ~tt = Jtt§ 1, hence we study ~tt for - 1 < t < 0.
Recall that a rhomb is always referred to as a translate of one of ten Pi, I ~ M.
Suppose a r h o m b P has one vertex in p(2-~5). Then all the four vertices are in p(7/5),
hence they each have an index (see Section 1.3). Let v and v' b e two vertices such
that the segment [v,v'] is an edge of P. Then the vector vv' is either one o f
e0, el, e2, e3, e 4 or one of - e o, - e l , - e 2 , - e 3 , - e 4. In the first case i n d e x ( v ' ) =
index(v) + 1, in the second index(v') = index(v) - 1. Hence the four vertices can
take only t h r e e values of the index i, i + 1, i + 2 (mod 5) for some i ~ {0, 1, 2, 3, 4}.
Definition. This number i is called the index of this rhomb.
In case P = p ( ~ + Cl) , ~ E 775, I ~ M , it is easy to see that i n d e x ( P ) =
index(p(~)). The index is important due to the following proposition which is a
consequence of Proposition 1.2(b).
Proposition 2.3. Suppose P, P ' are congruent rhombs, P = P ' + v. Then v belongs to
p ( A ) if and only if index(P) = index( P ' ).
Proposition 2.4. For regular ct ~ E 1 the tiles o f T~ have indices 1 or 2 if t = 0 and
indices O, 1, 2 if - 1 < t < O. (The case t = 0 has been proved in [dB1].)
Proof. Suppose P is a tile of Ta. Then P = p ( ~ + C1) for some prism ~ + C z
meeting E + a , by the cut method. Since E + a does not meet the parallel
boundary of the prism ~: + C t, it meets the interior of this prism. Projecting on A,
we have pa(E + a ) = t~ and pA(C1) = [0, --38]. Let P a ( ~ ) = m ~ , m ~ 7/. Then
m8 + [0, - 3 / ~ ] contains t8 as an interior point. It follows that, when t = 0, m must
be 1 or 2, and when - 1 < t < 0, then m must be 0, 1, or 2. By Proposition 1.2(a) we
have m = index(p(~)) = index(P). [ ]
2.3. The Cut Method in Four-Dimensional Planes
W e have seen that every tiling T~ is the slice of a periodic structure in ~5 by a
2-plane. F o r a fixed t every tiling Ta with a ~ Et can be obtained by a slice of a
periodic structure in the 4-plane R 4 as follows.
The intersection of R 4 with a tile ~ + C I of ~' is
[P(~) + P1] § let n (p ( ~ ) + p/i)],
by formula (
1
). The first term is a translate of the rhomb P~, the second term is
either a point or a polygon, if it is not empty (see Fig. 3). Hence the intersection of
E7 with a tile of e~ is either of dimension 2 or 4. Let evt be the family of all the
four-dimensional intersections of ~4 with tiles of ~. The family : t is a tiling of Rt4
and has two fundamental properties:
(i) It is invariant under translations from A (recall that A = ~4 ~ 7/5 is a lattice
in R4).
(ii) All its members are prisms; there are a finite number of them up to
congruence.
Definition. Any family of four-dimensional polyhedra in ~4 covering R 4 without
overlaps and satisfying conditions (i) and (ii) above is called an oblique periodic tiling
of R 4
It follows immediately from the construction that, for a regular a ~ Et, the tiling
T~ is obtained by intersecting E + a with members of the family 6~t and then
projecting onto E.
Definition. Suppose C is a prism in Rt4 such that p (C) is a polygon. The sum of
p(C) and an edge of polygon p (C) is called a small wall of C. The union of all the
small walls of C is its parallel boundary.
Suppose ~' is an oblique periodic tiling of R 4 whose tiles are colored polyhedra.
Let B ( ~ ) be the union of all the small walls of all the tiles of ~. If ct ~ E t is such
that E + a does not meet B(~'), then by intersecting E + a with the colored tiles of
g/ and then projecting onto E we get a colored tiling, called the colored tiling
defined by ~' and a. Denote b y O ' ( ~ ) the set of all such colored tilings and their
translates, and byY(~/) the closure of ~q(~). When ~' is a noncolored tiling we use a
similar definition, the set ~ ( ~ ' ) is then a set of noncolored tilings.
Of course ff'(@t) = fftt- The projection on Et of the parallel boundary ~ ' ( G t) is the
set of singular points lying in E r Hence we get
Proposition 2.5. The projection o f the parallel boundary B(~'t) onto Et is I r Et:
p (B(Gt)) = Ir f~ F't.
2.4. Lifting a Tiling
Suppose that ~" is an oblique periodic tiling of ~4 whose tiles are colored polyhedra,
and that T is a colored tiling of E.
Definition. A lift of a tile P of T into ~ is a tile C of ~/such that p(C) = P and the
colors of P and C are the same. A lift of T into ~/is a map l: {tiles of T} ~ {prisms
of ~/} such that for every tile P of T the tile l ( P ) is a lift of P into ~'.
Of course the lift does not always exist, and when one does exist it may not be
unique.
Definition. A lift l of a tiling T into ~' is strongly connected if, for every pair of tiles
P1, P2 sharing a c o m m o n edge, the polygons p (I(P1)), p (l(P2)) have a common
interior point.
A simple but important example is the case when T is a colored tiling defined by
a and ~', where E + a does not meet the parallel boundary B(~'). Then T has a lift
into ~ : I(P), where P is a tile of T, is the tile of ~ which meets E + a and projects
into P. This lift is strongly connected because all the polygons p (l(P)) contain
as an interior point.
Let P be a rhomb having vertices in p(7/5). There may be no lift of P into ~ but
there are always lifts o f P into ~'; they are of the form C + k56, k ~ 2~, where C is
congruent to one of ten prisms Cr The projection pa(Cl) is the segment [0, - 3 3 ] .
Hence there is at most one prism from the set C + k56 which meets ~7. Thus we
have:
Proposition 2.6. Suppose P is a rhomb having vertices in p(7/5). There is at most one
tile C o f t~t such that p(C) = P.
This means the lift into ~'t is always unique.
2.5. Singular Cases
For completeness we describe the "singular tilings," i.e., all the tilings in f f ~ ,
~ \ J t t . The results here are not used in what follows.
Suppose a ~ Ir. Recall that Ir is a family of 2-planes in E . There are several
2-planes from I r going through a ; they partition E into many pairs, called
"corners."
Proposition 2.7. Suppose o~i, i = 1, 2. . . . are regular, lie in one comer o f E
separated by 2-planes from I r going through a, and converge to c~ when i ~ ~. Then
the sequence o f tilings T~i converges to a tiling, called the tiling defined by ct and this
comer. This tiling depends only on the comer containing oti, but not on the choice o f
points a i.
Proof. We have to prove that for r > 0 a number N exists such that if i > N, then
all the tilings T,, coincide inside the disk Ur with center at 0 and radius r. Consider
the union Z of parallel boundaries of all prisms (from the family @) which projects
onto E into rhombs intersecting Ur. l e t Y = p i (Z). Of course Y c Ir, but it is not a
dense set in E l : for large r it divides a small neighborhood of a just as the
2-planes from Ir going through c~ do. Hence for large i all the points a i lie in
one part of E divided by Y. Obviously for these a i the tilings To, are the same
inside Ur. []
It can be proved that no two tilings defined by an irregular ~ and two different
corners are the same, and that the closure 2 of 3 - is the set of:
(a) All T~ with regular a, and their translates.
(b) All tilings defined by singular a and a corner of E divided by 2-planes from
I r going through a, and their translates.
The singular cases in Jtt are similar. Suppose a ~ Et is singular. The set I r n Et
is a family of lines in Et. There are several lines in I r n Et going through c~; they
partition E t into many parts. If a; is a sequence of regular points lying in one part
and converging to a, then the corresponding tilings converging to a tiling, called the
tiling defined by a and this part. Of course this tiling belongs to ~ and it can be
proved that ~tt, up to translations, is just the set of all such filings plus "regular"
ones. Using this picture the topology of J,,Jtt can be easily described.
3. The Penrose Tilings
Theorem 3.1. The set 5-0 admits a local rule o f type 2.
This theorem was first proved by de Bruijn [dB1]. The proof here is different from
that of de Bruijn and will be soon generalized for other cases.
We present a concrete local rule which is actually equivalent to the Penrose local
rule. We make use of the Main Technical Theorem that is proven in Section 6.
3.1. Coloring and the Local Rule for Joo
We describe in more detail the tiles (or members) of the family @0. Each tile is an
intersection (C 1 + ~) n R 4.
Definition. For a tile C of ~o let index(C) = index(p(C)).
So far we have defined the index for points in p(7/5), for rhombs with vertices in
p(ZS), and for tiles of ~0.
Proposition 3.2. Up to translations by vectors from A the family Go has 20 different
members.
Proof. Fix I ~ M . T h e intersections C ] = ( C 1+ G0 ) ~ l ~ g and C2 = ( C t + 2 6 0 )
C3 ~4 are not congruent: their projections on F, are two triangles which are not
congruent. W e prove that every m e m b e r of 6% is a translate of one of twenty C],
j = 1, 2, I ~ M, translated by a vector in A.
A m e m b e r of 6% is the intersection (r + C1) n R 4, where R 4 meets the interior
of ~: + CI. Then PA(r must be j6 where j = 1 or 2 (see Proposition 2.4). By
Proposition 1.2, 7 / = s r - J ~ 0 belongs to A. The translation by ~7 transforms C] into
the intersection (~ + Ct) n ~4. []
The index of a tile CJ + ~ of G0 is j. In this section we consider the index of a
tile of ~0 to be its color. Now G 0 is an oblique periodic tiling of ~4 whose tiles are
colored prisms. We use the notation G~ for this colored oblique periodic tiling. If
c~ ~ E is regular, let T~ be the colored tiling defined by c~ and 0'8; it has 20
prototiles: (Px, 1), (PI, 2), I ~ M. The color of a tile P of T f is its index. The set
f f ' ( ~ ) (see the definition in Section 2.3) is the closure of the set of all Tf. If we
ignore the color, then f f ( ~ ) = ~0.
We define the following local rule of type 2. Let ~'0 denote the set of all
edge-configurations of tilings from f f ( ~ ) . U p to congruence there are only a finite
n u m b e r of elements in ~ 0 . Then a colored tiling T satisfies the local rule ~q~0if and
only if every edge-configuration of T is congruent to an edge-configuration of a
colored tiling T~, a ~ E. A concrete description of this local rule is given later in
Section 3.4.
Theorem 3.3. If T is a colored tiling satisfying the local rule ~ o , then T belongs to
T h e o r e m 3.1 is a consequence of this theorem; we see later that this theorem is
equivalent to de Bruijn's theorem. We prove this theorem after some preparation.
Let P be an arbitrary rhomb congruent to one of 191,I ~ M, and having vertices
in ~7/5). T h e index(P) is defined (see Section 2.2) and equals index(u) of some
vertex v o f P. Then index(P + x) = index(v + x ) for every x ~ p(775). Hence
x ~ ~7/s) always exists such that index(P + x) is a given n u m b e r in {0, 1, 2, 3, 4}.
Consider a colored tiling T satisfying the local rule ~ 0 . Every tile o f T has color
1 or 2. I f one vertex of T is in p(7/5), then all the vertices of T are in p(7]5). After a
translation we m a y assume that there is a tile P o f T such that its color is the s a m e
as its index. F r o m now on fix such T.
Proposition 3.4.
The tiling T has a strongly connected lift into ~ .
Proof. First we prove that for every tile Q of T the index of Q and the color of Q
are the same. This is true for Q = P.
Suppose Q1, Q2 are two tiles of T sharing a c o m m o n edge, and index(Q 1) = color
of Q v By definition of the local rule, there is a pair of tiles Q'I, Q'2 of a colored tiling
T~, a ~ E, such that the colored pair (Q1, Q2) is congruent to the colored pair
(Q~, Q~), where the colors of Q~, Q~ are their indices. Since index(Q 1) = color of
Q1 = index(Q~), by Proposition 2.3 the pair Q1, Q2 is the translation o f Q'I, Q'2, by a
vector v = p ( ~ ) , ~ ~ A. T h e n again by Proposition 2.3 i n d e x ( Q 2 ) = index(Q~).
H e n c e index(Q2) = color of Q2.
Each of Q], Q~ has lift into ~ . Let the lift of Q~ be C[, i = 1, 2. T h e n the
colored tile C~ + ~ of ~ is the lift of Qi. T h e lift of a colored tile is unique
(Proposition 2.6). T h e projections of C~ and C[ on E have n o n e m p t y interior
intersection, because both contain a as the interior point. The lifts of Q1, Q2 are
translations of the lifts of Q'I, Q[, hence their projections on E also have n o n e m p t y
intersection. So we can lift every colored tile of T into ~f~ and the lift is strongly
connected. [ ]
The next step is to prove that if a colored tiling T has a connected lift into ~ ,
then T belongs to ff(~f~).
N o t e that the family ~'0 of prisms has the following symmetries:
(a) Translations by vectors from A.
(b) Actions of cyclic group 7/5.
(c) Symmetry with respect to 0 (this maps Ca to C~ + 3 e 0 - ez - e3 - g4 ).
U p to these symmetries in G0 there are only two tiles: their representatives are
C~o,1), C~o.z). This will cut significantly the n u m b e r of verifications which a p p e a r later
in the proofs.
3.2
Properties o f ~f~
Consider five 2-planes, Fi spanned by e i + 1 -- ~176 1 and e i + 2 -- ~176 2 for i = 0, 1, 2, 3, 4
(indices are taken m o d 5). T h e y are rational 2-planes contained in 4. A n important
r e m a r k is that the intersection of these 2-planes with E and F, are lines, i.e., they are
not in generic position with respect to E and F,. L e t f / = F i r E, f / = F / n E. T h e n
both fi and fi are lines and F,. = f / + fi. All o f the 2-planes F i are prisms and
P(Fi) = fi, p (Fi) = fi.
Proposition 3.5. I f two 2-planes F, F ' in S r have the same projection onto E, then
F = F ' .
Proof. We may assume that F = F / and F ' = F / + ~ where ~ A , for some i.
Then p i) = p i + ~), or, equivalently, ~ ( Fi + E). Note that Fi + E is a
homogeneous 3-plane. If ~ does not belong to Fi, then the 3-plane spanned by ~ and
Fi must coincide with F i + E. However, the 3-plane spanned by ~ and Fi is rational,
so it could not contain E (Proposition 1.2). Hence ~ ~ F/. This means F = F ' . []
For each i = 0, 1, 2, 3, 4 the set qbi = p (~//) = fi + P (A) is a dense family of
parallel lines in E. Denote by 9 the union of five sets ePi. Then dp = p (~-).
Proposition 3.6. The projection o f the parallel boundary B(~e~) onto E is dp:
Proof. By Propositions 2.2 and 2.5, p (B(~'~)) = p (B(~0)) is the intersection of
with ten families of planes H I + p (~_s), I ~ M. These ten families can be
grouped into five pairs, each pair has the same intersection with E, and the
intersections are exactly the five families (I)i of parallel lines, because f / = H(i +1,i-1) n
= H(i+2, i_2) f3 E (indices are taken mod5). []
We suppose the length of vector e0 is 1. Let Ud be the disk in E of radius d
centered at 0. The set Fi + Ud is a three-dimensional prism, its projection onto E is
the line f / ( j u s t as the projection of F/), and its projection onto E is fi + Ud which is
a strip of width 2d.
Definition. In this section any set of the type Fi + Udo + r where i = 0, 1, 2, 3, 4,
A, d o = (3 - v ~ ) / 4 , is called a big wall.
The union ~i~a0 of all the big walls is 3 - + Udo. T h e next proposition is a specific
property of the case t = 0.
Proposition 3.7. A l l the big walls are contained in the paraUel boundary: 7 fao c B(~'~).
Proof. We have to prove that the intersection of every tile of e~0 with every big wall
is contained in B(~'~). The intersection is of two types: of dimension 3 or of
dimension less than 3, because the dimension of a wall is 3.
First we prove that all the intersections of the first type are in B ( ~ ) . Since B(ee~)
and ~d0 are invariant under the symmetries listed in (a)-(c) in Section 3.1, it suffices
to prove that the intersection of dimension 3 of the prism C} with a big wall lies on
the parallel boundary of the prism C} where I = (
0, 1
) or (
0, 2
). In fact it is easy to
check that, for each I = (
0, 1
) or (
0, 2
), the prism C] intersects at most three big
walls if the intersection is of dimension 3, and in this case the intersection lies on the
parallel boundary of C]. Hence the intersection of the first type of every big wall
with every tile of C O is in B(C~).
Now let W be a big wall. Since W and B(C~) are closed subsets of N 4, the
complement W \ B(C~), as a subset of W, is open. By the above result, this
complement is the union of at most a countable number of sets of dimension less
than 3 and hence must be empty. That is, W c B(C~). []
We recall that a small wall of a prism C is the sum of p(C) and one edge of
p 9 (C). The projection of a small wall of a tile of ~f~ on F, is a segment lying in
dO = p 9 (,gr), hence it lies in the projection of a big wall. Proposition 3.5 says that
this big wall is unique.
Proposition 3.8. If the projection of a small wall w on E lies in the projection of a big
wall W, then the small wall has nonempty intersection with the big wall.
Proof. It suffices to consider the cases when w is a small wall of one of the C~,
where I = (
0, 1
) or (
0, 2
). In each case C has three small walls and the assertion can
be verified straightforwardly by finding the unique big wall involved and checking
that the projections of the small walls and the big wall onto E have nonempty
intersection. []
3.3. Proof of Theorem 3.3 Using the Main Technical Theorem
The following theorem is proved in Section 6.
Theorem 3.9 (Main Technical Theorem). Let ~ be an oblique periodic colored tiling
of ~4 satisfying the following conditions for some constant d > 0:
(a) The projection of the parallel boundary onto E is dO: p 9 (B(~)) = dO.
(b) 7 fa = 5 r + Ud is a subset o f the paralM boundary.
(c) I f w is a small wall of a tile of ~d and the segment p (w) lies on the line p 9 (F),
where F is a 2-plane in ~,, then w and F + Ud have nonempty intersection.
Then if T is a colored tiling having a strongly connected lift into ?/, T belongs
to ~(~').
By Propositions 3.6-3.8, the oblique periodic colored tiling C~ satisfies conditions
(a)-(c) of the Main Technical Theorem. By this theorem if T has a connected lift,
then T belongs to ff(C~). Theorem 3.3 is proved.
3
/ /
2
2
3
4
2
3
3
4
3
1
2
2
3
3.4. Connection to Penrose Local Rule
First we describe the local rule ~'0. The prototiles of this local rule are (1"l, J),
I ~ M, j = 1, 2. We have to point out all the possible edge-configurations in J(@~).
For every P / = p(CJ) we mark the vertices of polygon P] by their indices. Then we
have 20 rhombs with marked vertices. Four of them are as in Fig. 4 (in fact two of
them are congruent to P(0.1), two are congruent to P(0.2)). All the other rhombs are
obtained from these four by rotations by 2 m l r / 5 , m ~ Z.
There is one-to-one corresponding between the 20 prototiles (Px, J), I ~ M,
j = 1, 2, and these 20 marked rhombs. We describe the local rule ~r in terms of the
20 marked rhombs.
Proposition 3.10. The local rule ~ o is the set of edge-configurations:
(a) Shown in Fig. 5 together with their reflections with respect to a horizontal
line.
(b) Obtained by rotating any edge-configurations in (a) by 2mTr/5, rn ~ E.
This can be verified by analyzing the edge-configurations of T~.
The local rule ~ 0 is equivalent to the Penrose local rule in the sense that every
colored tiling satisfying this local rule can be equipped with arrows such that the
arrowed tiling satisfies the Penrose local rule, and, conversely, every tiling satisfying
the Penrose local rule can be colored into a colored tiling satisfying the local rule
~'0- This can be seen as follows.
For every marked rhomb we put a double arrow on every edge with vertices 2 and
1 such that the arrow directs to 1. We also put a double arrow on every edge with
vertices 3 and 4 such that the arrow directs to 4. Then every rhomb of the 20 rhombs
has double arrows on two or four its edges. On the other two edges we put single
arrows such that they direct to obtuse angles of the rhomb. By this way we get the 20
arrowed rhombs of Penrose. Then it can be checked easily that an
edge-configuration belongs to the local rule ~r if and only if arrows on the common edge coming
from different rhombs are the same. Conversely, every tiling satisfying the Penrose
local rule can be marked (indexed) at vertices such that (when forgetting about
arrows) this tiling satisfies the local rule oqr0 (see [dB1]). Hence the local rule
constructed here and the Penrose local rule are equivalent.
3
Fig. 5. The local rule forff(~).
4. A Local Rule of Type 2 for t = (m + n C ~ ) / q
Theorem 4.1. I f t = ( m + n z ) / q , where m, n, q are integers and T = (1 + v~-)/2,
then the set ~tt admits a local rule o f type 2.
The proof is in fact simpler than the previous case. The reason is that in the
previous case we wanted to establish not only the existence of a local rule but also
the coincidence with the Penrose local rule. Here we may simplify the proof but the
local rules we obtain are much more complicated. In the previous sections we had to
prove that all the big walls are contained in the parallel boundary and if a small wall
has projection onto E lying in the projection of a big wall, then the small wall and
the big wall have nonempty intersection. Here these properties are fulfilled
automatically from the construction of the refined oblique periodic tiling. A n example of the
local rule for the case t = 21 is presented in Section 7.
4.1. The Big Walls
CJ, t = (C l + j e o) n ~4.
(
2
)
For - 1 < t < 0 none of the three sets Cj, t, j = 0, 1, 2, is empty, and no two of them
are congruent. Using the same proof of Proposition 3.2 it can be proved that every
tile of @t is the translation of one of 30 prisms CJ, t by a vector in A:
~'t = {CJl,t + ~, I ~ m , j
= 0 , 1 , 2 , ~:~ A}.
Definition. For a tile D of @t let index(D) = index(p(D)).
Then the index of a tile CJ, t + ~ of @t, where ~ ~ A, is j.
By Propositions 2.2 and 2.5 the projection of the parallel boundary B(@t) o n Et is
the union of ten families of lines [ H1 n Et] + A. When t = (m + n r ) / q all the ten
families of lines are subsets of a system of lines which is as "good" as q~ (this is the
place where we use t = (m + nT)/q)):
Proposition 4.2. I f t = (m + n'r)/q, then the projection of B(~'t) onto Et, after
translation by - t e d - , is contained in ~ / q , where r / q = {xlqx ~ qb}:
p (B(Gt)) c (ted- + d~/q).
Proof. Using Propositions 2.2 and 2.5, it suffices to verify that each line H I n Et is
lying in the set te d- + ~ / q . This can be done straightforwardly. []
Let ~ i = Fi + A / q + ted-, and i f = U 4 0 ~//. Each ~ / i s a locally finite family of
parallel 2-planes in R 4. We have p (if-) = ~ / q + ted-.
Proposition 4.3. I f p ( F ) = p ( F ' ) where F, F ' are 2-planes from ~ , then F = F'.
Proof. After translation by - ted- and rescaling by q from the system of 2-planes . ~
we get the system ~ . Hence this proposition follows from Proposition 3.4. []
The projection of a small wall of a tile of @t onto Et is a segment lying in the
projection of some 2-plane in f f o n t o E r Up to translations by vectors in A there are
only a finite number of small wails of the tiles of ~'r Hence there is a constant d > 0
such that if w is a small wall of a tile of ~t and the segment p J- (w) is contained in
the line p (F), where F is a 2-plane in ~,, then the Hausdorff distance between the
line p ( F ) and the rhomb p(w) is less than d. In other words, the sets p ( F ) + Ua and
p(w) must have a c o m m o n point, where Ua is the disk in E with center at 0 and
radius d. We fix such a d.
Definition. In this section a set F + Ud where F is a 2-plane in ~ is called a big
wall.
4.2. The Refinement of eet
Each big wall F + Ud is contained in a unique 3-plane F + E. We construct an
oblique periodic tiling of 4 such that all the big walls are contained in the parallel
boundary of this olique periodic tiling.
Every compact set in E7 meets only a finite number of 2-planes oar and hence
meets only a finite number o f big walls. Each prism CJ, t meets a finite n u m b e r of big
walls. The 3-planes going through these big walls divide C],t into many pieces. Each
piece is also a prism, and the projection of each piece onto E is the same as the
projection o f C], t onto E. Suppose there are s pieces; we denote them by
c1 9't, ci t, . . . , ClJ,;t', where s depends on I and j, s = s(I, j).
So far we have divided every C],t, I ~ M, j = 0, 1, 2, into smaller prisms. Every
tile of eet is a translate of one of C], t, I ~ M, j = 0, 1, 2, and we repeat the division
in every tile of eel, by translation. We get a new family of (smaller) prisms, each of
the form C[itk + ~ where I ~ M, j ~ {0, 1, 2}, k ~ {1. . . . . s(I, j)}, ~ ~ A. We define
the color of the prism C1itk + s as the pair (j; k). D e n o t e this new family of colored
prisms by ~ t . It is an oblique periodic colored tiling o f R 4. The color of a tile
contains the index of this tile. F r o m the construction we have:
Proposition 4.4. The parallel boundary B(6~tc) contains all the big walls and has the
same projection o n Et as the big walls:
pX (B(6~t)) = p i (~-) = gp/q + t e ~ .
Proposition 4.5.
Every colored rhomb in E has at most one lift onto ~ .
This follows from Proposition 2.6 and the coloring.
If w is a small wall of a tile C of ~_f, then p (w) is a segment lying on some line
p (W), where W is a big wall.
Proposition 4.6. I f the projection of a small wall w of a prism C of ~ onto Et lies on
the line p (W), where W is a big wall, then w and W have nonempty intersection.
Proof. C is contained in a prism D of the original family eet with p(C) = p(D). Let
Q = p i (D). Then p (C) is a polygon lying in Q, and p (w) is an edge o f p (C).
There are two possibilities:
(a) p (w) does not lie in any edge of Q. Then by the construction of the family
~t~ there must be a big wall meeting D and projecting into the line containing
p (w). This big wall must be W because it has the same projection on Et as
W has. Hence W and D have nonempty intersection, so p(W) and p(D) =
p(w) have nonempty intersection. Then w and W also have nonempty
intersection because they are prisms and their projections both on E and on E
have nonempty intersection.
(b) p (w) is a segment lying in a side of Q. By definition of d, p(W) and
p ( D ) = p(w) have nonempty intersection. We can apply the last phrase of the
previous paragraph. []
4.3. Proof of Theorem 4.1
The oblique periodic colored tiling ~te defines a set of colored tilings 6T'(~t). If
does not belong to ~ / q + te~ = p (B(~tc)), then a and ~t~ define a colored tiling
Tf which is the same as T~ if we ignore the colors.
We define ~'t as the set of all edge-configurations of all colored tilings in J ( : t c ) .
Up to translation there are only a finite number of edge-configurations in ~'t. By
definition, ~'t is a local rule of type 2.
Theorem 4.7. Every colored tiling satisfying the local rule ~q~tis a tiling of f f ( ~ t ).
Theorem 4.1 is a consequence of this theorem. Again we use the Main Technical
Theorem to prove Theorem 4.7.
Suppose T is colored tiling satisfying the local rule ~ ' r Recall that a lift of T into
~ is a map l: {tiles of T} ~ {prisms of ~tc} such that p(l(P)) = P and the color of
P is the color of I(P) for every tile P of T.
Proposition 4.8. I f T is a tiling satisfying the local rule ~ t , then, after a translation, T
has a strongly connected lift onto d~f.
Proof. Using the uniqueness of the lift of a colored tile (Proposition 4.5), the proof
is the same as in the case t = 0 (see Proposition 3.4). []
^
The oblique periodic tiling ~tc does not satisfy the conditions of the Main
Technical Theorem, because it is not a tiling of Rg. However, after translation by
-te~- and rescaling by q from ~tr we get an oblique periodic colored tiling of R 4
satisfying all the conditions of the Main Technical Theorem, by Propositions 4.4 and
4.6. Hence we can use this theorem to conclude that T belongs to ff'(~p). Theorem
4.7 is proved.
5. Local Rules of Type 1 for t = m + n'r
At the outset fit is a set of tilings without any colors. Hence the question of finding
local rules of type 1 (which does not involve any coloring) for these sets is the most
natural.
Theorem 5.1. I f t = m + n'c, where m, n are integers and ~-= (1 + v/-5)/2, then the
set ~ admits a local rule o f type 1.
p ( B ( ~ t e ) ) = p i (B(@,t)) = ~ + t e ~ = Ir N Et.
(3)
If c~ ~ Et is regular we denote, as in the previous section, the colored tiling
defined by a and @tc by T~.
5.1. r-Maps and Coloring
Fix t = m + m-. For a rhomb P with vertices in p(7/5), index(P) is the index of a
unique vertex of P, called the base vertex of P. For a regular a ~ Et we have the
colored tiling T~ and the noncolored underlying tiling T~.
Proposition 5.2. There/s ro > 0 such that, for every two congruent tiles P, P ' o f T~, if
the ro-maps at the base vertices o f these tiles are congruent, then the colors o f P and P '
in T f are the same.
First we show how to deduce Theorem 5.1 from this proposition. Fix a tiling T
satisfying ~r with r > ro + 2. A priori there are no colors on tiles of T. We color
them as follows.
Let P be a tile of T with base vertex v. The r0-map of T at v is congruent to an
r0-map of T~; let Q be the tile of T~ corresponding to P under this congruence. We
color P by the color of Q in T~. This way we color all the tiles of T.
If P, P ' are tiles of T sharing a c o m m o n edge, then the colored edge
configuration (P, P ' ) is congruent to a colored edge-configuration of T~, since r > r 0 + 2. By
Theorem 4.7 the colored tiling T belongs to ff(~tc). Hence the noncolored tiling T
belongs to ffTr []
It remains to prove Proposition 5.2. The idea is as follows. We define r-volumes
which correspond to vertices with the same r-maps, and the volume of colored tiles
which corresponds to tiles of the same colors of Tf. For large r we prove that every
r-volume cannot meet two different volumes of colored tiles.
5.2. r-Volumes
First we recall the concept of r-volumes. In this subsection a is not necessarily
in F"t.
For a point x ~ p " (Z 5) there is a unique ~ ~ 7]5 such that p~ ( ~ ) = x. Define
q~(x) = p(~). The map q~ is defined only for points in p ( y s ) c tA k ~ zF'k and is not
one-to-one. If qffx) = v, then q~-l(v) = {x + k 5 8 , k ~ 7]}. For a set X in E we
define i f ( X ) as the set ~p(X n p (7]5)).
Let K = p ( y ) = Pol(e~ . . . . . e~); it is a polyhedron in E with 22 vertices, 40
edges, and 20 facets (Fig. 6). The projection of K on A is the segment [0,56]. For
O
every x E p(7/5), the set K - r is the union of infinitely m a n y translates of K,
any two of t h e m differ by a multiple of 58, hence at most two of t h e m m e e t s K.
F o r a r e g u l a r a ~ E " the set of vertices of T~ is ff(c~ + K ) . T h e following
proposition follows easily.
Proposition 5.3.
to (o{a + [ K N ( K
Two points x, x + v are vertices o f a tiling T~ if and only if x belongs
q~ l(v))]}.
Now fix r > 0. Let ~ ' be an r - m a p at some vertex x. If x + v is a vertex of ~ ' ,
then there is p a t h connecting x and x + v by edges of .~'. E a c h edge is a translate
of el, i = 0. . . . . 4, hence using the p a t h we can write v = ET=0 niei, ni E ~. This
p r e s e n t a t i o n o f v as an integer linear combination of the ei's does n o t d e p e n d
on the p a t h in ~a~ connecting x and x + v. Let O(v) = E4=0 niei and K ~ = K n
(Nv K - $ ( v ) ) , where v runs the set of all vertices of .gL N o t e that ~ ( v ) is a point
in ~ - l(v). T h e set Km is called the volume of ~ ' . A n r-volume is the volume of an
r-map. It follows from the previous proposition:
Proposition 5.4.
(~(a + Km).
The r-map o f T~ at v & congruent to ~q~ if and only if v belongs to
T h e r e are only a finte n u m b e r of r-maps; and the p o l y h e d r o n K is p a r t i t i o n e d
into r-volumes, each c o r r e s p o n d s to an r-map. The partition can be described as
follows. L e t cr(r) be the set of all vectors in p(7/5) whose length is less t h a n r. A l l
the p o l y h e d r a K - p l(v), when v ~ t r ( r ) together divide K into r-volumes. N o t e
that there are only a finite n u m b e r of points in o-(r) such that K - ~ - l ( v ) meets K.
The significance of the r-volumes is expressed in the following:
Proposition 5.5. For a regular ~ ~ E 1 , two r-maps o f T~ at vertices v, v' are
congruent if and only if there is an r-volume Y such that both v, v' belong to (p(a + Y ) .
This follows from P r o p o s i t i o n 5.4.
The p o l y h e d r o n K has 20 facets, each a translate of one of ten p a r a l l e l o g r a m s
Pol(eiI , e j l ) , (i, j ) ~ M, by vectors in p i (7/5). If we use translations n l e 7 + n2e ff ,
nl, n 2 ~ 7/, t h e n the translates of Pol(eiI , e / ) cover the whole plane Ho.j) s p a n n e d
by eiI , e ~ . M o r e o v e r , every c o m p a c t set in H(i' j) can be covered by a finite n u m b e r
of such translates o f P o l ( e i l , e j l ) . R e c a l l that I r = (.JIEM(Hj + p 1 ( 7 / 5 ) ) ( s e e
Proposition 2.2).
Proposition 5.6. I f X & a polyhedron lying in K and having all facets lying in Ir, then,
for sufficiently large r, every r-volume is either contained in X or does not meet the
interior o f X .
Proof. By the observation p r e c e d i n g the proposition, there a r e a finite n u m b e r o f
points Yl . . . . . yq e p (7/5) such that all the facets o f X lie in the union of all the
facets of the p o l y h e d r a K, K - Y l . . . . . K --yq. Let r > maxj= 1..... ql,p(yi)l. T h e n by
construction every r - m a p is e i t h e r c o n t a i n e d in X or d o e s not have i n t e r i o r
intersection with X. [ ]
Proposition 5.7. There is an r 1 such that, for every regular a 9 E , the rl-maps of T~
at vertices o f different indices are not congruent.
Proof. Using the previous proposition we see that when r ~ 0% the maximal
diameter of r-volumes tends to zero. Hence there is r 1 such that the projection of
every q-volume on A is a segment having length less than the length o f [0, 6 ]. The
rl-maps of vertices of different indices in T~ are different, because if two integer
points having different index, then they project on A into two points whose distance
is greater or equal to the distance between 0 and 8. []
Remark. In fact one can take r I = 3.
Proposition5.8. I f X is a polygon tying in a 2-plane F,,, u 9 R, and having edges ~ying
in I r (~ E , , then for sufficiently large r the intersection o f every r-volume with Eu is
either contained in X or does not meet the interior of X.
Proof. This is a straight corollary of Proposition 5.6.
[]
Proposition 5.9 [IS]. I f ~tt admits local rules of type 1, then t is o f the form m + nr for
some integers m, n.
Proof. The r-maps which apear in T~, a 9 F-~t,are those which correspond to the
r-volumes which meets the 2-planes k6 - Et, k 9 Y. Suppose t is not of the form
m + n~'. Fix r > 0. The facets of every r-volume are in Ir, hence every vertex of it is
the intersection point of three different 2-planes in It. It is easy to check that the
projection of every intersection point of three 2-planes in I r onto A is of the form
(m + m-)6 (see the description of I r in Proposition 2.2). Hence the planes k8 - Et,
k 9 2v, can never meet vertices of any r-maps. It follows that, for t ' very close to t,
the plane k6 - E't, meets the same r-volumes as k6 - Et does. This means every
tiling in ~t, has the same r-maps as tilings in ffTr []
5.3. Proof of Proposition 5.2
We fix t = m + nz. Recall that the prototiles of ~^tc are Clj,;kt, where I 9 M,
j = 0,1, 2, k = 1. . . . . s(I, j ) (see Section 4.2). In what follows we drop the subscript
t. Let Q ~ ' k = p and Pj'k=p(C~"~). Then P ] ; k = p I + j e o is a rhomb
congruent to PI and has base vertex jeo; we color p/;k by (j, k). It follows from
formula (
2
) that Q~J;k = j'e~ - Q~; k is a polygon lying in the intersection of K with
j 6 - E r The following proposition describes Q~;k as the "volume" of p];k.
Proposition 5.10. Let o t e F't be regular. A vertex v of T f is a base vertex of a colored
tile congruent to P]; k if and only if
(since ~0(e0~) = e0)
A I p ( ~ ) = v - j e o and a + E - Qi; k contains ~c
(since a - Q ~ ; k c R4o)
= v - j e
o and a + E m e e t s
s
;k
o a n d p ( s
3 ~
3 s 1 6 3
However, p( s + C]' k) is the colored tile congruent to Pi' k with base point v.
[]
Now we can prove Proposition 5.2. Every edge of Qi 'k lies in p (B(~'~), hence
every edge of Q~' k lies in I r N (j~ -- E't), by (3). By Proposition 5.8 there is r 0 such
that for every polygon QJ' k the intersection of every r0-volume with every plane
k8 - Et, k ~ ;7, either lies in this polygon or does not have interior intersection
with it.
Now suppose P, P ' are two congruent tiles of T,, with base vertices v, v',
respectively. By Proposition 5.4 both v, v' belong to ff(a + Y) for some ro-volume
Y. Let the color of P be (j, k). Then v belongs to ff(a + 0 i 'k) by Proposition 5.10.
Hence Y c O],k. This, in turn, implies v' E ff(a + Qi'k). By Proposition 5.10 the
tile P ' has color (j, k), the same as P. []
Together with Proposition 2 we have proved T h e o r e m 5.1.
6. The Main Technical Theorem
Now we prove the Main Technical T h e o r e m (Theorem 3.8). We reduce the problem
to investigating a set of oriented lines on a plane. Similar proofs have appeared in
[LPS1] and [LPS2].
Definition. An oriented line on a plane is a line equipped with an open half-plane
divided by this line, called the positive half-plane of the oriented line. A set X of
points is greater than an oriented line if all points of X are in the positive
half-plane. A set of oriented lines is compatible if there is a point greater than all of
them. A set of oriented lines is weakly compatible if every finite subset is compatible.
In order to prove the Main Technical T h e o r e m we have to prove that every
strongly connected list into ~" has to define a tiling in J ( ~ ' ) . We need to prove that
there is a plane parallel to E meeting all the members of the lift (actually with some
extra conditions for the singular case). This is equivalent to the fact that the
projections of all the members of the lift onto E have a c o m m o n point. Note that the
projection of evey tile of ~ onto E is a polygon. If we orient all the lines going
through an edge of the projection of a m e m b e r of the lift so that the interior of the
projection is greater than this line, then it is enough to prove that there is a point
greater than any line or at least lying on this line. We prove an even stronger
statement: these lines are weakly compatible.
In what follows we introduce orientations for lines of an even larger set (it is 49),
since it is easier to deal with this larger set. We then prove that this set of oriented
lines is weakly compatible, using the "bootstrapped" property and the compatibility
of every three lines meeting at a point.
6.1. Plan of the Proof of the Main Technical Theorem
Fix a tiling T having a strongly connected lift into ~'. First we prove:
Proposition 6.1. For every finite set of tiles P1. . . . . Pm of T, the projections
p (I(P1)) . . . . . p (I(Pm)) have nonempty interior intersection.
Then we use the following:
Proposition 6.2. Suppose that the lift l of a colored tiling T satisfies Propositon 6.1.
Then T is a colored tiling belonging to ff( ~').
T h e o r e m 3.8 followed from these propositions. For a regular a ~ E, the colored
tiling defined by a and ~' is denoted by T ( a , ~').
Proof of Proposition 6.2. We number the tiles of T, T = {P1, P2 . . . . } so that for
every r the disk Ur is covered by the first N tiles, here N depends on r. Since the
polygons p (I(P1)) . . . . . P (l(PN)) have nonempty interior intersection, the
intersection of these polygons is a polygon (with nonempty interior). There is a regular
point c~r ~ E belonging to this polygon. Then the colored tiling T ( a r, ~') is the same
as T inside the disk U~. Of course t~r ~ a when r ~ ~ and T(ar, ~ ) ~ T when
r .---~o9. [ ]
It remains to prove Proposition 6.1. Note that every edge of a polygon p
(I(P)), P E T, is lying on a line in qb. To prove Proposition 6.1 we introduce
orientations on all the lines in 49 such that:
( * ) If the polygon p (l(P)), where P is a tile of T, has an edge lying on a line
h, then the interior of this polygon is greater than h.
(* *) The set 49 of oriented lines is weakly compatible.
Proposition 6.1 follows immediately from ( * ) and (* *).
O u r task is to find, for every connected lift l of a colored tiling T, a way to orient
all the lines in 49 satisfying conditions ( * ) and (* *).
Definition. In this section a set F + Ud where F is a 2-plane in 3r is called a big
wall.
6.2. Orientations of Lines
We define a multivalued map p: E ~ E as follow. F o r a point x which is an interior
point of a tile P of T let p ( x ) be the interior of the polygon p (l(P)). Suppose x is
not an interior point of any tile, let Pa,'", Pk be the tiles of T containing x and let
Y l , . . . , Yk be, respectively, interior points of P1 . . . . . Pk" Define p ( x ) = I.J ki=1 P(Yi).
If x is an interior point of a tile of T, then the set x + O(x) does not meet the
parallel boundary B ( ~ ) . F o r a set X in E we define p ( X ) as the union of all p ( x )
where x ~ X. The following is a consequence of the strong connectedness of the lift.
Proposition 6.3. For every path-connected set X in E the set p ( X ) is connected.
Suppose W is a big wall: W = F i + ~ + U d, ~ A . Then it is a prism, its
projection on E is a line contained in qb, and its projection on E is a strip,
p ( W ) =f,. + p(~:) + Ud.
Lemma 6.4. I f W is a big wall, then p ( p ( W ) ) does not meet the line p ( W ) .
Proof. Suppose y is a point lying in both p ( p ( W ) ) and p (W). Then y ~ p ( x ) for
x ~ p(W). Since d > 0 we can choose x such that it is an interior point of a tile of
T. Then we see that x + p ( x ) contains x + y which belongs to W because W is a
prism. This means that set x + p ( x ) meets W, but by condition (c) all the big walls
are contained in B 0, and hence does not meet x + p(x). The proposition follows. []
Now we can introduce orientations for the lines in qb. F o r every line h in qb there
is a unique big wall W which projects into h, p ( W ) = h. By the previous lemma,
the set p ( p ( W ) ) does not meet the line h. Since p ( p ( W ) ) is a connected set, it lies in
one half-plane of E separated by h. W e define the orientation of h such that
p ( p ( W ) ) is greater than h.
Proposition 6.5. Suppose p (F), p (F'), and p (F"), where F, F', and F" are
2-planes in ~ , have a common point. Then F, F', F" also have a common point.
Proof. It is easy to see that no two of F, F ' , F " are parallel. Let vl, v2, v 3 be three
intersection-points of three pairs from F, F ' , F". Then they have rational
coordinates and have the same projection on E, hence they must be coincident by
Proposition 1.2(c). [ ]
6.3. Property ( * )
Suppose C = l ( P ) is a lift of a tile of T. Then p (C) is a polygon having edges lying
in qb.
Proposition 6.6. Suppose Q = p (C) has an edge s lying on a line f in ~P. Then the
interior o f Q is greater than f.
Proof. T h e r e is a big wall W such that f = p ~-(W). The set p(C) + s is a small
wall. By condition (c) of the Main Technical T h e o r e m the small wall p ( C ) + s and
the big wall W have a c o m m o n point y. Let x = p(y). Then p(x) contains the
interior of p (C) and by definition of the orientation p(x) > f, hence the interior
of Q is greater than f. []
6.4. Property ( * * )
The following is crucial.
Proposition 6.7. If several lines in ~P have a common point, then they are
compatible.
Proof. Suppose p ( F ) , p ( F ' ) . . . . . p (F") have a c o m m o n point where
F, F ' , . . . . F " are 2-planes from ~ , then by Proposition 6.5 F, F', . . . . F" also have a
c o m m o n point y. Let x = p(y). By definition the set p(x) is greater than all the
lines p ( F ) , p ( F ' ) . . . . . p (F"). []
Proposition 6.8. The set p ( E ) / s bounded in E.
The boundedness of p(E) is an expression of the " w e a k local rule" in the sense of
Levitov [Levi. The p r o o f is presented in the Appendix.
T h r e e sets X, Y, Z are called bootstrapped if X f3 Y = X n Z = Y f3 Z.
Proposition 6.9. Every three of five sets ~0,o~11,~z,~3,J4 are bootstrapped. In other
words, through every intersection point of 2-planes from two of three families ~i, ~jj,~
passes a 2-plane from the third family.
Proof. Every ~ contains A. It suffices to prove that if 0 _< i < j _< 4, then ~ n
= A. W e assume that i = 0 and j = 1 (the other cases are similar). Consider two
2-planes F o + ~:o and F 1 + ~1 where ~o, ~1 ~ A. Note that F o is spanned by
Vl = ~4 -- 1 and v 2 = e 3 - e 2, and F 1 is spanned by v 3 = E0 - e 2, v4 = ,~4 -- 63"
These four vectors vl, v 2, v 3, v4 generate A as a 7J-module, so we have ~0 = E4= 1 aivi,
~1 = E4i=1 bivi where ai, bi ~ 7/. Then the point (a3v 3 + a4v4) + (biv I + b2v 2)
obviously belongs to both F 0 and F 1 and it also belongs to A. []
D e f i n i t i o n . Two parallel oriented lines have the same orientation if the intersection
of their positive haft-planes is a haft-plane.
The p r o p e r t y of the same direction is an equivalent relation between oriented
parallel lines.
The family q~ o f oriented lines satisfies the following:
(i) If several lines from ~ intersect at a point, then they are compatible.
(ii) Every three of five sets qbi are bootstrapped.
(iii) Each family of parallel lines (I) i a r e not of the same direction.
(i) is Proposition 6.7, (ii) follows from Propositions 6.9 and 6.6, and (iii) follows
from Proposition 6.8.
Proposition 6.10. There is a point ot ~ E such that f o r every line f in 9 either a ~ f
or a is greater than f .
For every pair o f lines in dpi = p i (~ii) there is a point greater than both o f
Proof. We may assume i = 0. We suppose the converse, then there are two lines
f, f ' in 4P0 as shown in Fig 7, where the positive half-plane is shadowed. A line h in
(I) 1 intersects f, f ' at x 0 and Y0 as in Fig. 7. By (ii) there are lines in qb2 going
through x 0 and Y0, and by applying this again and again, we get a system of lines as
in Fig. 8 where two horizontal lines are f and f ' , all the others are lines in ~1 or
qb2, and all intersection-points are triple. There are two possible cases of orientation
of h as indicated in Fig. 9(a) and (b).
Consider the case in Fig. 9(a). By (i) there is a point greater than all the lines
going through x 0. Hence the orientation of the line x o y o can be found easily. Then
applying (i) to points x 1, Y2, x2, etc., we see that all the lines XlYl, x2Y2, x3Y3, etc.,
have the same orientation (Fig. 10). This contradicts (iii).
In the case in Fig. 9(b) all the lines XoYo, x _ l Y _ l , x 2Y_2, etc., have the same
orientation, which is also a contradiction. []
Fig. 7. Two noncompatible parallel lines.
Fig. 8. Systemof bootstrapping lines.
Step 2. A line h i parallel to all the lines in t~ i exists such that f o r every line f in (~i
either h i = f o r h i is greater than f .
Proof.
(I)i c a n n o t
This trivially follows from the previous step and the fact that all the lines in
have the same orientation. [ ]
Let a be the intersection point of h 0 and h 1.
Step 3.
For every line f in dp either a ~ f or a is greater than f .
Proof. Suppose a does not lie in f. Let Xo, x a denote the intersection-points of f
and, respectively, h 0 and h r Because the set of projections (on E) of points from A
is dense in f there is a point y ~ f lying between x 0 and x I and y is the projection
of a point from A.
There are lines from ~ 0 and qbI going through y (Fig. 11), and a is greater than
them. By using (i) to point y we see that a > f. [ ]
Now we can prove property ( * *).
f
f,
(b)
Fig. 9. Two cases in the proof of Step L
f
f,
~.,.
Fig. ll}. Orientation of bootstrapping line.
Proposition 6.11. For every finite number o f lines gl, g2 . . . . . gm in dp there is a point
greater than all o f them.
P r o p e r t y (* * ) is established, a n d T h e o r e m 3.8 is proved.
Concluding Remarks, Example: t =
~17.1. On Local Rules o f Type 2
W e have a c r i t e r i o n for the existence of local rules of type 1. As for the local rules
of type 2 we have p r o v e d o n l y the sufficient c o n d i t i o n for the existence: t =
X0
Y
( m + nv/'5)/q. Is this also a necessary condition? Note that there are only a
countable number of local rules of type 2, hence only a countable number of sets ffTt
can admit local rules of type 2.
Conjecture. I f gt admits a local rule o f type 2, then t = ( m + nv/5 ) / q for some
integers m, n, q.
Note also that Theorem 4.7 is much stronger than Theorem 4.1. Every colored
tiling satisfying oq~t' is a quasi-periodic colored tiling. That is, together with colors this
tiling is still quasi-periodic, since it is the slice of a periodic colored tiling of E4.
7.2. On Coloring
When t = (m + n ~ / 5 ) / q but is not of the form m ' + n%-, then the set ~ admits
local rules of type 2 but does not admit local rules of type 1. Proposition 5.2 means
that in this case the coloring is not a "local" operation.
We noted in the Introduction that, for such t, T ~ ffTt exist which have two
different colorings such that both of the colored tilings we obtain satisfy the local
rule -~'t- They can be constructed as follows. Take a point a belonging to p i (B(d~,tc))
but not belonging to p (lit). Such a point always exists in this case. Then a is
regular, but E + a meets the parallel boundary of the refinement 6~. There are
several lines in p j- (B(@t)) going through a , they divide F-'t into many parts. Each
part defines a colored tiling in f f ( ~ c ) by taking the limit as in Section 2.5. Different
parts define different colored tilings, but these tiling, when ignoring the color, are
coincident with T~. It follows from the proof that only those tiling described above
can be colored in different ways. Hence there is a subset of measure 0 in fit whose
tilings can be colored in different ways, other tilings can be colored uniquely.
When t = m + m-, each tiling T ~ fftt has a unique coloring as a colored tiling in
7.3. Generalizations
Our method can be applied to more general cases. First we introduce the following
local rule of type 3 which contains both types 1 and 2 as special cases. Let a local
rule of type 3 of radius r be any finite set of colored r-maps. A noncolored tiling
satisfies this local rule if it can be colored so that every colored r-map of the colored
tiling is congruent to a colored r-map in this set.
Suppose E is a quadratic and totally irrational n-plane in the Euclidean space
N, equipped with a standard base. Here "quadratic" means that E is spanned by n
vectors v 1. . . . . vn with coordinates belonging to 7lye- for some natural number d,
and "totally irrational" means that E des not contain any rational point except 0. Let
be the n-plane spanned by n vectors which are the conjugates of v 1. . . . . v,. Then
E n F, = {0} and R u = E 9 F,, 9 A where A is a rational ( N - 2n)-plane. For each
t ~ A let ~-t be the closure (under the operation "limit") of the set of all tilings To
and their translates, where a is regular and belongs to E, + t.
T h e o r e m 7.1. I f t has coordinates lying in Ovrd, then ~tt admits a local rule of type 3.
This is a generalization of T h e o r e m 4.1 to higher-dimensional cases, the p r o o f is
essentially the same. As for local rules of type 1, the situation is more complicated
because of the b o o t s t r a p p e d condition. Some necessary conditions for the existence
of local rules of type 1 are presented in [Lev] and [Lel].
7.4. The Case t = 1
We present here a local rule of type 3 for the case t = ~I-. This case enjoy as much
symmetry as the case t = 0: both have the dihedral group D 5 as the symmetry group
in the sense that D 5 preserves the same local isomorphism class. However, the latter
has a local rule of type 1 while the case t = ~1- does not.
In order to reduce the number of prototiles we have to refine the p r o o f of
T h e o r e m 4.7. The price for this is that we do not get a local rule of type 2, but we get
a local rule (of type 3) in which star-configurations play the role of
edge-configurations.
Consider the six colored rhombs in Fig. 12(a). A tiling of the plane by images of
the six colored rhombs under isometries of the plane (reflections are allowed) is said
to be admissible if arrows on edges and marks at vertices are matched. H e r e
matching at edges m e a n s every edge has a definite arrow, and matching at vertices
m e a n s only m a r k e d vertices can m e e t and at every m a r k e d vertex the m a r k s f r o m all
the r h o m b s incident to this vertex f o r m a " k e y " like in Fig. 12(b).
T h e claim is that every admissible tiling, after some rotation, belongs to Jll/2"
Conversely, every tiling in ~9~1/2 can be colored to b e c o m e an admissible tiling.
O f course this is a local rule o f type 3. This looks similar to the A m m a n n
matching rules for the set of tilings having eight-fold symmetry (see [AGS] and
[Le2]) and Socolar local rule for the set of tilings having twelve-fold s y m m e t r y [Sol].
However, the m e t h o d s to derive t h e m are quite different. I f we ignore the marks at
vertices we get the K l e m a n - P a v l o v i t c h matching rule (see [KP]).
T h e r e is a subset of m e a s u r e 0 in ~1/2 such that every tiling in this subset can be
colored to b e c o m e an admissible tiling in two or four different ways, while every
other tiling can be colored in a unique way. T h e key of Fig. 12(b) can a p p e a r in ten
directions.
R e m a r k . J. Socolar suggests that there is an interpretation of the vertex key in terms
of A m m a n n ' s lines as in the cases o f eight-fold and twelve-fold tilings (see [Sol]).
H e r e is a sketch o f the way we arrive at this local rule. First we refine the oblique
periodic tiling #'1/2 to the oblique periodic colored tiling G1~2 whose parallel
b o u n d a r y contains all the five sets ~qr_ e i / 2 ' i = 0. . . . . 4. Geometrically the
refinement looks as follows. Every tile of @1/2 is the sum of a r h o m b in E and a
polygon in El/2. T h e latter is either a triangle or a hexagon; each hexagon has two
long and one short diagonals. T h e r e f i n e m e n t corresponds to dividing the hexagon
into four parts by the two long diagonals and keeping all the triangles untouched.
This r e f i n e m e n t defines a set of colored tilings 3 ( ~ / 2 ).
T h e following fact, which relates admissible tilings to the refinement, can be
verified by analyzing the vertex-configuration o f colored tilings in ff'(6e~ 2) and by
inspection.
Proposition 7.2. Every admissible tiling is a colored tilings whose vertex-configurations
are translates o f vertex-configurations o f a colored tiling in J-(-delC/C2).
U p to isometry and colors there are two rhombs: a thick one and a thin one. For
a colored tiling let the thick p a r t be the union of all the thick tiles. T h e following is
trivial:
I f two admissible tilings are coincident on the thick part, they they are the
W e consider the 2-planes in 9 - _ ~ i / 2 as the big wall of ~ff2; all o f t h e m have
width 0. W e call a tile o f Ga/2 a thick tile if its projection on E is a thick rhombs.
Note that the refinement ~1c>2 is weaker than the refinement ~'~/2 in Section 4.
Now Proposition 4.6 is not valid for @1c~2 but it is still valid if the small wall w
projects onto a thick rhomb in E:
Lemma 7.4. The parallel boundary o f ~1c~2 contains all the 2-plane from U ~=o(Jr
~i/2). If the projection on E l o f a small wall w of a thick tile o f ~1/2 lies on the
projection o f a 2-plane F from IJ 4=0(9-- el~2) , then w and F have nonempty
intersection.
(a) All lines li are parallel, all lines l~ are parallel, none of them is parallel to
either of h 1, h 2.
(b) li, lI, and h I have a common point and li, l'i+1 have a common point for every
i ~ 7 7 .
Now the proof of Theorem 4.1 shows that there is a colored tiling T ' of J(~'f~2)
such that T = T ' on the thick part. L e m m a 7.3 says that T and T ' are the same.
This means every admissible tiling belongs to J(Gf~2).
Acknowledgments
The author is sincerely grateful to his adviser, Professor S. P. Novikov for attention
to this work. He would like ot thank L. Danzer, C. Radin, and V. Sadov for fruitful
discussions. H e is especially indebted to M. Senechal and J. Socolar who carefully
checked the manuscript and suggested many things for the revised version. The
author acknowledges the Max-Planck Institut fiir Mathematik for hospitality and
support.
Appendix
We prove Proposition 6.8. This proof is a modification of an (unpublished) proof of
Levitov. The author learned of this proof through V. Sadov, but the proof that he
learned from V. Sadov (for the class of quasi-crystals having eight-fold symmetry)
contains a gap. In what follows we overcome the mistake and at the same time apply
the proof to our cases.
Lemma A.1. For every two points x, y ~ E, if Ix - y [ < 1, then the Hausdorff
distance between p ( x ) and p ( y ) is less than a number Q, not depending on x and y.
Proof. This follows form the finiteness of the prototiles of U.
[]
Lemma A.2. There is a constant c 2 such that for every line h parallel to p ( F ) where F
is a 2-plane from ~ , the set p(h) is contained in the c2-neighborhood o f a line h'
parallel to p ( F ).
Proof. Suppose F c ~0- Let V be the set p 0 n p - l ( h + U1/2)). This set V is a
discrete family of lines parallel to p-L (F). There is a constant d such that the least
distance between two lines in V is greater than d. The set X = p(x + U1/2), where
x is a point of h, is bounded, and the diameter of X is less than 2c 1. If H is a
2-plane in ~o such that p ( H ) intersects x + Ur, then there is a point of X greater
than p ( H ) . Hence if X does not meets p ( H ) , then X is greater than p x (H).
Let h i, h'2 be two lines from V such that X lies between them and every line
from V lying between them must intersect X. Then the distance between h' 1 and
h ' 2 is less than 2c i + 2d (see Fig. 13).
Suppose H i , / / 2 are 2-planes from ~0 such that p ( H i ) = h'~, p i (Ha) = h'2. Let
h i = p ( H 1) and h 1 = p(H2).
By definition p(h 1) > h'l, p(hz) > h'2. Since the distance between h and h 1 (or
h 2) is less than 89 from Lemma A.1 it follows that p(h) lies between two lines l i and
12 where l 1 (resp. l 2) is the line in E parallel to h'1 and lying in the nonpositive
half-plane of h'1 (resp. h'2) and having distance to h'l (resp. h'2) equal to Q. The
distance between l I and 12 is less than c2 = 3 q + 2d. []
Fig. 13. Proof of Lemma A.2.
Proof of Boundedness of p(E). Choose the coordinate systems (a 1, a 2) in E and
(bl, b2) in F, such that f l is given by {a 1 = 0}, f2 by {a2 = 0}, f4 by {a 1 - a 2 = 0}, f l
is given by {b 1 = 0}, f2 by {b 2 = 0}, f4 by {b 1 - b E = 0}.
T h e n it can be checked that f3 is given by {a 1 + "ra2 = 0} a n d f3 by {TbI + b 2 =
F o r a pair of functions ~ and u o n E, let the sign /z - v m e a n that I/x - vl <
const. In o t h e r words, the distance b e t w e e n /~(x) and v ( x ) is less t h a n a constant for
every x ~ E. If t~ is a multivalued function o n E (i.e., tz(x) is a subset of ~), then
the distance b e t w e e n /z(x) a n d v ( x ) is u n d e r s t o o d as the H a u s d o r f f distance.
Note that O is a multivalued m a p from E to E. Put
bl(X) = {bal3b 2 such that ( b l , b 2) E p ( x ) } ,
b 2 ( x ) = {b213b 1 such that ( b 1, b 2) ~ p ( x ) } .
Both bl(X) , b2(x) are multivalued function on E. By applying L e m m a A.2 to lines
parallel to f l we see that b l ( a l , a 2) - /z(a 1) for some function tx o n E, d e p e n d i n g
only o n al, a n d h e n c e can be regarded as a function o n ~. Similarly b2(al, a 2) =
v(a2). By applying L e m m a A.2 to f4 we h a v e / ~ ( a l) - v(a z) =- tr(a 1 - a 2) for some
function tr.
Putting a 2 = 0 we see that /~ - tr. Putting a~ = 0 we get v -=- tr. H e n c e p, --- v
--In addition /~ also satisfies L e m m a A.1. F r o m this it is not difficult to prove that
for a fixed c o n s t a n t c we have ~(cx) -= cry(x).
Finally, by applying L e m m a A.2 to f3, we h a v e / z ( a 1) + rlx(a z) = O("ra1 + a 2) for
some function 0; putting a 2 = 0 we get jz - 0. T h e n
(1 - "r)bt(a 1) ------( r - 1)/.t(a2).
Because ~"4 : 1 (this is the expression of "completeness" defined in [LPS1] a n d [Lel])
it follows that /z --- 0. This m e a n s that p is b o u n d e d . []
B o u n d e d n e s s of p m e a n s the local rule is a "weak local rule" in the
sense of Levitov. F o r every t ~ ~ o n e can easily construct a local rule of type 1 such
that every tiling satisfying this local rule has a strongly connected lift into 4 - Using
a similar p r o o f as the above it can be proved that this local rule is a " w e a k local
rule" in the sense of Levitov. A similar m e t h o d works in the higher-dimensional case
as well. The existence of weak local rules for ff~-t,has also b e e n proved by Levitov
[Lev] by a different m e t h o d .
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