Definition 2.4 If X is a subspace of Y, both topological spaces, and
A C X, then we write IntxA to mean the interior of A in the subspace
topology of X.
Likewise we write dxA for the boundary of A taken in the sub-
space topology of X.
Lemma 2.5 a/Ifu G NS, then (Q + u)nIntK = Int(Q+u}nE±((Q +
u) n K) and (Q + u) n ^ - L K = 9(Q+u)ni^ ((Q + w)flif).
b/IfueNS, then((Q + u)nE±)\NS = d(Q+u)nE±((Q + u)n
Proof a/ To show both facts, it is enough to show that (dE±K) H
(Q + u) has no interior as a subspace of (Q + u) n
Suppose otherwise and that U is an open subset of dK D (Q + u)
in (Q + u)nE±. By the density of ^(ZN) in Q n ^ , we find i; G ZN
such that wEtZ + Tr
^). But this implies that u G dK +
so u 0 iVS - a contradiction.
b/ By defintion the left-hand side of the equation to be proved
is equal to (dE±K +
T T ^ Z ^ ) )
Ci(Q + u) which equals (dE±Kn (Q +
u)) +7r±(ZN) since 7rJ-(ZJV) is dense in Qn E±. By part a/ therefore
we obtain the right-hand side of the equation.
Condition 2.6 We exclude immediately the case (Q + u)nIntK = 0
since, when u G NS, this is equivalent to Pu = 0.
Examples 2.7 We note the parameters of two well-studied examples,
both with canonical acceptance domain (2.3).
The octagonal tiling [Soc] has N = 4 and d = 2, where i? is a
vector subspace of
invariant under the action of the linear map
which maps orthonormal basis vectors e\ H- e2, ^2 ^- e3, e3 i— e4,
e4 H-• —ei. Its orthocomplement, E1"1, is the other invariant subspace.
Here Q = R4 and so many of the distinctions made in subsequent
sections are irrelevant to this example.
The Penrose tiling [Pe] [dBl] has N = 5 and d = 2 (although
we note that there is an elegant construction using the root lattice of
A4 in
[BJKS]). The linear map which maps e^ H- e^+i (indexed
modulo 5) has two 2 dimensional and one 1 dimensional invariant
subspaces. Of the first two subspaces, one is ^hosen &&JE and the
other we name V. Then in fact Q = E © V © A, where A = \{e\ +
^2 + 63 + e4 + es)Z, and Q is therefore a proper subset of
a fact
which allows the construction of generalized Penrose tilings using a
parameter u G NS \ Q.
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