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these tilings from the viewpoints of weak cross-polytope imaginary cubes and
double imaginary cubes.
We set positive integers
n
≥
3and
k
≥
2. Consider a subset
Z
of the
n
-
dimensional cubic lattice
n
Z
=
{
x
∈
Z
|
x
·
1
≡
0
(mod
k
)
}
.
n
)a
lattice-cube
. In each lattice-cube
C
, take an MCI of
C
whose set of v-vertices is
Z
n
)+
We call a cube conv(
{
0
,
1
}
{
v
}
(
v
∈
Z
∩
vert(
C
). Such an MCI is a
translation of one of
M
r
for 0
≤
r<k
defined as
n
V
(
M
r
)=
{
x
∈{
0
,
1
}
|
x
·
≡
r
(mod
k
)
}
.
1
Note that every pair of these MCIs which are placed in adjacent
n
-cubes
share the faces on their intersection. After placing such MCIs, there remain
holes around lattice points
n
{
x
∈
Z
|
x
·
1
≡
0
(mod
k
)
}
.
These holes are weak cross-polytopes because all of the vertices are on the lattice
edges. Therefore, for every
n
and
k
, we have a tiling of
n
-dimensional space by
translations of
M
r
for 0
r<k
and
n
-dimensional weak cross-polytopes of
several shapes. In the case
n
=3and
k
= 2, this tiling is the three-dimensional
tiling by regular tetrahedra and regular octahedra. In the case
n
=3and
k
=3,
M
r
(
r
=0
,
1
,
2) are
H
,
T
,and
T
, respectively, and each hole is a T. Therefore,
we have the three-dimensional tiling by Hs and Ts. In the case
n
=4and
k
=2,
not only MCIs placed in lattice-cubes but also the holes are 16-cells, and we get
the four-dimensional tiling by 16-cells. Since T and 16-cell are the only weak
cross-polytope imaginary
n
-cube shapes for
n
≤
3 (Proposition
6
), among these
tilings, there are only two tilings by imaginary cubes.
These two tilings are related to the fact that H and 16-cell are double imag-
inary cubes. The three-dimensional tiling by Hs and Ts can be characterized
as follows [
1
]. Let
σ
3
be the isometry on three-dimensional Euclidean space to
rotate by 180 degrees around the axis
x
=
y
=
z
. Then, the tiling is a Voronoi
tessellation of the union
≥
3
σ
3
(
3
) of the two cubic lattices such that Voronoi
Z
∪
Z
3
) have the shape H and those of other points have
the shape T. See [
6
], for example, for the notion of Voronoi tessellations.
This construction can be extended to higher-dimensional cases. In the
n
-
dimensional Euclidean space, let
σ
n
3
σ
3
(
cells of points in
Z
∩
Z
n
be the orthogonal transformation on
R
that satisfies
σ
n
(
1
)=
1
and
σ
n
(
n
v
)=
−
v
for
v
∈
R
with
v
·
1
= 0. Then,
n
σ
n
(
n
). The Voronoi cell of the origin is
take the Voronoi tessellation of
Z
∪
Z
n
)and
σ
n
(conv(
n
)),
the intersection of two
n
-cubes conv(
{−
1
/
2
,
1
/
2
}
{−
1
/
2
,
1
/
2
}
n
) are its translations.
In the case
n
=4,
σ
4
maps the set
V
1
to
V
3
,
V
3
to
V
1
,and
V
2
to itself, where
n
σ
n
(
and Voronoi cells of points in
Z
∩
Z
the
sets
V
1
,V
2
,
and
V
3
are
defined
in
Sect.
3.2
.
Therefore,
the
cube
4
)) = conv(
V
2
∪
conv(
V
1
)and
their intersection conv(
V
2
) is the Voronoi cell at the origin. One can show that
{−
1
/
2
,
1
/
2
}
V
3
) is mapped to the cube conv(
V
2
∪
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