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the other Voronoi cells are also 16-cells, and therefore this tiling is the four-
dimensional tiling by 16-cells.
For
n
3, if the intersection
E
n
n
)and
≥
of two cubes conv(
{−
1
/
2
,
1
/
2
}
σ
n
(conv(
n
)) is an imaginary cube of an
n
-cube
C
, then it must
also be an imaginary cube of
σ
n
(
C
). It is easy to show that
C
and
σ
n
(
C
)are
different
n
-cubes and thus
E
n
is a double imaginary cube. Since H and 16-cell
are the only double imaginary
n
-cube shapes for
n
{−
1
/
2
,
1
/
2
}
3 (Theorem
9
), among
these Voronoi tessellations there are only two tilings by imaginary cubes.
≥
4 Fractal Imaginary Hypercubes
4.1 Fractal Imaginary Cubes
From a regular tetrahedron, one can form a fractal (i.e., self-similar) object
known as a Sierpinski tetrahedron (Fig.
5
(a)). It has similarity dimension two
and it is also an imaginary cube.
(b)
(a)
(c)
Regular tetrahedron
Fig. 5.
The first two approximations of (a) Sierpinski tetrahedron, (b) H
∞
,and
(c) T
∞
.
n
with the
Hausdorff metric. According to the theory of IFS (iterated function system) frac-
tals developed by Hutchinson [
7
], for contractions
f
i
:
n
be the metric space of non-empty compact subsets of
Let
H
R
n
n
(
i
=1
,...,m
),
R
ₒ
R
an IFS
I
=
{
f
i
|
i
=1
,
2
,...,m
}
defines a fractal object as the fixedpoint of the
n
:
following contraction map on
H
m
F
I
(
X
)=
f
i
(
X
)
.
(4)
i
=1
As for a Sierpinski tetrahedron, let
S
be a regular tetrahedron and let
I
S
=
3
3
{
be an IFS where
f
i
(
i
=1
,
2
,
3
,
4) are homothetic
transformations (i.e., similitudes that perform no rotations) with the scale 1/2
whose centers are vertices of
S
. The induced fractal is a Sierpinski tetrahedron.
It is an imaginary cube of the cube
C
of which
S
is an imaginary cube. Note
that this fractal object is minimal among imaginary cubes of
C
.
f
i
:
R
ₒ
R
|
i
=1
,
2
,
3
,
4
}
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