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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
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
R
|
i =1 , 2 , 3 , 4
}
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