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Imaginary Hypercubes
B
Hideki Tsuiki
(
)
and Yasuyuki Tsukamoto
Graduate School of Human and Environmental Studies,
Kyoto University, Kyoto, Japan
{
tsuiki,tsukamoto
}
@i.h.kyoto-u.ac.jp
Abstract.
Imaginary cubes are three-dimensional objects that have
square projections in three orthogonal ways, just like a cube has. In this
paper, we introduce higher-dimensional extensions of imaginary cubes
and study their properties.
1
Introduction
Imaginary cubes are three-dimensional objects that have square projections in
three orthogonal ways, just like a cube has [
1
]. A regular tetrahedron and a
cuboctahedron are examples of imaginary cubes (Fig.
1
(a,b)). There are two
imaginary cubes with remarkable geometric properties: a hexagonal bipyramid
imaginary cube (Fig.
1
(c); we simply call it an H) and a triangular antipris-
moid imaginary cube (Fig.
1
(d); we call it a T). Figure
2
shows how they can be
considered as imaginary cubes. The first author of this paper has studied imagi-
nary cubes, in particular minimal convex imaginary cubes and fractal imaginary
cubes. He has also designed sculptures and puzzles based on them [
1
-
4
].
In this paper, we study higher-dimensional extensions of imaginary cubes. In
particular, we study
n
-dimensional counterparts of regular tetrahedron, H, and
Tforeach
n
2, which we call S
n
,
H
n
, and T
n
, respectively. We also study
fractal imaginary cubes that correspond to these three series of polytopes.
In Sect.
2
, we review properties of imaginary cubes based on [
1
]. Then, we
study higher-dimensional extensions of them in Sect.
3
, and fractal imaginary
hypercubes in Sect.
4
.
≥
Objects and Polytopes
n
. Therefore,
Here, we only study imaginary cubes that are compact subsets of
R
n
in this paper. We say that
two objects are
similar
if one can be transformed to the other by scaling and
isometry. We call this equivalence class a
shape
. Each shape S is also regarded
as a name of an object, and we say that an object
A
is an S if
A
belongs to the
class S. We use roman font to denote a shape, but italic font is used for objects.
A
polytope
is a convex hull of a finite set of points in
an
object
means a non-empty compact subset of
R
n
. We denote by
conv(
A
) the convex hull of an object
A
, and by vert(
P
) the set of vertices of a
polytope
P
.A
facet
of an
n
-dimensional polytope
P
is an (
n
R
−
1)-dimensional
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