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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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