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(a)
(b)
(c)
(d)
Fig. 1.
Examples of imaginary cubes: (a) regular tetrahedron, (b) cuboctahedron, (c)
H: hexagonal bipyramid with 12 isosceles triangle faces with a height 3/2 of the base,
(d) T: triangular antiprismoid obtained by
t
runcating the three vertices of a base of a
regular triangular prism whose height is
√
6
/
4ofanedge.
(a)
(b)
(c)
(d)
Fig. 2.
Imaginary cubes in Fig.
1
placed in cubes.
face of
P
. We simply call an
n
-dimensional hypercube an
n
-
cube
. We refer the
reader to [
5
] for background material on polytopes.
For any two objects
A
and
B
, and for any scalar
c
∈
R
, we set their Minkowski
n
sum
A
+
B
=
{
a
+
b
∈
R
|
a
∈
A,
b
∈
B
}
, and scaling
cA
=
{
c
a
|
a
∈
A
}
.In
n
, and “
n
.
this paper,
1
is the vector (1
,...,
1)
∈
R
·
” is the dot product on
R
2
Imaginary Cubes
Imaginary cubes are three-dimensional objects with square projections in three
orthogonal ways. Note that a regular octahedron also has square projections in
three orthogonal ways, but its square projections are arranged differently. We
exclude such a case by defining an imaginary cube more precisely as follows.
Definition 1.
Let
C
be a 3-cube, and
A
be an object.
1.
A
is an
imaginary cube of C
if
A
has the same three square projections as
C
has.
2.
A
is an
imaginary cube
if it is an imaginary cube of a cube.
3.
A
is a
minimal convex imaginary cube (MCI for short) of C
if
A
is minimal
among convex imaginary cubes of
C
.
4.
A
is an
MCI
if it is an MCI of a cube.
It is clear that a convex object
A
is an imaginary cube of
C
if and only if
each edge of
C
contains at least one point of
A
. Therefore, an MCI of
C
is a
convex hull of some points of the edges of
C
, and thus it is a polytope.
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