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d
c
a
b
(2)
(1)
(3)
(4)
(5)
Fig. 6.
(1)
d
(
P
) of a rectangular parallelepiped
P
; (2) a cut 2-complex which includes
arectangle
abcd
in
P
∗
; (3) the unfolding of
d
(
P
) by the cut 2-complex showed in (2);
(4) another cut 2-complex of
d
(
P
)when
abcd
in (2) is a parallelogram whose vertices
are on edges of
P
; (5) the unfolding of
d
(
P
) by the cut 2-complex showed in (4).
edge is parallel to an edge of
P
(see Figs.
6
(1) and (2)), and Fig.
6
(3)isthe
corresponding unfolding of
d
(
P
)
. The rectangle
R
=
abcd
may degenerate
to a line segment if
a
=
b
and
c
=
d
, and to a point if
a
=
b
=
c
=
d
.
Otherwise,
is composed of four-sided faces and the intersection of
P
∗
with
a plane intersecting all four side faces (see Figs.
6
(4) and (5)).
(2) If
P
is a triangular prism,
C
C
may include a triangle
abc
which is similar
to the triangular face of
P
.If
a
=
b
=
c
,
may include a line segment (
ab
possible
a
=
b
) parallel to side edges of
P
(see Fig.
7
). Otherwise,
C
is com-
posed of three side faces and the intersection of
P
∗
with a plane intersecting
all three side faces. The resulting figures are similar to the ones shown in
Figs.
6
(4) and (5).
(3) If
P
is a tetrahedron,
C
C
may have one vertex in the interior of
P
∗
(see Fig.
5
).
In Proposition, any unfolding
W
of
d
(
P
) is a space-filler, and the type of the
tiling depends on
P
. For example, if
P
is a rectangular parallelepiped, the set of
W
and three congruent copies of
W
obtained by rotations with angle
ˀ
about
three edges of
P
, tiles the space by its translations (see [
8
]).
3 Theorems and Corollaries
In this section, we assume an unfolding
W
of
d
(
P
) of a polyhedron
P
includes
the original
P
(i.e., the corresponding cut 2-complex of
W
is contained in
P
∗
).
Theorem 1.
Let
P
be a reflective space-filler whose mirror image is strictly
congruent to
P
, that is,
P
be one of the following;
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