WOLFRAM|DEMONSTRATIONS PROJECT

Dehn Invariant of Some Disjoint Unions of Polyhedra with Icosahedral Symmetry

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combination
dodecahedron
icosahedron
icosidodecahedron
truncated dodecahedron
truncated icosahedron
smallrhombicosidodecahedron
greatrhombicosidodecahedron
adjacent axes of m-fold, n-fold symmetry
tangent
dihedral angle
numerical
5, 5
2
π-
-1
tan
(2)
2.03444
3, 3
2
5
π
2
+
-1
tan
5
2
2.41186
3, 5
3-
5
1
4
3π+2
-1
tan
(2)-2
-1
tan
5
2
2.48923
2, 5
1
2
(-1+
5
)
π-
1
2
-1
tan
(2)
2.58802
2, 3
1
2
(3-
5
)
1
4
3π+2
-1
tan
5
2
2.77673
Dehn invariant
1
-30v(
-1
tan
(2))
2
30v
-1
tan
5
2
3
30v(
-1
tan
(2))-v
-1
tan
5
2
4
5
6
7
total: 0
5
2
Two sets
A
and
B
are equidecomposable (can be dissected into each other) if there are two families of sets
A
i
and
B
i
,
i=1,2,…,n
, such that
A=
n
⋃
i=1
A
i
, the interiors of the
A
i
are disjoint,
B=
n
⋃
i=1
B
i
, the interiors of the
B
i
are disjoint, and
A
i
≡
B
i
(
A
i
is congruent to
B
i
). More intuitively,
A
can be cut up into finitely many pieces that can be rearranged to form
B
; here the pieces should be polyhedra.
The Dehn invariant of a polyhedron is
n
∑
i=1
l
i
v(
ϕ
i
)
, where
l
i
is the length of the edge
i
,
ϕ
i
is the corresponding dihedral angle, and
v
is an additive functional defined on a certain finite-dimensional vector space of reals over the rationals for which
v(π)=0
[1]. A polyhedron has Dehn invariant 0 if and only if it is equidecomposable with a cube of same volume.
This Demonstration calculates Dehn invariants for disjoint unions of Platonic and Archimedean solids with icosahedral symmetry and edge length 1. In this case, dihedral angles are supplementary to angles between suitable pairs of adjacent axes of rotational symmetry. The disjoint union of an icosahedron, a dodecahedron, and an icosidodecahedron is an example of a combination of polyhedra with Dehn invariant 0. Find some other combinations with Dehn invariant 0.