WOLFRAM|DEMONSTRATIONS PROJECT

Dehn Invariant of Some Disjoint Unions of Polyhedra with Octahedral Symmetry

​
polyhedron P
Dehn invariant
copies of P
contribution
1
tetrahedron
-12v
-1
tan

2

0
1
2
0
2
octahedron
24v
-1
tan

2

0
1
2
0
3
truncated tetrahedron
12v
-1
tan

2

0
1
2
0
4
cuboctahedron
-24v
-1
tan

2

0
1
2
0
5
truncated cube
-24v
-1
tan

2

0
1
2
0
6
truncated octahedron
0
0
1
2
0
7
small rhombicuboctahedron
24v
-1
tan

2

0
1
2
0
8
great rhombicuboctahedron
0
0
1
2
0
total
0
adjacent axes of m-fold, n-fold symmetry
tangent
dihedral angle
numerical
4, 4
∞
π
2
1.5708
3, 3, A
2
2
2
-1
tan

2

1.91063
3, 3, B
-2
2
π-2
-1
tan

2

1.23096
4, 3
2
π-
-1
tan

2

2.18628
4, 2
1
3π
4
2.58802
3, 2
1
2
π
2
+
-1
tan

2

2.52611
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 having octahedral symmetry (and edge length 1). In this case, the dihedral angles are supplementary to the angles between suitable axes of rotational symmetry.
The disjoint union of a tetrahedron and a truncated tetrahedron has Dehn invariant 0.