​
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