Dehn invariant

Defining the Dehn invariant in a way that can apply to all polyhedra simultaneously involves infinite-dimensional vector spaces (see § Full definition, below). However, when restricted to any particular example consisting of finitely many polyhedra, such as the Platonic solids, it can be defined in a simpler way, involving only a finite number of dimensions, as follows:[7]

• Determine the edge lengths and dihedral angles (the angle between two faces meeting along an edge) of all of the polyhedra.
• Find a subset of the angles that forms a rational basis. This means that each dihedral angle can be represented as a linear combination of basis elements, with rational number coefficients. Additionally, no rational linear combination of basis elements may sum to zero. Include ${\displaystyle \pi }$ (or a rational multiple of ${\displaystyle \pi }$) in this basis.
• For each edge of a polyhedron, represent its dihedral angle as a rational combination of angles from the basis. Discard the coefficient for the rational multiple of ${\displaystyle \pi }$ in this combination. Interpret the remaining coefficients as the coordinates of a vector whose dimensions represent basis angles, and scale this vector by the edge length.
• Sum the vectors for all edges of a polyhedron to produce its Dehn invariant.

Although this method involves arbitrary choices of basis elements, these choices affect only the coefficients by which the Dehn invariants are represented. As elements of an abstract vector space, they are unaffected by the choice of basis. The vector space spanned by the Dehn invariants of any finite set of polyhedra forms a finite-dimensional subspace of the infinite-dimensional vector space in which the Dehn invariants of all polyhedra are defined. The question of which combinations of dihedral angles are related by rational linear combinations is not always straightforward, and may involve nontrivial methods from number theory.[7]

For the five Platonic solids, the dihedral angles are:

• ${\displaystyle \theta _{\mathrm {tet} }=\arccos {\tfrac {1}{3}}\approx 70.5^{\circ }}$ for the tetrahedron.
• ${\displaystyle \theta _{\mathrm {cube} }=\pi /2=90^{\circ }}$, a right angle, for the cube.
• ${\displaystyle \theta _{\mathrm {oct} }=\arccos(-{\tfrac {1}{3}})\approx 109.5^{\circ }}$ for the octahedron.
• ${\displaystyle \theta _{\mathrm {dodec} }=2\arctan \varphi \approx 116.6^{\circ }}$ for the dodecahedron, where ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$ is the golden ratio.
• ${\displaystyle \theta _{\mathrm {icos} }=\arccos(-{\tfrac {1}{3}}{\sqrt {5}})\approx 138.2^{\circ }}$ for the icosahedron.

The dihedral angle of a cube is a rational multiple of ${\displaystyle \pi }$, but the rest are not. The dihedral angles of the regular tetrahedron and regular octahedron are supplementary angles: they sum to ${\displaystyle \pi }$. Omitting either the tetrahedron or the octahedron from these five angles produces a rational basis: there are no other rational relations between these angles.[7] If, for instance, the basis that omits ${\displaystyle \theta _{\mathrm {oct} }}$ is used, and ${\displaystyle \theta _{\mathrm {cube} }}$ is used as a basis element but then omitted (as a rational multiple of ${\displaystyle \pi }$) from the Dehn invariant calculation, then the remaining angle basis elements are ${\displaystyle \theta _{\mathrm {tet} }}$, ${\displaystyle \theta _{\mathrm {dodec} }}$, and ${\displaystyle \theta _{\mathrm {icos} }}$. The resulting Dehn invariants will have one dimension for each basis element. With this basis, for Platonic solids with edge length ${\displaystyle s}$, the Dehn invariants are:[lower-alpha 1]

• ${\displaystyle (6s,0,0)}$ for the tetrahedron. It has six edges of length ${\displaystyle s}$, with tetrahedral dihedral angles.
• ${\displaystyle (0,0,0)}$ for the cube. Its edges have dihedral angles that are expressed only in terms of ${\displaystyle \theta _{\mathrm {cube} }}$, omitted from the Dehn invariant.
• ${\displaystyle (-12s,0,0)}$ for the octahedron. Its twelve edges have dihedrals ${\displaystyle \theta _{\mathrm {oct} }=2\theta _{\mathrm {cube} }-\theta _{\mathrm {tet} }}$. In this combination, the coefficient for ${\displaystyle \theta _{\mathrm {cube} }}$ is discarded, leaving only a coefficient of ${\displaystyle -1}$ for ${\displaystyle \theta _{\mathrm {tet} }}$.
• ${\displaystyle (0,30s,0)}$ for the dodecahedron. It has 30 edges with dodecahedral dihedral angles.
• ${\displaystyle (0,0,30s)}$ for the icosahedron. It has 30 edges with icosahedral dihedral angles.

The cube is the only one of these whose Dehn invariant is zero. The Dehn invariants of each of the other four Platonic solids are unequal and nonzero. The Dehn invariant of the octahedron is ${\displaystyle -2}$ times the Dehn invariant of a tetrahedron of the same edge length.[7]

The Dehn invariant of any parallelepiped is zero, just as it is for the cube. Each set of four parallel edges in a parallelepiped have the same length and have dihedral angles summing to ${\displaystyle 2\pi }$, so their contributions to the Dehn invariant cancel out to zero.[8] The Dehn invariants of the other Archimedean solids can also be expressed as rational combinations of the invariants of the Platonic solids.[7] In terms of the same basis as before, with the same assumption that these shapes have edge length ${\displaystyle s}$, the Dehn invariants are:[lower-alpha 1]

• ${\displaystyle (-6s,0,0)}$ for the truncated tetrahedron.
• ${\displaystyle (12s,0,0)}$ for the truncated cube, rhombicuboctahedron, and cuboctahedron.
• ${\displaystyle (0,0,0)}$ for the truncated octahedron, which tiles space as the bitruncated cubic honeycomb.[9]
• ${\displaystyle (0,0,-30s)}$ for the truncated dodecahedron.
• ${\displaystyle (0,-30s,0)}$ for the truncated icosahedron.
• ${\displaystyle (0,-30s,-30s)}$ for the icosidodecahedron.
• ${\displaystyle (0,30s,30s)}$ for the rhombicosidodecahedron.
• ${\displaystyle (0,0,0)}$ for the truncated icosidodecahedron. This does not tile space directly, but as a zonohedron it can be partitioned into parallelepipeds, which do.[9][10]

The definition of the Dehn invariant requires a notion of a polyhedron for which the lengths and dihedral angles of edges are well defined. Most commonly, it applies to the polyhedra whose boundaries are piecewise linear manifolds, embedded on a finite number of planes in Euclidean space. However, the Dehn invariant has also been considered for polyhedra in spherical geometry or in hyperbolic space,[5] and for certain self-crossing polyhedra in Euclidean space.[18]

The values of the Dehn invariant belong to an abelian group[19] defined as the tensor product

$${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /2\pi \mathbb {Z} .}$$

The left factor of this tensor product is the set of real numbers (in this case representing lengths of edges of polyhedra) and the right factor represents dihedral angles in radians, given as numbers modulo rational multiples of 2π.[12] (Some sources take the angles modulo π instead of modulo 2π,[5][19][20] or divide the angles by π and use ${\displaystyle \mathbb {R} /\mathbb {Z} }$ in place of ${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$,[21] but this makes no difference to the resulting tensor product, as any rational multiple of π in the right factor becomes zero in the product.)

The Dehn invariant of a polyhedron with edge lengths ${\displaystyle \ell _{i}}$ and edge dihedral angles ${\displaystyle \theta _{i}}$ is the sum[12]

$${\displaystyle \sum _{i}\ell _{i}\otimes \theta _{i}.}$$

Its structure as a tensor gives the Dehn invariant additional properties that are geometrically meaningful. In particular, it has a tensor rank, the minimum number of terms ${\displaystyle \ell \otimes \theta }$ in any expression as a sum of such terms. Since the expression of the Dehn invariant as a sum over edges of a polyhedron has exactly this form, the rank of the Dehn invariant gives a lower bound on the minimum number of edges possible for any polyhedron resulting from a dissection of a given polyhedron.[22]

An alternative but equivalent description of the Dehn invariant involves the choice of a Hamel basis, an infinite subset ${\displaystyle B}$ of the real numbers such that every real number can be expressed uniquely as a sum of finitely many rational multiples of elements of ${\displaystyle B}$. Thus, as an additive group, ${\displaystyle \mathbb {R} }$ is isomorphic to ${\displaystyle \mathbb {Q} ^{(B)}}$, the direct sum of copies of ${\displaystyle \mathbb {Q} }$ with one summand for each element of ${\displaystyle B}$. If ${\displaystyle B}$ is chosen to have π (or a rational multiple of π) is one of its elements, and ${\displaystyle B'}$ is the rest of the basis with this element excluded, then the tensor product ${\displaystyle \mathbb {R} \otimes \mathbb {R} /2\pi \mathbb {Z} }$ can be described as the (infinite dimensional) real vector space ${\displaystyle \mathbb {R} ^{(B')}}$. The Dehn invariant can be expressed by decomposing each dihedral angle ${\displaystyle \theta _{i}}$ into a finite sum of basis elements

$${\displaystyle \theta _{i}=\sum _{j=0}^{k_{i}}q_{i,j}b_{i,j}}$$

where ${\displaystyle q_{i,j}}$ is rational, ${\displaystyle b_{i,j}}$ is one of the real numbers in the Hamel basis, and these basis elements are numbered so that ${\displaystyle b_{i,0}}$ is the rational multiple of π that belongs to ${\displaystyle B}$ but not ${\displaystyle B'}$. With this decomposition, the Dehn invariant is

$${\displaystyle \sum _{i}\sum _{j=1}^{k_{i}}\ell _{i}q_{i,j}e_{i,j},}$$

where each ${\displaystyle e_{i,j}}$ is the standard unit vector in ${\displaystyle \mathbb {R} ^{(B')}}$ corresponding to the basis element ${\displaystyle b_{i,j}}$. The sum here starts at ${\displaystyle j=1}$, to omit the term corresponding to the rational multiples of π.[23]

This alternative formulation shows that the values of the Dehn invariant can be given the additional structure of a real vector space.[24] Although, in general, the construction of Hamel bases involves the axiom of choice, this can be avoided (when considering any specific finite set of polyhedra) by restricting attention to the finite-dimensional vector space generated over ${\displaystyle \mathbb {Q} }$ by the dihedral angles of the polyhedra.[4]

For an ideal polyhedron in hyperbolic space, the edge lengths are infinite, making the usual definition of the Dehn invariant inapplicable. Nevertheless, the Dehn invariant can be extended to these polyhedra by using horospheres to truncate their vertices, and computing the Dehn invariant in the usual way for the resulting truncated shape, ignoring the extra curved edges created by this truncation process. The result does not depend on the choice of horospheres for the truncation, as long as each one cuts off only a single vertex of the given polyhedron.[25]