Dehn invariant

Dissection of a square and equilateral triangle into each other. No such dissection exists for the cube and regular tetrahedron.

In two dimensions, the Wallace–Bolyai–Gerwien theorem states that any two polygons of equal area can be cut up into polygonal pieces and reassembled into each other. David Hilbert became interested in this result as a way to axiomatize area, in connection with Hilbert's axioms for Euclidean geometry. In Hilbert's third problem, he posed the question of whether two polyhedra of equal volumes can always be cut into polyhedral pieces and reassembled into each other. Hilbert's student Max Dehn, in his 1900 habilitation thesis, invented the Dehn invariant in order to prove that this is not always possible, providing a negative solution to Hilbert's problem. Although Dehn formulated his invariant differently, the modern approach is to describe it as a value in a tensor product, following Jessen (1968).[1][2]

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 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,[1] and for certain self-crossing polyhedra in Euclidean space.[3]

The values of the Dehn invariant belong to an abelian group[4] 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 2π.[5] (Some sources take the angles modulo π instead of modulo 2π,[1][4][6] or divide the angles by π and use ${\displaystyle \mathbb {R} /\mathbb {Z} }$ in place of ${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$[7] 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[5]

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

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 carefully so that 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 ${\displaystyle \mathbb {R} \otimes \mathbb {R} /2\pi \mathbb {Z} }$ is 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 ${\displaystyle b_{i,j}}$. Note that the sum here starts at ${\displaystyle j=1}$, to omit the term corresponding to the rational multiples of π.[8]

Although the Hamel basis formulation appears to involve 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.[9] This alternative formulation shows that the values of the Dehn invariant can be given the additional structure of a real vector space.

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 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.[10]

The Platonic solids each have uniform edge lengths and dihedral angles, none of which are rational multiples of each other. The dihedral angle of a cube, π/2, is a rational multiple of π, but the rest are not. The dihedral angles of the regular tetrahedron and regular octahedron are supplementary: they sum to π.[11]

In the Hamel basis formulation of the Dehn invariant, one can choose four of these dihedral angles as part of the Hamel basis. The angle of the cube, π/2, is the basis element that is discarded in the formula for the Dehn invariant, so the Dehn invariant of the cube is zero. More generally, the Dehn invariant of any parallelepiped is also zero.[12] Only one of the two angles of the tetrahedron and octahedron can be included, as the other one is a rational combination of the one that is included and the angle of the cube. The Dehn invariants of each of the other Platonic solids will be a vector in ${\displaystyle \mathbb {R} ^{B'}}$ formed by multiplying the unit vector for that solid's angle by the length and number of edges of the solid. No matter how they are scaled by different edge lengths, the tetrahedron, icosahedron, and dodecahedron all have Dehn invariants that form vectors pointing in different directions, and therefore are unequal and nonzero.[13]

The negated dihedral angle of the octahedron differs from the angle of a tetrahedron by an integer multiple of π, and in addition the octahedron has two times as many edges as the tetrahedron (twelve instead of six). Therefore, the Dehn invariant of the octahedron is −2 times the Dehn invariant of a tetrahedron of the same edge length. The Dehn invariants of the other Archimedean solids can also be expressed as rational combinations of the invariants of the Platonic solids.[13]

Unsolved problem in mathematics: Is there a dissection between every pair of spherical or hyperbolic polyhedra with the same volume and Dehn invariant as each other? (more unsolved problems in mathematics)

As Dehn (1901) observed, the Dehn invariant is an invariant for the dissection of polyhedra, in the sense that cutting up a polyhedron into smaller polyhedral pieces and then reassembling them into a different polyhedron does not change the Dehn invariant of the result. Another such invariant is the volume of the polyhedron. Therefore, if it is possible to dissect one polyhedron P into a different polyhedron Q, then both P and Q must have the same Dehn invariant as well as the same volume.[14] Sydler (1965) extended this result by proving that the volume and the Dehn invariant are the only invariants for this problem. If P and Q both have the same volume and the same Dehn invariant, it is always possible to dissect one into the other.[5][15]

Dehn's result continues to be valid for spherical geometry and hyperbolic geometry. In both of those geometries, two polyhedra that can be cut and reassembled into each other must have the same Dehn invariant. However, as Jessen observed, the extension of Sydler's result to spherical or hyperbolic geometry remains open: it is not known whether two spherical or hyperbolic polyhedra with the same volume and the same Dehn invariant can always be cut and reassembled into each other.[16] Every hyperbolic manifold with finite volume can be cut along geodesic surfaces into a hyperbolic polyhedron, which necessarily has zero Dehn invariant.[17]

The Dehn invariant also controls the ability of a polyhedron to tile space (part of the subject of Hilbert's eighteenth problem). Every space-filling tile has Dehn invariant zero, like the cube.[18][19] The reverse of this is not true – there exist polyhedra with Dehn invariant zero that do not tile space, but they can always be dissected into another shape (the cube) that does tile space.

More generally, if some combination of polyhedra jointly tiles space, then the sum of their Dehn invariants (taken in the same proportion) must be zero. For instance, the tetrahedral-octahedral honeycomb is a tiling of space by tetrahedra and octahedra (with twice as many tetrahedra as octahedra), corresponding to the fact that the sum of the Dehn invariants of an octahedron and two tetrahedra (with the same side lengths) is zero.[20]

Although the Dehn invariant takes values in ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /2\pi \mathbb {Z} ,}$ not all of the elements in this space can be realized as the Dehn invariants of polyhedra. The Dehn invariants of Euclidean polyhedra form a linear subspace of ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /2\pi \mathbb {Z} }$: one can add the Dehn invariants of polyhedra by taking the disjoint union of the polyhedra (or gluing them together on a face), negate Dehn invariants by making holes in the shape of the polyhedron into large cubes, and multiply the Dehn invariant by any scalar by scaling the polyhedron by the same number. The question of which elements of ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /2\pi \mathbb {Z} ,}$ (or, equivalently, ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /\mathbb {Z} }$) are realizable was clarified by the work of Dupont and Sah, who showed the existence of the following short exact sequence of abelian groups (not vector spaces) involving group homology:[21]

${\displaystyle 0\to H_{2}(\operatorname {SO} (3),\mathbb {R} ^{3})\to {\mathcal {P}}(E^{3})/{\mathcal {Z}}(E^{3})\to \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /\mathbb {Z} \to H_{1}(\operatorname {SO} (3),\mathbb {R} ^{3})\to 0}$

Here, the notation ${\displaystyle {\mathcal {P}}(E^{3})}$ represents the free abelian group over Euclidean polyhedra modulo certain relations derived from pairs of polyhedra that can be dissected into each other. ${\displaystyle {\mathcal {Z}}(E^{3})}$ is the subgroup generated in this group by the triangular prisms, and is used here to represent volume (as each real number is the volume of exactly one element of this group). The map from the group of polyhedra to ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /\mathbb {Z} }$ is the Dehn invariant. ${\displaystyle \operatorname {SO} (3)}$ is the Euclidean point rotation group, and ${\displaystyle H}$ is the group homology. Sydler's theorem that volume and the Dehn invariant are the only invariants for Euclidean dissection is represented homologically by the statement that the group ${\displaystyle H_{2}(\operatorname {SO} (3),\mathbb {R} ^{3})}$ appearing in this sequence is actually zero. If it were nonzero, its image in the group of polyhedra would give a family of polyhedra that are not dissectable to a cube of the same volume but that have zero Dehn invariant. By Sydler's theorem, such polyhedra do not exist.[21]

The group ${\displaystyle H_{1}(\operatorname {SO} (3),\mathbb {R} ^{3})}$ appearing towards the right of the exact sequence is isomorphic to the group ${\displaystyle \Omega _{\mathbb {R} }^{1}}$ of Kähler differentials, and the map from tensor products of lengths and angles to Kähler differentials is given by

${\displaystyle \ell \otimes \theta /\pi \mapsto \ell {\frac {d\cos \theta }{\sin \theta }},}$

where ${\displaystyle d}$ is the universal derivation of ${\displaystyle \Omega _{\mathbb {R} }^{1}}$. This group ${\displaystyle H_{1}(\operatorname {SO} (3),\mathbb {R} ^{3})=\Omega _{\mathbb {R} }^{1}}$ is an obstacle to realizability: its nonzero elements come from elements of ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /\mathbb {Z} }$ that cannot be realized as Dehn invariants.[22]

Analogously, in hyperbolic or spherical space, the realizable Dehn invariants do not necessarily form a vector space, because scalar multiplication is no longer possible, but they still form a subgroup. Dupont and Sah prove the existence of the exact sequences[21]

${\displaystyle 0\to H_{3}(\operatorname {SL} (2,\mathbb {C} ),\mathbb {Z} )^{-}\to {\mathcal {P}}({\mathcal {H}}^{3})\to \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /\mathbb {Z} \to H_{2}(\operatorname {SL} (2,\mathbb {C} ),\mathbb {Z} )^{-}\to 0}$

and

${\displaystyle 0\to H_{3}(\operatorname {SU} (2),\mathbb {Z} )\to {\mathcal {P}}(S^{3})/\mathbb {Z} \to \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /\mathbb {Z} \to H_{2}(\operatorname {SU} (2),\mathbb {Z} )\to 0.}$

Here ${\displaystyle \operatorname {SL} }$ denotes the special linear group, and ${\displaystyle \operatorname {SL} (2,\mathbb {C} )}$ is the group of Möbius transformations; the superscript minus-sign indicates the (−1)-eigenspace for the involution induced by complex conjugation. ${\displaystyle \operatorname {SU} }$ denotes the special unitary group. The subgroup ${\displaystyle \mathbb {Z} }$ in ${\displaystyle {\mathcal {P}}(S^{3})/\mathbb {Z} }$ is the group generated by the whole sphere.[21] Again, the rightmost nonzero group in these sequences is the obstacle to realizability of a value in ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} /\mathbb {Z} }$ as a Dehn invariant.

This algebraic view of the Dehn invariant can be extended to higher dimensions, where it has a motivic interpretation involving algebraic K-theory.[17]

An approach very similar to the Dehn invariant can be used to determine whether two rectilinear polygons can be dissected into each other only using axis-parallel cuts and translations (rather than cuts at arbitrary angles and rotations). An invariant for this kind of dissection uses the tensor product ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} }\mathbb {R} }$ where the left and right terms in the product represent height and width of rectangles. The invariant for any given polygon is calculated by cutting the polygon into rectangles, taking the tensor product of the height and width of each rectangle, and adding the results. Again, a dissection is possible if and only if two polygons have the same area and the same invariant.[6][9]

Flexible polyhedra are a class of polyhedra that can undergo a continuous motion that preserves the shape of their faces. By Cauchy's rigidity theorem, they must be non-convex, and it is known (the "bellows theorem") that the volume of the polyhedron must stay constant throughout this motion. A stronger version of this theorem states that the Dehn invariant of such a polyhedron must also remain invariant throughout any continuous motion. This result is called the "strong bellows theorem". It has been proven for all non-self-intersecting flexible polyhedra.[23] However, for more complicated flexible polyhedra with self-intersections the Dehn invariant may change continuously as the polyhedron flexes.[24]

The total mean curvature of a polyhedral surface has been defined as the sum over the edges of the edge lengths multiplied by the exterior dihedral angles. Thus (for polyhedra without rational angles) it is a linear function of the Dehn invariant, although it does not provide full information about the Dehn invariant. It has been proven to remain constant for any flexing polyhedron.[25]

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