In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space.
It can be defined as the convex hull of a finite set of ideal points.
Using linear programming, it is possible to test whether a polyhedron has an ideal version, in polynomial time.
The surface of an ideal polyhedron forms a hyperbolic manifold, topologically equivalent to a punctured sphere, and every such manifold forms the surface of a unique ideal polyhedron.
An ideal polyhedron can be constructed as the convex hull of a finite set of ideal points of hyperbolic space, whenever the points do not all lie on a single plane.
Alternatively, any Euclidean convex polyhedron that has a circumscribed sphere can be reinterpreted as an ideal polyhedron by interpreting the interior of the sphere as a Klein model for hyperbolic space.
However, another highly symmetric class of polyhedra, the Catalan solids, do not all have ideal forms.
Catalan solids that cannot be ideal include the rhombic dodecahedron and the triakis tetrahedron.
[6] If a simplicial polyhedron (one with all faces triangles) has all vertex degrees between four and six (inclusive) then it has an ideal representation, but the triakis tetrahedron is simplicial and non-ideal, and the 4-regular non-ideal example above shows that for non-simplicial polyhedra, having all degrees in this range does not guarantee an ideal realization.
, an ideal regular octahedron or cuboctahedron, with four edges per vertex, has dihedral angles
, and an ideal regular icosahedron, with five edges per vertex, has dihedral angles
This difficulty can be avoided by using a horosphere to truncate each vertex, leaving a finite length along each edge.
Because of the way the Dehn invariant is defined, and the constraints on the dihedral angles meeting at a single vertex of an ideal polyhedron, the result of this calculation does not depend on the choice of horospheres used to truncate the vertices.
[14] For example, the rhombic dodecahedron is bipartite, but has an independent set with more than half of its vertices, and the triakis tetrahedron has an independent set of exactly half the vertices but is not bipartite, so neither can be realized as an ideal polyhedron.
The geometric characterization of inscribed polyhedra was attempted, unsuccessfully, by René Descartes in his c.1630 manuscript De solidorum elementis.
, while the supplementary angles crossed by any Jordan curve on the surface of the polyhedron that has more than one vertex on both of its sides must be larger.
but the four angles crossed by a curve midway between two opposite faces sum to
When such an assignment exists, there is a unique ideal polyhedron whose dihedral angles are supplementary to these numbers.
As a consequence of this characterization, realizability as an ideal polyhedron can be expressed as a linear program with exponentially many constraints (one for each non-facial cycle), and tested in polynomial time using the ellipsoid algorithm.
from the graph leaves a number of connected components that is strictly smaller than
[17] Because the ideal regular tetrahedron, cube, octahedron, and dodecahedron all have dihedral angles that are integer fractions of
[18] In this they differ from the Euclidean regular solids, among which only the cube can tile space.
[20] The universal cover of the manifold inherits the same decomposition, which forms a honeycomb of ideal polyhedra.
[22] These two honeycombs, and three others using the ideal cuboctahedron, triangular prism, and truncated tetrahedron, arise in the study of the Bianchi groups, and come from cusped manifolds formed as quotients of hyperbolic space by subgroups of Bianchi groups.
[23] The surface of an ideal polyhedron (not including its vertices) forms a manifold, topologically equivalent to a punctured sphere, with a uniform two-dimensional hyperbolic geometry; the folds of the surface in its embedding into hyperbolic space are not detectable as folds in the intrinsic geometry of the surface.
Because this surface can be partitioned into ideal triangles, its total area is finite.
Conversely, and analogously to Alexandrov's uniqueness theorem, every two-dimensional manifold with uniform hyperbolic geometry and finite area, combinatorially equivalent to a finitely-punctured sphere, can be realized as the surface of an ideal polyhedron.
(As with Alexandrov's theorem, such surfaces must be allowed to include ideal dihedra.
)[24] From this point of view, the theory of ideal polyhedra has close connections with discrete approximations to conformal maps.
In this respect, ideal polyhedra are different from Euclidean polyhedra (and from their Euclidean Klein models): for instance, on a Euclidean cube, any geodesic can cross at most two edges incident to a single vertex consecutively, before crossing a non-incident edge, but geodesics on the ideal cube are not limited in this way.