An ideal regular octahedron in the Poincaré ball model of hyperbolic space (sphere at infinity not shown). All dihedral angles of this shape are right angles.
Animation of an ideal icosahedron in the Klein model of hyperbolic space
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. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.
The Platonic solids and Archimedean solids have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra can be represented as ideal polyhedra – a polyhedron can be ideal only when it can be represented in Euclidean geometry with all its vertices on a circumscribed sphere. Using linear programming, it is possible to test whether a given polyhedron has an ideal version, in polynomial time.
Every two ideal polyhedra with the same number of vertices have the same surface area, and it is possible to calculate the volume of an ideal polyhedron using the Lobachevsky function. 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.
three-dimensional hyperbolic geometry, an idealpolyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than...
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many', and ἕδρον (-hedron) 'base, seat') is a three-dimensional shape...
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive...
geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere...
or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform...
approach it). In the hyperboloid model there are no ideal points. Ideal triangle Idealpolyhedron Points at infinity for uses in other geometries. Sibley...
to determine whether one polyhedron can be cut into pieces and reassembled ("dissected") into another, and whether a polyhedron or its dissections can tile...
star-honeycombs in H3: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical. Ideal vertices now appear when the vertex figure...
complexes in which every polyhedron is a simplex. Voronoi diagrams. Splines. A fan is a polyhedral complex in which every polyhedron is a cone from the origin...
submersion Regular polygons, polygons with all sides and angles equal Regular polyhedron, a generalization of a regular polygon to higher dimensions Regular polytope...
way of enlarging a polyhedron Augmentation (algebra), a certain algebra homomorphism Augmentation ideal, in mathematics, an ideal in a group ring Breast...
Automan's invulnerability. Cursor was his sidekick, a floating, shifting polyhedron which could "draw" and generate physical objects as needed. The most common...
uniformity when changing a spherical polyhedron to its planar counterpart can push faces through the centre of the polyhedron and back out the other side, changing...
the hyperplanes. A hyperplane H is called a "support" hyperplane of the polyhedron P if P is contained in one of the two closed half-spaces bounded by H...
semi-regular and stellated polyhedra have been made. A cuboid is a rectilinear polyhedron. That is, all its edges form right angles. Or in other words (in the majority...
construct a regular polyhedron is via a fold-out net. To obtain a fold-out net of a polyhedron, one takes the surface of the polyhedron and cuts it along...
near that of the virtual orthographic camera. Cube mapping and other polyhedron mappings address the severe distortion of sphere maps. If cube maps are...
flattened hexagons, like these Goldberg polyhedron G(2,0): There are also 9 Johnson solids with regular hexagons: The ideal crystalline structure of graphene...
replacement’. It uses a unique algorithm based on three multi-faceted polyhedrons floating in RGB colorspace that are used to isolate color regions in...
polyhedron with octahedral symmetry: Klein modeled the quartic by a shape with octahedral symmetries and with points at infinity (an "open polyhedron")...
uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families...
rather than each quarter-turn. The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the...
holomorphically convex. D is the union of an increasing sequence of analytic polyhedrons in D. D is pseudoconvex. D is Locally pseudoconvex. The implications...
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