Polyhedron

An abstract polyhedron is a certain kind of partially ordered set (poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, etc.) of a polyhedron.

An extra axial straw is added which doesn't exist in the simple polyhedron An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.

A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface.

If a polyhedron is self-dual, then the compound of the polyhedron with its dual will comprise congruent polyhedra.

If we reciprocate such a polyhedron about its intersphere, the dual polyhedron will share the same edge-tangency points and so must also be canonical; it is the canonical dual, and the two together form a canonical dual pair.

In 1750 the German Leonhard Euler for the first time considered the edges of a polyhedron, allowing him to discover his polyhedron formula relating the number of vertices, edges and faces.

One modern approach treats a geometric polyhedron as an injection into real space, a realisation, of some abstract polyhedron.

Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.

Specifically, for any problem, the convex hull of the solutions is an integral polyhedron; if this polyhedron has a nice/compact description, then we can efficiently find the optimal feasible solution under any linear objective.

The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces; the dual of an isotoxal polyhedron (having equivalent edges) is also isotoxal.

Polyhedron: green, 14-hedron; red, 15-hedron.

A complex polyhedron is mathematically more closely related to configurations than to real polyhedra.

A global polyhedron of permanently operating GPS stations under the auspices of the IERS is used to define a single global, geocentric reference frame which serves as the "zero order" global reference to which national measurements are attached.

All of them can be seen to be related to a regular or uniform polyhedron by gyration, diminishment, or dissection.

Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities.

Another convex polyhedron is formed by the small central space common to all members of the compound.

Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset.

As observed by Edmonds and Giles in 1977, one can equivalently say that the polyhedron is integral if for every bounded feasible integral objective function c, the optimal value of the linear program is an integer.

A spherical 180° × 360° panorama can be projected onto any polyhedron; but the rhombicuboctahedron provides a good enough approximation of a sphere while being easy to build.

A symmetrical polyhedron can be rotated and superimposed on its original position such that its faces and so on have changed position.