Orthogonal polyhedron with no vertex visible from center

An orthogonal polyhedron is a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

in which all edges are parallel to the axes of a

Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...

, resulting in the

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

faces and implying the dihedral angle between faces are

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

s. The angle between Jessen's icosahedron's faces is right, but the edges are not axis-parallel, which is not an orthogonal polyhedron.

Polycube image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricube ...

s are a special case of orthogonal polyhedra that can be decomposed into identical cubes and are three-dimensional analogs of planar

polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popu ...

es. Orthogonal polyhedra can be either

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

(such as

rectangular cuboid A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these ...

s) or non-convex. Orthogonal polyhedra were used in in which he showed that any polyhedron is equivalent to a cube: it can be decomposed into pieces which later can be used to construct a cube. This showed the requirements for the polyhedral equivalence conditions by Dehn invariant. Orthogonal polyhedra may also be used in computational geometry, where their constrained structure has enabled advances in problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net. The simple orthogonal polyhedra, as defined by , are the three-dimensional polyhedra such that three mutually perpendicular edges meet at each vertex and that have the topology of a sphere. By using

Steinitz's theorem In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedron, convex polyhedra: they are exactly the vertex connect ...

, there are three different classes: the arbitrary orthogonal polyhedron, the skeleton of its polyhedron drawn with hidden vertex by the

isometric projection Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and ...

, and the polyhedron wherein each axis-parallel line through a vertex contains other vertices. All of these are polyhedral graphs that are

cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

and bipartite.

References

Further reading