Octahedron

One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

Many types of irregular octahedra also exist, including both convex and non-convex shapes.

A regular octahedron is convex, meaning that for any two points within it, the line segment connecting them lies entirely within it.

[4] This ancient set of polyhedrons was named after Plato who, in his Timaeus dialogue, related these solids to nature.

[5] Following its attribution with nature by Plato, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids.

[5] In his Mysterium Cosmographicum, Kepler also proposed the Solar System by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets.

These pyramids cover their square bases, so the resulting polyhedron has eight triangular faces.

of sixteen: the appearance is symmetrical by rotating around the axis of symmetry that passing through apices and base's center vertically, and it has mirror symmetry relative to any bisector of the base; it is also symmetrical by reflecting it across a horizontal plane.

of a regular octahedron can be ascertained by summing all of its eight equilateral triangles, whereas its volume

It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size.

The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.

One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of a regular icosahedron.

[18] The regular octahedron can be considered as the antiprism, a prism like polyhedron in which lateral faces are replaced by alternating equilateral triangles.

[19] Therefore, it has the property of quasiregular, a polyhedron in which two different polygonal faces are alternating and meet at a vertex.

[20] Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space.

The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement.

The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex.

The octahedron and its dual polytope, the cube, have the same symmetry group but different characteristic tetrahedra.

around its exterior right-triangle face (the edges opposite the characteristic angles 𝟀, 𝝉, 𝟁),[a] plus

Some of the polyhedrons do have eight faces aside from being square bipyramids in the following: The following polyhedra are combinatorially equivalent to the regular octahedron.

They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: A space frame of alternating tetrahedra and half-octahedra derived from the Tetrahedral-octahedral honeycomb was invented by Buckminster Fuller in the 1950s.

It is commonly regarded as the strongest building structure for resisting cantilever stresses.

A regular octahedron can be augmented into a tetrahedron by adding 4 tetrahedra on alternated faces.

It is also one of the simplest examples of a hypersimplex, a polytope formed by certain intersections of a hypercube with a hyperplane.

The octahedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.

The regular octahedron can also be considered a rectified tetrahedron – and can be called a tetratetrahedron.

Compare this truncation sequence between a tetrahedron and its dual: The above shapes may also be realized as slices orthogonal to the long diagonal of a tesseract.

The octahedron as a tetratetrahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane.

[29][30] As a trigonal antiprism, the octahedron is related to the hexagonal dihedral symmetry family.

The octahedron can be generated as the case of a 3D superellipsoid with all exponent values set to 1.