Regular octahedron
In geometry, a regular octahedron is an eight-sided polyhedron with equilateral triangles as its faces. Known for its highly symmetrical form, the regular octahedron is a Platonic solid, and more generally, a regular polyhedron. If the faces are isosceles triangles, the regular octahedron becomes a square bipyramid. The regular octahedron is an example of many classifications as deltahedron and simplicial polyhedron.
Regular octahedra occur in nature and science, such as the crystal structures and in stereochemistry as a resemblance of a chemical molecule known as octahedral molecular geometry. Other appearances are popular culture and music theory. It can be the core of polyhedra construction, and it can tile with different polyhedra to create a honeycomb.
The vertices and edges of a regular octahedron give rise to a graph, a discrete structure drawn in a plane. It is an example of a four-connected simplicial well-covered graph. It is also one of the six connected graphs in which the neighborhood of every vertex is a cycle of length four or five. Within this structure, the graph forms a topological surface called a Whitney triangulation.
Description
The regular octahedron is a polyhedron with eight equilateral triangles, where each vertex is the meet of four edges and four faces. It is one of the Platonic solids, a set of convex polyhedra whose faces are congruent regular polygons. Platonic solids are the ancient set of five polyhedra named after Plato, relating them to classical elements in his Timaeus dialogue. The regular octahedron represents wind. Following his attribution with nature, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids. 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. The ordered solids started from the innermost to the outermost: regular octahedron, regular icosahedron, regular dodecahedron, regular tetrahedron, and cube.A regular octahedron is the cross-polytope in 3-dimensional space. It can be oriented and scaled so that its axes align with Cartesian coordinate axes and its vertices have coordinates,, and. Such an octahedron has edge length.
Symmetry and duality
The regular octahedron has three-dimensional symmetry groups, namely the octahedral symmetry. The regular octahedron has thirteen axes rotatonal symmetry: three axes of four-fold rotational symmetry passing through a pair of vertices oppositing each other, four axes of three-fold rotational symmetry passing through the center of opposite triangular faces, and six axes of two-fold rotational symmetry passing through the pair of opposite edges at their midpoints. Additionally, the regular octahedron has nine reflectional planes.The dual polyhedron can be obtained from each of the polyhedra's vertices tangent to a plane by a process known as polar reciprocation. One property of dual polyhedra is that the polyhedron and its dual share their three-dimensional symmetry point group. In the case of a regular octahedron, its dual polyhedron is the cube, and they have the same three-dimensional symmetry groups. Like its dual, the regular octahedron has three properties: any two faces, two vertices, and two edges are transformed by rotation and reflection under the symmetry orbit, such that the appearance remains unchanged; these are isohedral, isogonal, and isotoxal respectively. Hence, it is considered a regular polyhedron. Four triangles surround each vertex, so the regular octahedron is by vertex configuration or by Schläfli symbol.
Other properties
Measurements
The surface area of a regular octahedron can be ascertained by summing the area of all its eight equilateral triangles, whereas its volume is twice the volume of a square pyramid; if the edge length is,The radius of a circumscribed sphere , the radius of an inscribed sphere , and the radius of a midsphere , are:
The dihedral angle of a regular octahedron between two adjacent triangular faces is, which is about 109.47°. This can be obtained from the dihedral angle of an equilateral square pyramid: its dihedral angle between two adjacent triangular faces is the dihedral angle of an equilateral square pyramid between two adjacent triangular faces, and its dihedral angle between two adjacent triangular faces on the edge in which two equilateral square pyramids are attached is twice the dihedral angle of an equilateral square pyramid between its triangular face and its square base.
The regular octahedron has two types of closed geodesics. The closed geodesics are the paths on a regular octahedron's surface that are locally straight. In other words, they avoid the vertices, follow line segments across the faces that they cross, and form complementary angles on the two incident faces of each edge that they cross. These geodesics have the length of and.
Combinatorial structure
The skeleton of a regular octahedron is the graph formed by its vertices and edges. Steinitz's theorem guarantees that the this graph can be drawn with no edge crossing another, which is 3-connected. Being -connected means a graph remains connected whenever vertices are removed. Its graph called the octahedral graph, a Platonic graph.The octahedral graph is a complete tripartite graph. It means that the octahedral graph is partitioned into three independent sets, each consisting of two opposite vertices, and there exists an edge between every pair of vertices from different independent sets. It is an example of a Turán graph.
As a 4-connected simplicial, the octahedral graph is one of only four 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.
The octahedral graph is one of only six connected graphs in which the neighborhood of every vertex is a cycle of length four or five, the others being the Fritsch graph, the icosahedral graph, and the edge graphs of the pentagonal bipyramid, snub disphenoid and gyroelongated square bipyramid. More generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form a topological surface called a Whitney triangulation. These six graphs come from the six Whitney triangulations that, when their triangles are equilateral, have positive angular defect at every vertex. This makes them a combinatorial analogue of the positively curved smooth surfaces. They come from six of the eight deltahedra—excluding the two that have a vertex with a triangular neighborhood.
Other appearances
Beyond its existence as a Platonic solid, the regular octahedron appears in many fields, such as nature and science, popular culture, and music theory.In nature and science
The natural crystals with octahedral structures are commonly discovered in diamond, alum, and fluorite. The plates of kamacite alloy in octahedrite-structural meteorites are arranged parallel to the eight faces of an octahedron. Many metal ions coordinate six ligands in an octahedral or distorted octahedral configuration. Widmanstätten patterns in nickel-iron crystals.Octahedral molecular geometry is a chemical molecule resembling a regular octahedron in stereochemistry. This structure has a main-group element without an active lone pair, which can be described by a model that predicts the geometry of molecules known as VSEPR theory.
The regular octahedron is the known solution of a six-electron case in Thomson problem, concerning the minimum-energy configuration of charged particles on a sphere. The solution is done by placing the vertices of a regular octahedron inscribed in a sphere.
If each edge of an octahedron is replaced by a one-ohm resistor, the resistance between opposite vertices is ohm, and that between adjacent vertices ohm.
In popular culture
In roleplaying games, this solid is known as a "d8", one of the more common polyhedral dice.In music theory
The hexany is the octahedron's orthogonal projection. Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad.As other special cases
A regular octahedron is one of the eight convex deltahedra, polyhedra whose faces are all equilateral triangles. It is a composite polyhedron constructed by attaching two equilateral square pyramids base-to-base. When the square pyramids are a right pyramid, the regular octahedron becomes a square bipyramid, wherein its faces are all isosceles triangles. In the case of a square bipyramid, its dual is a square prism. Regardless of the different types of triangles, both a regular octahedron and a square bipyramid are examples of a simplicial polyhedron.The regular octahedron is a type of trigonal antiprism, formed by taking a trigonal prism with equilateral triangle bases and rectangular lateral faces, and replacing the rectangles by alternating isosceles triangles. In the case of the regular octahedron, all of the resulting faces are congruent equilateral triangles.
The regular octahedron can also be considered a rectified tetrahedron, sometimes called a tetratetrahedron ; if alternate faces are considered to have different types, the octahedron can be considered a type of quasiregular polyhedron, a polyhedron in which two different types of polygonal faces alternate around each vertex. It exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations 2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are Wythoff constructions within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.
A regular octahedron is the three-dimensional case of the more general concept of a cross-polytope.