Regular octahedron

Measurements

The surface area of a regular octahedron ${\displaystyle A}$ can be ascertained by summing the area of all its eight equilateral triangles. For its volume ${\displaystyle V}$, one can cut the regular octahedron into two equilateral square pyramids (see § As other special cases), hence the volume is twice as the pyramids' volume by adding together. Let ${\displaystyle a}$ be the edge length of a regular octahedron, then its surface area and volume can be formulated as:^[4] $${\displaystyle A=2{\sqrt {3}}a^{2}\approx 3.464a^{2},\qquad V={\frac {1}{3}}{\sqrt {2}}a^{3}\approx 0.471a^{3}.}$$ The radius of a circumscribed sphere ${\displaystyle r_{u}}$ (one that touches the octahedron at all vertices), the radius of an inscribed sphere ${\displaystyle r_{i}}$ (one that tangent to each of the octahedron's faces), and the radius of a midsphere ${\displaystyle r_{m}}$ (one that touches the middle of each edge), are:^[5] $${\displaystyle r_{u}={\frac {\sqrt {2}}{2}}a\approx 0.707a,\qquad r_{i}={\frac {\sqrt {6}}{6}}a\approx 0.408a,\qquad r_{m}={\frac {1}{2}}a=0.5a.}$$ The dihedral angle of a regular octahedron is the angle between its two adjacent triangular faces. The angle can be obtained from the dihedral angle of an equilateral square pyramid. One can construct a regular octahedron by attaching two equilateral square pyramids base-to-base (see § As other special cases). For the pyramid, the dihedral angle between a triangle and a square is ${\displaystyle \arctan({\sqrt {2}})\approx 54.7^{\circ }}$. Therefore, for the regular octahedron, the dihedral angle between two adjacent triangles that can be made up by such an attachment is twice the square pyramid's square-to-triangle angle. The angle measurement is also equal to the square pyramid's two adjacent triangles' angle. That is:^[6] $${\displaystyle \arctan({\sqrt {2}})+\arctan({\sqrt {2}})=2\arctan({\sqrt {2}})=\arccos \left(-{\frac {1}{3}}\right)\approx 109.5^{\circ }.}$$

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 ${\displaystyle 3}$ and ${\displaystyle 2{\sqrt {3}}\approx 3.464}$.^[7]

The regular octahedron has a Rupert property, meaning another regular octahedron with the same or larger size can pass through a hole in it. The original name of this property is from Prince Rupert of the Rhine, who wagered whether a cube can pass through a hole in it. English mathematician John Wallis, who recounted the story, answered that it is possible, and the solution was improved by Dutch mathematician Pieter Nieuwland. His solution led to the geometric measurement of the largest polyhedron's hole, known as the "Nieuwland constant". Scriba (1968) discovered that both a regular octahedron and a regular tetrahedron have the Rupert property.^[8] The Nieuwland constant for the regular octahedron with a unit edge length is equal to the cube's, approximately ${\displaystyle 1.06}$.^[9]

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 (0°, 90°, 180°, and 270°) passing through a pair of vertices oppositing each other, four axes of three-fold rotational symmetry (0°, 120°, and 240°) passing through the center of opposite triangular faces, and six axes of two-fold rotational symmetry (0° and 180°) passing through the pair of opposite edges at their midpoints.^[10] Additionally, the regular octahedron has nine reflectional planes. Each of the three planes passes through four vertices on each equator, and each of the six planes passes through the pair of opposite vertices and the center of the pair of opposite edges.^[11]

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.^[12] Like its dual, the regular octahedron has three properties: any two faces, two vertices, or 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 ${\displaystyle 3.3.3.3}$ by vertex configuration or ${\displaystyle \{3,4\}}$ by Schläfli symbol.^[13]

Combinatorial structure

The regular octahedron can be drawn into a graph, a structure in graph theory consisting of a set of vertices that are connected with an edge. This is possible because of Steinitz's theorem, which states that a graph can be represented as the vertex-edge graph of a polyhedron, provided it satisfies the following properties. It must be planar (where no edges are crossing each other) and 3-connected (being ${\displaystyle k}$-connected means a graph remains connected whenever ${\displaystyle k-1}$ vertices are removed).^[14][15] Its graph called the octahedral graph, a Platonic graph.^[1] It has the same number of vertices and edges as the regular octahedron, six vertices and twelve edges.

Six vertices of the octahedral graph can be partitioned into three independent sets, which contain different pairs of two opposite vertices. Hence, it is a complete tripartite graph, designated as ${\displaystyle K_{2,2,2}}$.^[16] It is an example of a Turán graph ${\displaystyle T_{6,3}}$.^[17] It has three boxicities that represent abstract structures' graph by the intersection of axis-parallel boxes in the minimum dimensional Euclidean space.^{[citation needed]}

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 (i.e., the same number of edges). 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.^[18]

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.^[19]

In nature and science

The natural crystals with octahedral structures are commonly discovered in diamond,^[20] alum,^[21] pyrite (although there are more polyhedral structures),^[22] 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.^{[citation needed]} 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.^[23]

The radiolarian Circoporus octahedrus has an octahedral shape.^[24]

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.^[25]

If each edge of an octahedron is replaced by a one-ohm resistor, the resistance between opposite vertices is ${\textstyle {\frac {1}{2}}}$ ohm, and that between adjacent vertices ${\textstyle {\frac {5}{12}}}$ ohm.^[26]

In popular culture

In roleplaying games, this solid is known as a "d8", one of the more common polyhedral dice.^[27]

In music theory

The regular octahedron is used in musical tuning systems known as the hexany. Invented by Mexican–American music theorist Erv Wilson, the hexany uses 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 (a set of two notes or pitch) and each face represents a consonant triad (a set of three dittos).^[28]

In construction of polyhedra

The regular octahedron represents the central intersection of two tetrahedra

The truncated octahedron by removing the vertices of a regular octahedron

The triakis octahedron by attaching triangular pyramids on each face

Several constructions of polyhedra commence from the regular octahedron.

• The interior of the compound of two dual tetrahedra is an octahedron, and this compound—called the stella octangula—is its first and only stellation.^[38] Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The regular octahedron's vertices lie at the midpoints of the edges of the tetrahedron, and in this sense, it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids.
• The truncated octahedron is an Archimedean solid, constructed by removing all of the regular octahedron's vertices, resulting in six squares and eight hexagons, leaving out six square pyramids.^[39]
• The triakis octahedron is a Catalan solid, the Kleetope of a regular octahedron, by attaching triangular pyramids onto its faces, topologically similar to the stellated octahedron.^[40]
• The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. It has four of the triangular faces and three central squares.^[41]
• One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. Five octahedra define any given icosahedron in this fashion, and together they define a regular compound. A regular icosahedron produced this way is called a "snub octahedron".^[42]

Honeycomb

Tetrahedral-octahedral honeycomb by regular octahedra and tetrahedra

Rectified cubic honeycomb by regular octahedra and cuboctahedra

The Dehn invariant of a regular octahedron can be defined as a tensor product of the edge length and the dihedral angle of a regular octahedron, $${\displaystyle 12a\otimes \arccos \left(-{\frac {1}{3}}\right),}$$ which is non-zero.^{[citation needed]} Every polyhedron with a Dehn invariant of zero can tile a space with its copy by attaching its faces to another, forming a honeycomb. The regular octahedron, nevertheless, cannot tile a space. Instead, two different polyhedra that join to tile the space can have the Dehn invariant of zero.^[43] In the case of a regular octahedron, it can tile alternately to regular tetrahedra to form a vertex, edge, and face-uniform tessellation of space, which is named tetrahedral-octahedral honeycomb.^[44] R. Buckminster Fuller, in the 1950s, applied these alternating polyhedra as a space frame, which developed the strongest building structure for resisting cantilever stresses.^[45] Another honeycomb is tesselating the regular octahedra alternately with cuboctahedra, named the rectified cubic honeycomb.^[46]

Miscellanea

The regular octahedron has eleven different nets. A net is the arrangement of eight equilateral triangles that can be folded by edge-joining to become the faces of a regular octahedron.^[47]

A regular octahedron is the cross-polytope in three-dimensional space. It can be oriented and scaled so that its axes align with Cartesian coordinate axes and its vertices have coordinates ${\displaystyle (\pm 1,0,0)}$, ${\displaystyle (0,\pm 1,0)}$, and ${\displaystyle (0,0,\pm 1)}$. Such an octahedron has edge length ⁠${\displaystyle {\sqrt {2}}}$⁠.^[48] The regular octahedron is a Hanner polytope, because it can be constructed by using the direct sum of three line segments. Its dual polyhedron, the cube, is constructed by the Cartesian product of three line segments.^[49] More generally, every cross-polytope and its dual, hypercube, in any higher-dimensional space are Hanner polytope.

Compound of octahedra

The polyhedral compounds, in which the regular octahedra share the same centre, are uniform polyhedron compounds, meaning they are polyhedral compounds whose constituents are identical—although possibly enantiomorphous—uniform polyhedra, in an arrangement that is also uniform. The list of compounds enumerated by Skilling (1976) for the compound of four octahedra (with its rotational freedom), eight octahedra, twenty octahedra (with its rotational freedom), two different compound of five octahedra, and compound of five octahedra.^[50] The compound of three octahedra appeared in the 15th-century manuscript De quinque corporibus regularibus by Piero della Francesca in which the compound is drawn circumscribed around a cube (although it does not depict the compound),^[51] in the model photograph of 1900 Max Brückner's mathematical literature alongside the explanation of the compound,^[52][53] and 1948 M. C. Escher's wood engraving Stars in which used as the central figure of the woodcut a cage in this shape that containing two chameleons and floating through space.^[54]

The spherical octahedron represents a regular octahedron projected to a sphere, a part of spherical polyhedron. There are eight spherical triangles,^[55] formed by the arc of great circles. Fuller identified that there are 25 great circles.

A regular octahedron is a 3-ball in the Manhattan (ℓ₁) metric.

Notes