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 in 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. The name is octahedral graph. The octahedral graph 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.
Properties A regular octahedron is a polyhedron with eight equilateral triangles. Each vertex is the meet of four edges and four faces. Hence, the regular octahedron has eight faces, twelve edges and six vertices. It is a convex polyhedron, and like any convex polyhedron, it has Euler's characteristic of 2, according to the formula V − E + F = 2 {\displaystyle V-E+F=2} ; the three letters denote respectively the number of vertices, edges, and faces.
A regular octahedron 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.
Measurements
The surface area of a regular octahedron A {\displaystyle A} can be ascertained by summing the area of all its eight equilateral triangles. For its volume V {\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 a {\displaystyle a} be the edge length of a regular octahedron, then its surface area and volume can be formulated as:
A = 2 3 a 2 ≈ 3.464 a 2 , V = 1 3 2 a 3 ≈ 0.471 a 3 . {\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 r u {\displaystyle r_{u}} (one that touches the octahedron at all vertices), the radius of an inscribed sphere r i {\displaystyle r_{i}} (one that tangent to each of the octahedron's faces), and the radius of a midsphere r m {\displaystyle r_{m}} (one that touches the middle of each edge), are:
r u = 2 2 a ≈ 0.707 a , r i = 6 6 a ≈ 0.408 a , r m = 1 2 a = 0.5 a . {\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 arctan ( 2 ) ≈ 54.7 ∘ {\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:
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