octahedron
octahedron  

Type of side surfaces  equilateral triangles 
Number of faces  8th 
Number of corners  6th 
Number of edges  12 
Schläfli icon  {3.4} 
dual to  Hexahedron (cube) 
Body net  
Number of different networks  11 
Number of edges in a corner  4th 
Number of corners of a surface  3 
The (also, especially Austrian : the) octahedron [ ɔktaˈeːdɐ ] (from ancient Greek ὀκτάεδρος oktáedros , German 'eightsided' ) is one of the five Platonic solids , more precisely a regular polyhedron ( polyhedron , polyhedron ) with
 8 congruent equilateral triangles as side surfaces
 12 edges of equal length and
 6 corners in which four sides meet
The octahedron is both an equilateral foursided double pyramid with a square base and an equilateral antiprism with an equilateral triangle as a base .
symmetry
Because of its high symmetry  all corners , edges and surfaces are similar to each other  the octahedron is a regular polyhedron . It has:
 3 fourfold axes of rotation (through opposite corners)
 4 threefold axes of rotation (through the centers of opposing surfaces )
 6 twofold axes of rotation (through the centers of opposite edges)
 9 levels of symmetry (3 levels through four corners each (e.g. red), 6 levels through two corners each and two edge centers (e.g. green))
 14 rotations (6 by 90 ° with planes through four corners each and 8 by 60 ° with planes through six edge centers each)
and is
 point symmetrical to the center.
In total, the symmetry group of the octahedron  the octahedron group or cube group  has 48 elements.
Relations with other polyhedra
The octahedron is the dual polyhedron to the hexahedron ( cube ) and vice versa.
Two tetrahedra can be inscribed in a cube in such a way that the corners are at the same time cube corners and the edges are diagonals of the cube faces (see illustration). The threedimensional intersection of the tetrahedron is an octahedron with half the side length. The union is a a star tetrahedron .
If, on the eight faces of the octahedron tetrahedron to also create a star tetrahedron .
With the help of octahedron and cube , numerous bodies can be constructed that also have the group of cube as a symmetry group . So you get for example
 the truncated octahedron with 8 hexagons and 6 squares
 the cuboctahedron with 8 triangles and 6 squares, i.e. with 14 faces , and 12 corners
 the truncated cube with 8 triangles and 6 octagons
as the intersection of an octahedron with a cube (see Archimedean solids ) and
 the rhombic dodecahedron with 8 + 6 = 14 corners and 12 lozenges as faces
as a convex hull of a union of an octahedron with a cube .
Formulas
Sizes of an octahedron with edge length a  

volume 


Surface area  
Umkugelradius  
Edge ball radius  
Inc sphere radius  
Ratio of volume to spherical volume 

Interior angle of the equilateral triangle 

Angle between adjacent faces 

Angle between edge and face 

3D edge angles  
Solid angles in the corners 
Calculation of the regular octahedron
volume
The octahedron basically consists of two assembled pyramids with a square base and edge length
For pyramids and thus for half the volume of the octahedron applies
therein is the base area (square)
and the height of the pyramid
with inserted variables and the factor 2
Surface area
For the surface area of the octahedron (eight equilateral triangles) applies
Pyramid height
The height of the pyramid can be determined using the following rightangled triangle.
The side lengths of this triangle are (see picture in formulas ): side height as a hypotenuse, pyramid height as a large side and half the edge length of the pyramid as a small side.
The following applies to the height of the equilateral triangle
and according to the Pythagorean theorem applies
Angle between adjacent faces
This angle, marked with (see picture in formulas ), has its apex at one edge of the octahedron. It can be determined using the following right triangle.
The side lengths of this triangle are: edge ball radius as a hypotenuse, incipple radius as a large leg and a third of the side height as a small leg. This value is determined by the position of the center of gravity of the triangular area, since the geometric center of gravity divides the height of the triangle in a ratio of 2: 1.
The following applies to the angle
Angle between edge and face
This angle, marked with , has its apex at one corner of the octahedron. Angle can be determined using the following right triangle.
The side lengths of this triangle are (see picture in formulas ): pyramid edge as hypotenuse, pyramid height as large cathetus and half the diagonal of a square with side length / edge as small cathetus.
The following applies to the angle
3D edge angle
This angle, marked with (see picture in formulas ), has its apex at one corner of the octahedron and corresponds to twice angle d. H. the interior angle of a square .
Thus applies to the 3D edge angle of the octahedron
Solid angles in the corners
The following formula, described in Platonic Solids, shows a solution for the solid angle
With the number of edges / faces at a corner and the interior angle of the equilateral triangle, the following applies
because of it
used in and formed
simplification
generalization
The analogs of the octahedron in any dimension n are called ndimensional crosspolytopes and are also regular polytopes . The n dimensional crosspolytope has corners and is bounded by (n − 1) dimensional simplexes (as facets ). The fourdimensional cross polytope has 8 corners, 24 edges of equal length, 32 equilateral triangles as side surfaces and 16 tetrahedra as facets. The onedimensional cross polytope is a segment , the twodimensional cross polytope is the square .
A model for the n dimensional cross polytope is the unit sphere with respect to the sum norm
 For
in vector space . The (closed) cross polytop is therefore
 the amount
 .
 the convex hull of the vertices , where are the unit vectors .
 the intersection of the halfspaces divided by the hyperplanes of the shape
 can be determined and contain the origin .
The volume of the ndimensional cross polytope is , where the radius of the sphere around the origin is with respect to the sum norm. The relationship can be proven using recursion and Fubini's theorem.
Networks of the octahedron
The octahedron has eleven nets . That means, there are eleven ways to unfold a hollow octahedron by cutting open 5 edges and spreading it out in the plane . The other 7 edges connect the 8 equilateral triangles of the mesh. To color an octahedron so that no neighboring faces are the same color, you need at least 2 colors.
Graphs, dual graphs, cycles, colors
The octahedron has an undirected planar graph with 6 nodes , 12 edges and 8 regions assigned to it, which is 4 regular , ie 4 edges start from each node, so that the degree is 4 for all nodes. In the case of planar graphs, the exact geometric arrangement of the nodes is not important. However, it is important that the edges do not have to intersect. The nodes of this octahedral graph correspond to the corners of the cube.
The nodes of the octahedral graph can be colored with 3 colors so that adjacent nodes are always colored differently. This means that the chromatic number of this graph is 3. In addition, the edges can be colored with 4 colors so that adjacent edges are always colored differently. This is not possible with 3 colors, so the chromatic index for the edge coloring is 4 (the picture on the right illustrates these coloring).
The dual graph (cube graph ) with 8 nodes , 12 edges and 6 areas is helpful to determine the required number of colors for the areas or areas . The nodes of this graph are assigned onetoone (bijective) to the areas of the octahedral graph and vice versa (see bijective function and figure above). The nodes of the cube graph can be colored with 2 colors in such a way that neighboring nodes are always colored differently, so that the chromatic number of the cube graph is 2. From this one can indirectly conclude: Because the chromatic number is equal to 2, 2 colors are necessary for such a surface coloring of the octahedron or a coloring of the areas of the octahedron graph.
The 5 cut edges of each network (see above) together with the corners ( nodes ) form a spanning tree of the octahedral graph . Each net corresponds exactly to a spanning tree and vice versa, so that there is a onetoone ( bijective ) assignment between nets and spanning trees. If you consider an octahedron network without the outer area as a graph, you get a dual graph with a tree with 8 nodes and 7 edges and the maximum node degree 3. Each area of the octahedron is assigned to a node of the tree. Not every graphtheoretical constellation (see isomorphism of graphs ) of such trees occurs, but some occur several times.
The octahedron graph has 32 Hamilton circles and 1488 Euler circles .
Room fillings with octahedra
The threedimensional Euclidean space can be completely filled with Platonic solids or Archimedean solids of the same edge length. Such threedimensional tiling is called room filling . The following space fills contain octahedra:
Room filling with octahedron and tetrahedron
Room filling with cuboctahedron and octahedron
Room filling with truncated hexahedron and octahedron
Applications
In chemistry , predicting molecular geometries using the VSEPR model can result in octahedral molecules . The octahedron also appears in crystal structures , such as the facecentered cubic sodium chloride structure (coordination number 6), in the unit cell , as well as in complex chemistry if 6 ligands are located around a central atom .
Some naturally occurring minerals , e.g. B. alum , crystallize in octahedral form.
In roleplaying games, octahedral game dice are used and are referred to as “D8”, ie a dice with 8 faces.
Web links
 Euclid: Stoicheia. Book XIII.14. Octahedron of a sphere ...
 Octahedron .  Math tinkering
Individual evidence
 ^ Wilhelm Pape , Max Sengebusch (arrangement): Concise dictionary of the Greek language . 3rd edition, 6th impression. Vieweg & Sohn, Braunschweig 1914 ( zeno.org [accessed on March 12, 2020]).
 ↑ Eric Weisstein: Regular Octahedron. Umkugelradius formula (12). In: MathWorld Wolfram. A Wolfram Web Resource, accessed June 27, 2020 .
 ↑ Harish Chandra Rajpoot: Solid angles subtended by the platonic solids (regular polyhedra) at their vertices. SlideShare, March 2015, accessed June 27, 2020 .
 ↑ Alternative expression for . WolramAlpha, accessed June 27, 2020 .
 ↑ Eric Weisstein: Regular Octahedron. Networks. In: MathWorld Wolfram. A Wolfram Web Resource, accessed June 27, 2020 .
 ↑ Mike Zabrocki: HOMEWORK # 3 SOLUTIONS  MATH 3260. (PDF) York University, Mathematics and Statistics, Toronto, 2003, p. 3 , accessed May 31, 2020 .
 ↑ Eric Weisstein: Octahedral Graph. In: MathWorld Wolfram. A Wolfram Web Resource, accessed June 27, 2020 .