Icosahedron
Regular icosahedron  

Type of side surfaces  equilateral triangles 
Number of faces  20th 
Number of corners  12 
Number of edges  30th 
Schläfli icon  {3.5} 
dual to  Dodecahedron 
Example of a body net  
Number of different networks  43380 
Number of edges in a corner  5 
Number of corners of a surface  3 
The (also, especially Austrian : the) icosahedron [ ikozaˈʔeːdɐ ] (from ancient Greek εἰκοσάεδρον eikosáedron "twenty flat ", "twenty flat ") is one of the five Platonic solids , more precisely a regular polyhedron ( polyhedron , polyhedron ) with
 20 congruent equilateral triangles as side surfaces
 30 edges of equal length and
 12 corners, in each of which five side surfaces meet.
symmetry
Because of its high symmetry  all corners , edges and surfaces are similar to each other  the icosahedron is a regular polyhedron . It has:
 6 fivefold axes of rotation (through opposite corners)
 10 threefold axes of rotation (through the centers of opposing surfaces )
 15 twofold axes of rotation (through the centers of opposing edges)
 15 planes of symmetry (due to opposite and parallel edges)
and is
 point symmetric (point reflection at the center of the polyhedron ).
In total, the symmetry group of the icosahedron  the icosahedral group or dodecahedral group  has 120 elements. The subgroup of rotations of the icosahedron has order 60 and is the smallest nonabelian simple group ( , alternating group of order 5). The symmetry of the icosahedron is not compatible with a periodic spatial structure because of the fivefold symmetry that occurs with it (see tiling ). There can therefore be no crystal lattice with icosahedral symmetry (see quasicrystals ).
Cartesian coordinates
The following Cartesian coordinates define the corners of an icosahedron with edge length a = 2, centered at the origin:
 (0, ± 1, ± )
 (± 1, ± , 0)
 (± , 0, ± 1)
with ( golden number ).
Relations with other polyhedra
The dodecahedron is the dual polyhedron to the icosahedron and vice versa.
With the help of icosahedron and dodecahedron , numerous bodies can be constructed that also have the icosahedron group as a symmetry group . So you get for example
 the truncated icosahedron (soccer ball) with 12 pentagons and 20 hexagons as the intersection of a dodecahedron with an icosahedron (see Archimedean solids , fullerenes ). It is created from the icosahedron by cutting the corners perpendicular to the straight lines connecting the corners with the center , with regular pentagons appearing as cut surfaces and the triangles mutating into hexagons. At a certain cutting height, the hexagons are regular.
 the icosidodecahedron with 20 triangles and 12 pentagons
 the truncated dodecahedron with 20 triangles and 12 decagons as the intersection of an icosahedron with a dodecahedron (see Archimedean solids )
 a rhombic triacontahedron with 20 + 12 = 32 corners and 30 rhombuses as surfaces as a convex hull of a union of an icosahedron with a dodecahedron and
 an icosahedron star by extending all the edges of an icosahedron beyond its corners until three of them intersect at a point .
The structure of the icosahedron
As the figure on the right shows, you can select 3 pairs of opposite edges from the edges of the icosahedron so that these pairs span 3 congruent rectangles that are orthogonal to each other . The lengths of the sides of these rectangles correspond to the golden ratio because they are sides or diagonals of regular pentagons . The icosahedron can therefore be inscribed in a cube in such a way that these 6 edges lie in the 6 faces of the cube and are parallel to the edges of the cube.
The 24 remaining edges delimit 8 triangles that lie in the surfaces of an octahedron  circumscribed by the icosahedron , with the corners of the icosahedron lying on its edges.
There are a total of five such positions, each edge of the icosahedron belonging to exactly one such group of orthogonal edge pairs, while each surface lies twice in the surface of a circumscribed octahedron . The symmetry group of the icosahedron causes all 5! / 2 = 60 even permutations of these five positions.
The edges of the icosahedron contain twelve flat pentagons , each edge belonging to two and every corner belonging to five of these pentagons. You can use this property to build a wireframe model.
The icosahedron can also be thought of as a combination of a uniformed pentagonal antiprism and a fivesided pyramid on each side .
Formulas
Sizes of an icosahedron 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 angle  
Solid angles in the corners 
Calculation of the regular icosahedron
volume
The icosahedron consists of twenty assembled threesided pyramids.
For a pyramid and thus for one twentieth of the icosahedron applies
therein is the base area (equilateral triangle)
and the height of the pyramid is equal to the radius of the sphere
it follows with inserted variables for the volume of the icosahedron
Surface area
For the surface area of the icosahedron (twenty equilateral triangles) applies
Angle between adjacent faces
This angle, marked with (see picture in formulas ), has its apex at one edge of the icosahedron. 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 icosahedron.
Is, as in the illustration (see picture in formulas ), the angle z. B. searched at the corner , then it can be determined with the help of the right triangles and . Their common hypotenuse , equal to the radius of the sphere , divides the angle into the angles that are not shown, or so applies
The legs of the rightangled triangle with an angle are :, equal to the radius of the sphere , as a large leg and , equal to two thirds of the side height of the icosahedral triangle, as a small leg. This value is determined by the position of the center of gravity of the equilateral triangular area, since the geometric center of gravity divides the height of the triangle in a ratio of 2: 1.
The legs of the rightangled triangle with angle are :, equal to the edge ball radius , as a large leg and , equal to half the side , as a small leg.
The following applies to angles of the triangle
for angles of the triangle applies
the addition theorem for the arc function gives the angle
after inserting the relevant values ( ) applies
3D edge angle
This angle, marked with (see picture in formulas ), has its apex at a corner of the icosahedron and corresponds to the interior angle of the regular pentagon .
Thus applies to the 3D edge angle of the icosahedron
algebraically applies
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
Significance of the icosahedral group in mathematics
The point group of the icosahedron, the icosahedron group , is widely used in mathematics. This goes back to the famous monograph by Felix Klein from 1884 lectures on the icosahedron and the solution of the equations of the fifth degree . According to Galois theory, the general equation of the fifth degree has no solution in radicals, since the alternating group A _{5} cannot be resolved. _{}
Significance of the icosahedron in cluster physics
The icosahedral shape is very important for clusters (accumulations of atoms in the order of 3 to 50,000 atoms) from a size of more than 7 atoms. The reason for this is Friedel's rule, which says that the structure has the lowest energy for which the number of closestneighbor bonds is maximum. In many free clusters this occurs from 7 atoms, although there are exceptions and other structures are preferred (e.g. cubes).
Furthermore, there are socalled magic numbers in cluster physics , which are closely related to the socalled Mackay icosahedron. Here shell closures (i.e. perfect atomic icosahedra) ensure particularly stable clusters. This occurs with clusters with the magic atomic numbers 1, 13, 55, 147, 309, 561, 923, and 1415. These very old findings by Alan Mackay play an important role in current cluster physics.
The cluster numbers can be calculated using the following formula:
C = total number of atoms in the cluster
n = number of atoms per edge
Applications
 The capsids of many viruses have an icosahedral symmetry. This can be explained by the fact that viruses pack their nucleic acids optimally. The icosahedron shape is favorable in this respect because the icosahedron has the largest volume of all regular polyhedra with a given diameter. Examples are rhinovirus (runny nose), hepatitis B virus , adenovirus and poliovirus .
 The closododeca boranate anion B _{12} H _{12 }^{2−} has the structure of the particularly stable B _{12} icosahedron.
 Rudolf von Laban used the icosahedron intensively for his theory of spatial harmony and thus influenced modern dance . This is continued today in the Laban Movement Studies.
 In his cybernetic management theory, Stafford Beer worked out the icosahedron structure as a model for optimal networking of employees in teams.
 In many pen & paper roleplaying games , icosahedra are used as 20 sided dice (D20).
 Climbing frames for children are particularly stable in the icosahedral shape.
 An icosahedron placed in the globe forms the core of the lattice structure in the weather forecast model ICON of the German Weather Service (similar to a geodesic dome or the Dymaxion world map draft by Richard Buckminster Fuller ).
 The Dogic is a variant of the Rubik's Cube in the form of an icosahedron as a threedimensional, mechanical puzzle.
 Inside a Magic 8 Ball there is an icosahedron on which the possible answers are written. It swims in a dark blue liquid inside the sphere.
 In the military as a sonar reflector in mine hunting to position a base weight near a base mine. Here, the 20 equilateral triangles are opened again in 3 inwardfacing tetrahedra in order to generate as many angles of reflection as possible.
Networks of the icosahedron
The icosahedron has 43380 networks . This means that there are 43380 possibilities to unfold a hollow icosahedron by cutting open 11 edges and spreading it out in the plane . The other 19 edges connect the 20 equilateral triangles of the mesh. In order to color an icosahedron so that no neighboring faces are the same color, you need at least 3 colors.
Graphs, dual graphs, cycles, colors
The icosahedron has an undirected planar graph with 12 nodes , 30 edges and 20 areas assigned to it, which is 5 regular , ie 5 edges start from each node, so that the degree is equal to 5 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 icosahedron graph correspond to the corners of the icosahedron.
The nodes of the icosahedral graph can be colored with 4 colors so that neighboring nodes are always colored differently. This means that the chromatic number of this graph is 4 (see node coloring ). In addition, the edges can be colored with 5 colors so that adjacent edges are always colored differently. This is not possible with 4 colors, so the chromatic index for the edge coloring is equal to 5 (the picture on the right illustrates these coloring).
The dual graph (dodecahedron graph ) with 20 nodes , 30 edges and 12 areas is helpful to determine the required number of colors for the surfaces or areas . The nodes of this graph are assigned onetoone (bijective) to the areas of the icosahedral graph and vice versa (see bijective function and figure above). The nodes of the dodecahedron graph can be colored with 3 colors so that neighboring nodes are always colored differently, but not with 2 colors, so that the chromatic number of the icosahedral graph is 3. From this one can indirectly conclude: Because the chromatic number is equal to 3, 3 colors are necessary for such a surface coloring of the icosahedron or a coloring of the areas of the icosahedron graph.
The 11 cut edges of each network (see above) together with the corners ( nodes ) form a spanning tree of the icosahedral 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 icosahedron network without the outer area as a graph, you get a dual graph with a tree with 20 nodes and 19 edges and the maximum node degree 5. Each area of the icosahedron 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 icosahedral graph has 2560 Hamilton circles , but no Euler circles .
Web links
 Euclid: Stoicheia. Book XIII.16. Icosahedron of a sphere ...
 Derivation of the formulas
 Math tinkering  icosahedron
Individual evidence
 ^ Wilhelm Pape : Concise dictionary of the Greek language . 3. Edition. Vieweg & Sohn, Braunschweig 1914 ( zeno.org ).
 ^ Mathematical handicrafts  football: truncated icosahedron .
 ↑ Eric Weisstein: Regular Icosahedron. Umkugelradius, formula (12). In: MathWorld Wolfram. A Wolfram Web Resource, accessed June 22, 2020 .
 ↑ Eric Weisstein: Regular Icosahedron. Edge ball radius, formula (16). In: MathWorld Wolfram. A Wolfram Web Resource, accessed June 22, 2020 .
 ↑ Eric Weisstein: Regular Icosahedron. In spherical radius, formula (14). In: MathWorld Wolfram. A Wolfram Web Resource, accessed June 22, 2020 .
 ↑ Alternative expression for . Wolfram Alpha, accessed June 22, 2020 .
 ↑ Harish Chandra Rajpoot: Solid angles subtended by the platonic solids (regular polyhedra) at their vertices. SlideShare, March 2015, accessed June 22, 2020 .
 ↑ Alternative expression for . Wolfram Alpha, accessed June 22, 2020 .
 ↑ Felix Klein: Lectures on the icosahedron and the solution of the equations of the fifth degree . Teubner, Leipzig 1884 (VIII, 260, online ).
 ↑ AL Mackay: A dense noncrystallographic packing of equal spheres. In: Acta Crystallographia. Volume 15, 1962, pp. 916918, doi: 10.1107 / S0365110X6200239X .
 ^ Wolfram MathWorld: Regular Icosahedron
 ↑ Mike Zabrocki: HOMEWORK # 3 SOLUTIONS  MATH 3260. (PDF) York University, Mathematics and Statistics, Toronto, 2003, p. 4 , accessed May 31, 2020 .
 ^ Wolfram MathWorld: Icosahedral Graph