Regular icosahedron

In geometry, the regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

Many polyhedrons are constructed from the regular icosahedron. For example, most of the Kepler–Poinsot polyhedron is constructed by faceting. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron has many relations with other Platonic solids, one of them is the regular dodecahedron as its dual polyhedron and has the historical background on the comparison mensuration. It also has many relations with other polytopes.

The appearance of regular icosahedron can be found in nature, such as the virus with icosahedral-shaped shells and radiolarians. Other applications of the regular icosahedron are the usage of its net in cartography, twenty-sided dice that may have been found in ancient times and role-playing games.

Construction
The regular icosahedron can be constructed like other gyroelongated bipyramids, started from a pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its faces. These pyramids cover the pentagonal faces, replacing them with five equilateral triangles, such that the resulting polyhedron has 20 equilateral triangles as its faces. This process construction is known as the gyroelongation, so the resulting polyhedron is also called gyroelongated pentagonal bipyramid..

Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint. These twelve vertices describe the three mutually perpendicular planes, with edges drawn between each of them. Because of the constructions above, the regular icosahedron is Platonic solid, a family of polyhedra with regular faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are only eight different convex deltahedra, one of which is the regular icosahedron.

One possible system of Cartesian coordinate for the vertices of a regular icosahedron, giving the edge length 2, is: $$ \left(0, \pm 1, \pm \varphi \right), \left(\pm 1, \pm \varphi, 0 \right), \left(\pm \varphi, 0, \pm 1 \right), $$ where $$\varphi = (1 + \sqrt{5})/2 $$ denotes the golden ratio.

Mensuration
The insphere of a convex polyhedron is a sphere inside the polyhedron, touching every face. The circumsphere of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. The midsphere of a convex polyhedron is a sphere tangent to every edge. Therefore, given that the edge length $$ a $$ of a regular icosahedron, the radius of insphere (inradius) $$ r_I $$, the radius of circumsphere (circumradius) $$ r_C $$, and the radius of midsphere (midradius) $$ r_M $$ are, respectively: $$ r_I = \frac{\varphi^2 a}{2 \sqrt{3}} \approx 0.756a, \qquad r_C = \frac{\sqrt{\varphi^2 + 1}}{2}a \approx 0.951a, \qquad r_M = \frac{\varphi}{2}a \approx 0.809a. $$

The surface area of polyhedra is the sum of its every face. Therefore, the surface area of regular icosahedra $$ A $$ equals the area of 20 equilateral triangles. The volume of a regular icosahedron $$ V $$ is obtained by calculating the volume of all pyramids with the base of triangular faces and the height with the distance from a triangular face's centroid to the center inside the regular icosahedron, the circumradius of a regular icosahedron. $$ A = 5\sqrt{3}a^2 \approx 8.660a^2, \qquad V = \frac{5 \varphi^2}{6}a^3 \approx 2.182a^3. $$ A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers. As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).

The dihedral angle of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism. The dihedral angle of a pentagonal antiprism and pentagonal pyramid between two adjacent triangular faces are approximately $$ 138.2^\circ $$. The dihedral angle of a pentagonal antiprism between pentagon-to-triangle is $$ 100.8^\circ $$, and the dihedral angle of a pentagonal pyramid between the same faces is $$ 37.4^\circ $$. Therefore, for the regular icosahedron, the dihedral angle between two adjacent triangles, on the edge where the pentagonal pyramid and pentagonal antiprism are attached is $$ 37.4^\circ + 100.8^\circ = 138.2^\circ $$.

Symmetry
The rotational symmetry group of the regular icosahedron is isomorphic to the alternating group on five letters. This non-abelian simple group is the only non-trivial normal subgroup of the symmetric group on five letters. Since the Galois group of the general quintic equation is isomorphic to the symmetric group on five letters, and this normal subgroup is simple and non-abelian, the general quintic equation does not have a solution in radicals. The proof of the Abel–Ruffini theorem uses this simple fact, and Felix Klein wrote a book that made use of the theory of icosahedral symmetries to derive an analytical solution to the general quintic equation.

The full symmetry group of the icosahedron (including reflections) is known as the full icosahedral group. It is isomorphic to the product of the rotational symmetry group and the group $$C_2$$ of size two, which is generated by the reflection through the center of the icosahedron.

Icosahedral graph
Every Platonic graph, including the icosahedral graph, is a polyhedral graph. This means that they are planar graphs, graphs that can be drawn in the plane without crossing its edges; and they are 3-vertex-connected, meaning that the removal of any two of its vertices leaves a connected subgraph. According to Steinitz theorem, the icosahedral graph endowed with these heretofore properties represents the skeleton of a regular icosahedron.

The icosahedral graph is Hamiltonian, meaning that it contains a Hamiltonian cycle, or a cycle that visits each vertex exactly once.

In other Platonic solids
Aside from comparing the mensuration between the regular icosahedron and regular dodecahedron, they are dual to each other. An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.

An icosahedron can be inscribed in an octahedron by placing its 12 vertices on the 12 edges of the octahedron such that they divide each edge into its two golden sections. Because the golden sections are unequal, there are five different ways to do this consistently, so five disjoint icosahedra can be inscribed in each octahedron.

An icosahedron of edge length $\frac{1}{\varphi} \approx 0.618$ can be inscribed in a unit-edge-length cube by placing six of its edges (3 orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges. Because there are five times as many icosahedron edges as cube faces, there are five ways to do this consistently, so five disjoint icosahedra can be inscribed in each cube. The edge lengths of the cube and the inscribed icosahedron are in the golden ratio.

Stellation
The icosahedron has a large number of stellations. stated 59 stellations were identified for the regular icosahedron. The first form is the icosahedron itself. One is a regular Kepler–Poinsot polyhedron. Three are regular compound polyhedra.

Facetings
The small stellated dodecahedron, great dodecahedron, and great icosahedron are three facetings of the regular icosahedron. They share the same vertex arrangement. They all have 30 edges. The regular icosahedron and great dodecahedron share the same edge arrangement but differ in faces (triangles vs pentagons), as do the small stellated dodecahedron and great icosahedron (pentagrams vs triangles).

Diminishment
The Johnson solids are the polyhedra whose faces are all regular, but not uniform. This means they do not include the Archimedean solids, Catalan solids, prisms, and antiprisms. Some of them are constructed involving the removal of the part of a regular icosahedron, a process known as diminishment. They are gyroelongated pentagonal pyramid, metabidiminished icosahedron, and tridiminished icosahedron, which remove one, two, and three pentagonal pyramids from the icosahedron, respectively. The similar dissected regular icosahedron has 2 adjacent vertices diminished, leaving two trapezoidal faces, and a bifastigium has 2 opposite sets of vertices removed and 4 trapezoidal faces.

Relations to the 600-cell and other 4-polytopes
The icosahedron is the dimensional analogue of the 600-cell, a regular 4-dimensional polytope. The 600-cell has icosahedral cross sections of two sizes, and each of its 120 vertices is an icosahedral pyramid; the icosahedron is the vertex figure of the 600-cell.

The unit-radius 600-cell has tetrahedral cells of edge length $\frac{1}{\varphi}$, 20 of which meet at each vertex to form an icosahedral pyramid (a 4-pyramid with an icosahedron as its base). Thus the 600-cell contains 120 icosahedra of edge length $\frac{1}{\varphi}$. The 600-cell also contains unit-edge-length cubes and unit-edge-length octahedra as interior features formed by its unit-length chords. In the unit-radius 120-cell (another regular 4-polytope which is both the dual of the 600-cell and a compound of 5 600-cells) we find all three kinds of inscribed icosahedra (in a dodecahedron, in an octahedron, and in a cube).

A semiregular 4-polytope, the snub 24-cell, has icosahedral cells.

Relations to other uniform polytopes
As has been mentioned above, the regular icosahedron is unique among the Platonic solids in possessing a dihedral angle is approximately $ 138.19^\circ $. Thus, just as hexagons have angles not less than 120° and cannot be used as the faces of a convex regular polyhedron because such a construction would not meet the requirement that at least three faces meet at a vertex and leave a positive defect for folding in three dimensions, icosahedra cannot be used as the cells of a convex regular polychoron because, similarly, at least three cells must meet at an edge and leave a positive defect for folding in four dimensions (in general for a convex polytope in n dimensions, at least three facets must meet at a peak and leave a positive defect for folding in n-space). However, when combined with suitable cells having smaller dihedral angles, icosahedra can be used as cells in semi-regular polychora (for example the snub 24-cell), just as hexagons can be used as faces in semi-regular polyhedra (for example the truncated icosahedron). Finally, non-convex polytopes do not carry the same strict requirements as convex polytopes, and icosahedra are indeed the cells of the icosahedral 120-cell, one of the ten non-convex regular polychora.

There are distortions of the icosahedron that, while no longer regular, are nevertheless vertex-uniform. These are invariant under the same rotations as the tetrahedron, and are somewhat analogous to the snub cube and snub dodecahedron, including some forms which are chiral and some with $ T_h $-symmetry, i.e. have different planes of symmetry from the tetrahedron.

Appearances
Dice are the common objects with the different polyhedron, one of them is the regular icosahedron. The twenty-sided dice was found in many ancient times. One example is the dice from the Ptolemaic of Egypt, which was later the Greek letters inscribed on the faces in the period of Greece and Roman. Another example was found in the treasure of Tipu Sultan, which was made out of golden and with numbers written on each face. In several roleplaying games, such as Dungeons & Dragons, the twenty-sided die (labeled as d20) is commonly used in determining success or failure of an action. It may be numbered from "0" to "9" twice, in which form it usually serves as a ten-sided die (d10); most modern versions are labeled from "1" to "20". Scattergories is another board game, where the player names the categories in the card with given the set time. The naming of such categories is initially with the letters contained in every twenty-sided dice.

The regular icosahedron may also appear in many fields of science. In virology, herpes virus have icosahedral shells. The outer protein shell of HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus. Several species of radiolarians discovered by Ernst Haeckel, described its shells as the like-shaped various regular polyhedra; one of which is Circogonia icosahedra, whose skeleton is shaped like a regular icosahedron. In chemistry, the closo-carboranes are compounds with a shape resembling the regular icosahedron. Icosahedral twinning also occurs in crystals, especially nanoparticles. Many borides and allotropes of boron such as α- and β-rhombohedral contain boron B12 icosahedron as a basic structure unit. In cartography, R. Buckminster Fuller used the net of a regular icosahedron to create a map known as Dymaxion map, by subdividing the net into triangles, followed by calculating the grid on the Earth's surface, and transferring the results from the sphere to the polyhedron. This projection was created during the time that Fuller realized that the Greenland is smaller than South America. In the Thomson problem, concerning the minimum-energy configuration of $$n$$ charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the minimum solution known for $$n = 12$$ places the points at the vertices of a regular icosahedron, inscribed in a sphere. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.