The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices.

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.

These twelve vertices describe the three mutually perpendicular planes, with edges drawn between each of them.

All triangular faces of a regular octahedron are breaking, twisting at a certain angle, and filling up with other equilateral triangles.

[5] One possible system of Cartesian coordinate for the vertices of a regular icosahedron, giving the edge length 2, is:

of a regular icosahedron is twenty times that of each of its equilateral triangle faces.

of a regular icosahedron can be obtained as twenty times that of a pyramid whose base is one of its faces and whose apex is the icosahedron's center; or as the sum of two uniform pentagonal pyramids and a pentagonal antiprism.

It has fifteen mirror planes as in a cyan great circle on the sphere meeting at order

[15] 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,[16] 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.

According to Steinitz theorem, the icosahedral graph endowed with these heretofore properties represents the skeleton of a regular icosahedron.

[20] The icosahedral graph is a graceful graph, meaning that each vertex can be labelled with an integer between 0 and 30 inclusive, in such a way that the absolute difference between the labels of an edge's two vertices is different for every edge.

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.

can be inscribed in a unit-edge-length cube by placing six of its edges (three orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges.

Coxeter et al. (1938) in their work stated fifty-nine stellations were identified for the regular icosahedron.

The final stellation includes all of the cells in the icosahedron's stellation diagram, meaning every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside it.

[27] The truncated icosahedron is an Archimedean solid constructed by truncating the vertices of a regular icosahedron; the resulting polyhedron may be considered as a football because of having a pattern of numerous hexagonal and pentagonal faces.

Some of their construction involves the removal of the part of a regular icosahedron, a process known as diminishment.

[2] The similar dissected regular icosahedron has two adjacent vertices diminished, leaving two trapezoidal faces.

Nonetheless, it is occasionally incorrectly known as Jessen's icosahedron because of the similar visual, of having the same combinatorial structure and symmetry as Jessen's icosahedron;[c] the difference is the non-convex one does not form a tensegrity structure and does not have right-angled dihedrals.

One example is the die from the Ptolemaic of Egypt, which later used Greek letters inscribed on the faces in the period of Greece and Rome.

[33] Another example was found in the treasure of Tipu Sultan, which was made out of gold and with numbers written on each face.

[34] 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".

[35] Scattergories is another board game in which the player names the category entires on a card within a given set time.

The regular polyhedra have been known since antiquity, but are named after Plato who, in his Timaeus dialogue, identified these with the five elements, whose elementary units were attributed these shapes: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube) and the shape of the universe as a whole (dodecahedron).

Euclid's Elements defined the Platonic solids and solved the problem of finding the ratio of the circumscribed sphere's diameter to the edge length.

[44] Following their identification with the elements by Plato, Johannes Kepler in his Harmonices Mundi sketched each of them, in particular, the regular icosahedron.

[45] In his Mysterium Cosmographicum, he also proposed a model of the Solar System based on the placement of Platonic solids in a concentric sequence of increasing radius of the inscribed and circumscribed spheres whose radii gave the distance of the six known planets from the common center.