Regular icosahedron

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Regular icosahedron
Icosahedron.svg
Type Gyroelongated bipyramid
Deltahedron
Faces 20
Edges 30
Vertices 12
Vertex configuration
Symmetry group Icosahedral symmetry
Dual polyhedron Regular dodecahedron
Net
Icosahedron flat.svg

In geometry, the regular icosahedron [1] (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 the Platonic solid and of the deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

Contents

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. [2] This process construction is known as the gyroelongation, so the resulting polyhedron is also called gyroelongated pentagonal bipyramid.[ citation needed ].

Three mutually perpendicular golden ratio rectangles, with edges connecting their corners, form a regular icosahedron. Icosahedron-golden-rectangles.svg
Three mutually perpendicular golden ratio rectangles, with edges connecting their corners, form a regular icosahedron.

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. [3] 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. [4]

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

where denotes the golden ratio. [5]

Properties

Mensuration

3D model of a regular icosahedron Regular icosahedron.stl
3D model of a regular icosahedron

The insphere of a convex polyhedron is a sphere inside the polyhedron, touching every its face. The circumsphere of a convex polyhedron is a sphere that contains the polyhedron and touches every edge. The midsphere of a convex polyhedron is a sphere tangent to its every edge. Therefore, given that the edge length of a regular icosahedron, the radius of insphere (inradius) , the radius of circumsphere (circumradius) , and the radius of midsphere (midradius) are, respectively: [6]

The surface area of polyhedra is the sum of its every face. Therefore, the surface area of regular icosahedra equals the area of 20 equilateral triangles. The volume of a regular icosahedron 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. [7]

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. [8] 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. [9] Both volumes have formulas involving the golden ratio, but taken to different powers. [10] As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%). [11]

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 . The dihedral angle of a pentagonal antiprism between pentagon-to-triangle is , and the dihedral angle of a pentagonal pyramid between the same faces is . 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 . [12]

Symmetry

Full Icosahedral symmetry has 15 mirror planes (seen as cyan great circles on this sphere) meeting at order
p
/
5
,
p
/
3
,
p
/
2
{\displaystyle \pi /5,\pi /3,\pi /2}
angles, dividing a sphere into 120 triangle fundamental domains. There are 6 5-fold axes (blue), 10 3-fold axes (red), and 15 2-fold axes (magenta). The vertices of the regular icosahedron exist at the 5-fold rotation axis points. Sphere symmetry group ih.png
Full Icosahedral symmetry has 15 mirror planes (seen as cyan great circles on this sphere) meeting at order angles, dividing a sphere into 120 triangle fundamental domains. There are 6 5-fold axes (blue), 10 3-fold axes (red), and 15 2-fold axes (magenta). The vertices of the regular icosahedron exist at the 5-fold rotation axis points.

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. [13]

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 of size two, which is generated by the reflection through the center of the icosahedron.

Icosahedral graph

Icosahedral graph Icosahedron graph.svg
Icosahedral graph

All of the Platonic graph, including icosahedral graph, are 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. [14]

The icosahedral graph is Hamiltonian, meaning that any vertex that passes through another one is exactly once in a cycleway. [15]

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. [16]

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. [17]

An icosahedron of edge length 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. [18] 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. [19]

Stellation

The icosahedron has a large number of stellations. Coxeter et al. (1938) 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. [20]

21 of 59 stellations
Stellation diagram of icosahedron.svg
The faces of the icosahedron extended outwards as planes intersect, defining regions in space as shown by this stellation diagram of the intersections in a single plane.
Zeroth stellation of icosahedron.svg First stellation of icosahedron.png Second stellation of icosahedron.png Third stellation of icosahedron.svg Fourth stellation of icosahedron.png Fifth stellation of icosahedron.png Sixth stellation of icosahedron.png
Seventh stellation of icosahedron.png Eighth stellation of icosahedron.png Ninth stellation of icosahedron.png Tenth stellation of icosahedron.png Eleventh stellation of icosahedron.png Twelfth stellation of icosahedron.png Thirteenth stellation of icosahedron.png
Fourteenth stellation of icosahedron.png Fifteenth stellation of icosahedron.png Sixteenth stellation of icosahedron.png Seventeenth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png

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 polyhedron whose all of their faces are 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. [21] 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 , 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 . 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 . 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 Th-symmetry, i.e. have different planes of symmetry from the tetrahedron.

Relation to the 6-cube and rhombic triacontahedron

6demicube-odd-icosahedron.png

The icosahedron can be projected to 3D from the 6D 6-demicube using the same basis vectors that form the hull of the Rhombic triacontahedron from the 6-cube. Shown here including the inner 20 vertices which are not connected by the 30 outer hull edges of 6D norm length . The inner vertices form a dodecahedron.

The 3D projection basis vectors [u,v,w] used are:

Applications and its natural form

Twenty-sided die (icosahedron) with faces inscribed with Greek letters MET 10.130.1158 001.jpg
Twenty-sided dice from Ptolemaic of Egypt, inscribed with Greek letters at the faces.
ScatDice.JPG
The Scattergories twenty-sided die, excluding the six letters Q, U, V, X, Y, and Z.

Dice are the common objects with the different polyhedron, one of them is the regular icosahedron. This twenty-sided dice was found in many ancient times. One example is the dice from the Ptolemaic of Egypt, [22] which was later the Greek letters inscribed on the faces in the period of Greece and Roman. [23] Another example was found in the treasure of Tipu Sultan, which was made out of golden and with the numbers written on each face. [24] 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. This die is in the form of a regular icosahedron. 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". [25] 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. [26]

Circogonia icosahedra.jpg
The radiolarian Circogonia icosahedra has a regular icosahedral structure
Structure of a Myoviridae bacteriophage 2.jpg
A myovirus typically has a regular icosahedral capsid (head) about 100 nanometers across

The regular icosahedron may also appear in many fields of biology. 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. [27] 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. [28]

The closo-carboranes are chemical compounds with shape very close to icosahedron. Icosahedral twinning also occurs in crystals, especially nanoparticles. Many borides and allotropes of boron contain boron B12 icosahedron as a basic structure unit.

Dymaxion map, created by the net of a regular icosahedron Dymaxion projection.png
Dymaxion map, created by the net of a regular icosahedron

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 the South America. [29]

Notes

  1. See Jones 2003 for the pronunciation: ( /ˌkɒsəˈhdrən,-kə-,-k-/ or /ˌkɒsəˈhdrən/ )
  2. Silvester 2001, p.  140141; Cundy 1952.
  3. Cromwell 1997, p.  70.
  4. Shavinina 2013, p.  333; Cundy 1952.
  5. Steeb, Hardy & Tanski 2012, p.  211.
  6. MacLean 2007 , p.  4344; Coxeter 1973, Table I(i), pp. 292–293. See column "", where is Coxeter's notation for the midradius, also noting that Coxeter uses as the edge length (see p. 2).
  7. MacLean 2007, p.  4344.
  8. Herz-Fischler 2013, p.  138–140.
  9. Simmons 2007, p.  50.
  10. Sutton 2002, p.  55.
  11. Numerical values for the volumes of the inscribed Platonic solids may be found in Buker & Eggleton 1969.
  12. Johnson 1966, See table II, line 4.; MacLean 2007, p.  4344.
  13. Klein 1884. See icosahedral symmetry: related geometries for further history, and related symmetries on seven and eleven letters.
  14. Bickle 2020, p.  147.
  15. Hopkins 2004.
  16. Herrmann & Sally 2013, p.  257.
  17. Coxeter et al. 1938, p. 4; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. Each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a compound of five octahedra, which comes under our definition of stellated icosahedron. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated triacontahedron.) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a stella octangula, thus forming a compound of ten tetrahedra. Further, we can choose one tetrahedron from each stella octangula, to derive a compound of five tetrahedra, which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be chiral ."
  18. Borovik 2006, pp. 8–9, §5. How to draw an icosahedron on a blackboard.
  19. Reciprocally, the edge length of a cube inscribed in a dodecahedron is in the golden ratio to the dodecahedron's edge length. The cube's edges lie in pentagonal face planes of the dodecahedron as regular pentagon diagonals, which are always in the golden ratio to the regular pentagon's edge. When a cube is inscribed in a dodecahedron and an icosahedron is inscribed in the cube, the dodecahedron and icosahedron that do not share any vertices have the same edge length.
  20. Coxeter et al. 1938, p. 826.
  21. Berman 1971.
  22. Smith 1958, p.  295.
  23. Minas-Nerpel 2007.
  24. Cromwell 1997, p.  4.
  25. "Dungeons & Dragons Dice". gmdice.com. Retrieved August 20, 2019.
  26. Flanagan & Gregory 2015, p.  85.
  27. Strauss & Strauss 2008, p. 3562.
  28. Haeckel 1904; Cromwell 1997, p.  6.
  29. Cromwell 1997, p.  7.

Related Research Articles

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<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

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<span class="mw-page-title-main">Kepler–Poinsot polyhedron</span> Any of 4 regular star polyhedra

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<span class="mw-page-title-main">Octahedron</span> Polyhedron with eight triangular faces

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<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Snub dodecahedron</span> Archimedean solid with 92 faces

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

<span class="mw-page-title-main">Regular polytope</span> Polytope with highest degree of symmetry

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<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

<span class="mw-page-title-main">Triakis icosahedron</span> Catalan solid with 60 faces

In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.

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<span class="mw-page-title-main">Rectified 600-cell</span>

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<span class="mw-page-title-main">Final stellation of the icosahedron</span> Outermost stellation of the icosahedron

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<span class="mw-page-title-main">Chamfer (geometry)</span> Geometric operation which truncates the edges of polyhedra

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<span class="mw-page-title-main">Icosahedron</span> Polyhedron with 20 faces

In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty', and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".

References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
(Convex) icosahedron Small triambic icosahedron Medial triambic icosahedron Great triambic icosahedron Compound of five octahedra Compound of five tetrahedra Compound of ten tetrahedra Great icosahedron Excavated dodecahedron Final stellation
Zeroth stellation of icosahedron.svg First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.svg Seventeenth stellation of icosahedron.png
Stellation diagram of icosahedron.svg Small triambic icosahedron stellation facets.svg Great triambic icosahedron stellation facets.svg Compound of five octahedra stellation facets.svg Compound of five tetrahedra stellation facets.svg Compound of ten tetrahedra stellation facets.svg Great icosahedron stellation facets.svg Excavated dodecahedron stellation facets.svg Echidnahedron stellation facets.svg
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.