Regular dodecahedron | |
---|---|

(Click here for rotating model) | |

Type | Platonic solid |

Elements | F = 12, E = 30V = 20 (χ = 2) |

Faces by sides | 12{5} |

Conway notation | D |

Schläfli symbols | {5,3} |

Face configuration | V3.3.3.3.3 |

Wythoff symbol | 3 | 2 5 |

Coxeter diagram | |

Symmetry | I_{h}, H_{3}, [5,3], (*532) |

Rotation group | I, [5,3]^{+}, (532) |

References | U _{23}, C _{26}, W _{5} |

Properties | regular, convex |

Dihedral angle | 116.56505° = arccos(−1⁄√5) |

5.5.5 (Vertex figure) | Regular icosahedron (dual polyhedron) |

Net |

A **regular dodecahedron** or **pentagonal dodecahedron** is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals (60 face diagonals, 100 space diagonals).^{ [2] } It is represented by the Schläfli symbol {5,3}.

- Dimensions
- Surface area and volume
- Two-dimensional symmetry projections
- Spherical tiling
- Cartesian coordinates
- Facet-defining equations
- Properties
- As a configuration
- Geometric relations
- Relation to the regular icosahedron
- Relation to the nested cube
- Relation to the regular tetrahedron
- Relation to the golden rectangle
- Relation to the 6-cube and rhombic triacontahedron
- History and uses
- In nature and supramolecules
- Shape of the universe
- Space filling with cube and bilunabirotunda
- Related polyhedra and tilings
- Vertex arrangement
- Stellations
- Dodecahedral graph
- See also
- References
- External links

If the edge length of a regular dodecahedron is , the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices) is

(sequence A179296 in the OEIS )

and the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces) is

while the midradius, which touches the middle of each edge, is

These quantities may also be expressed as

where *ϕ* is the golden ratio.

Note that, given a regular dodecahedron of edge length one, *r _{u}* is the radius of a circumscribing sphere about a cube of edge length

The surface area *A* and the volume *V* of a regular dodecahedron of edge length *a* are:

Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio. A dodecahedron with an edge length of one unit has the properties:^{ [3] }

The *regular dodecahedron * has two high orthogonal projections, centered, on vertices and pentagonal faces, correspond to the A_{2} and H_{2} Coxeter planes. The edge-center projection has two orthogonal lines of reflection.

Centered by | Vertex | Face | Edge |
---|---|---|---|

Image | |||

Projective symmetry | [[3]] = [6] | [[5]] = [10] | [2] |

In perspective projection, viewed on top of a pentagonal face, the regular dodecahedron can be seen as a linear-edged Schlegel diagram, or stereographic projection as a spherical polyhedron. These projections are also used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3-dimensions.

Projection | Orthogonal projection | Perspective projection | |
---|---|---|---|

Schlegel diagram | Stereographic projection | ||

Regular dodecahedron | |||

Dodecaplex (120-cell) |

The regular dodecahedron can also be represented as a spherical tiling.

Orthographic projection | Stereographic projection |
---|

Vertex coordinates: | |

The orange vertices lie at (±1, ±1, ±1) and form a cube (dotted lines). | |

The green vertices lie at (0, ±ϕ, ±1/ϕ) and form a rectangle on the yz-plane. | |

The blue vertices lie at (±1/ϕ, 0, ±ϕ) and form a rectangle on the xz-plane. | |

The pink vertices lie at (±ϕ, ±1/ϕ, 0) and form a rectangle on the xy-plane. | |

The distance between adjacent vertices is 2/ϕ, and the distance from the origin to any vertex is √3.ϕ = 1 + √5/2 is the golden ratio. |

The following Cartesian coordinates define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented:^{ [4] }

- (±1, ±1, ±1)
- (0, ±
*ϕ*, ±1/*ϕ*) - (±1/
*ϕ*, 0, ±*ϕ*) - (±
*ϕ*, ±1/*ϕ*, 0)

where *ϕ* = 1 + √5/2 ≈ 1.618 is the golden ratio. The edge length is 2/*ϕ* = √5 − 1. The circumradius is √3.

Similar to the symmetry of the vertex coordinates, the equations of the twelve facets of the regular dodecahedron also display symmetry in their coefficients:

*x*±*ϕy*= ±*ϕ*^{2}*y*±*ϕz*= ±*ϕ*^{2}*z*±*ϕx*= ±*ϕ*^{2}

- The dihedral angle of a regular dodecahedron is 2 arctan(
*ϕ*) or approximately 116.565° (where again*ϕ*= 1 + √5/2, the golden ratio). OEIS: A137218 Note that the tangent of the dihedral angle is exactly −2. - If the original regular dodecahedron has edge length 1, its dual icosahedron has edge length
*ϕ*. - If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges. It is the
*roundest*of the five Platonic solids, enclosing the most volume within the same radius. - It has 43,380 nets.
- The map-coloring number of a regular dodecahedron's faces is 4.
- The distance between the vertices on the same face not connected by an edge is
*ϕ*times the edge length, because the diagonal of a pentagon is*ϕ*times its edge length. - If two edges share a common vertex, then the midpoints of those edges form a 36-72-72 golden triangle with the body center.

This configuration matrix represents the dodecahedron. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole dodecahedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{ [5] }^{ [6] }

Here is the configuration expanded with *k*-face elements and *k*-figures. The diagonal element counts are the ratio of the full Coxeter group H_{3}, order 120, divided by the order of the subgroup with mirror removal.

H_{3} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | k-fig | Notes | |
---|---|---|---|---|---|---|---|---|

A_{2} | ( ) | f_{0} | 20 | 3 | 3 | {3} | H_{3}/A_{2} = 120/6 = 20 | |

A_{1}A_{1} | { } | f_{1} | 2 | 30 | 2 | { } | H_{3}/A_{1}A_{1} = 120/4 = 30 | |

H_{2} | {5} | f_{2} | 5 | 5 | 12 | ( ) | H_{3}/H_{2} = 120/10 = 12 |

The *regular dodecahedron* is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron.

The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra.

A rectified regular dodecahedron forms an icosidodecahedron.

The regular dodecahedron has icosahedral symmetry I_{h}, Coxeter group [5,3], order 120, with an abstract group structure of *A*_{5} × *Z*_{2}.

The dodecahedron and icosahedron are dual polyhedra. A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges.

When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%).

A regular dodecahedron with edge length 1 has more than three and a half times the volume of an icosahedron with the same length edges (7.663... compared with 2.181...), which ratio is approximately 3.51246117975, or in exact terms: 3/5(3*ϕ* + 1) or (1.8*ϕ* + 0.6).

A cube can be embedded within a regular dodecahedron, affixed to eight of its equidistant vertices, in five different positions.^{ [7] } In fact, five cubes may overlap and interlock inside the regular dodecahedron to result in the compound of five cubes.

The ratio of the edge of a regular dodecahedron to the edge of a cube embedded inside such a regular dodecahedron is 1 : *ϕ*, or (*ϕ* − 1) : 1.

The ratio of a regular dodecahedron's volume to the volume of a cube embedded inside such a regular dodecahedron is 1 : 2/2 + *ϕ*, or 1 + *ϕ*/2 : 1, or (5 + √5) : 4.

For example, an embedded cube with a volume of 64 (and edge length of 4), will nest within a regular dodecahedron of volume 64 + 32*ϕ* (and edge length of 4*ϕ* − 4).

Thus, the difference in volume between the encompassing regular dodecahedron and the enclosed cube is always one half the volume of the cube times *ϕ*.

From these ratios are derived simple formulas for the volume of a regular dodecahedron with edge length *a* in terms of the golden mean:

*V*= (*aϕ*)^{3}· 1/4(5 + √5)*V*= 1/4(14*ϕ*+ 8)*a*^{3}

As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair).

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. In fact, 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, so as 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.^{ [8] }

Golden rectangles of ratio (*ϕ* + 1) : 1 and *ϕ* : 1 also fit perfectly within a regular dodecahedron.^{ [9] } In proportion to this golden rectangle, an enclosed cube's edge is *ϕ*, when the long length of the rectangle is *ϕ* + 1 (or *ϕ*^{2}) and the short length is 1 (the edge shared with the regular dodecahedron).

In addition, the center of each face of the regular dodecahedron form three intersecting golden rectangles.^{ [10] }

It can be projected to 3D from the 6-dimensional 6-demicube using the same basis vectors that form the hull of the rhombic triacontahedron from the 6-cube. Shown here including the inner 12 vertices, which are not connected by the outer hull edges of 6D norm length √2, form a regular icosahedron.

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

*u*= (1,*ϕ*, 0, −1,*ϕ*, 0)*v*= (*ϕ*, 0, 1,*ϕ*, 0, −1)*w*= (0, 1,*ϕ*, 0, −1,*ϕ*)

Regular dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy.

Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons".^{ [11] } In * Theaetetus *, a dialogue of Plato, Plato hypothesized that the classical elements were made of the five uniform regular solids; these later became known as the platonic solids. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven". Timaeus (c. 360 BC), as a personage of Plato's dialogue, associates the other four platonic solids with the four classical elements, adding that there is a fifth solid pattern which, though commonly associated with the regular dodecahedron, is never directly mentioned as such; "this God used in the delineation of the universe."^{ [12] } Aristotle also postulated that the heavens were made of a fifth element, which he called aithêr (*aether* in Latin, *ether* in American English).

Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist. Euclid completely mathematically described the Platonic solids in the *Elements*, the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.

Regular dodecahedra have been used as dice and probably also as divinatory devices. During the Hellenistic era, small, hollow bronze Roman dodecahedra were made and have been found in various Roman ruins in Europe. Their purpose is not certain.

In 20th-century art, dodecahedra appear in the work of M. C. Escher, such as his lithographs * Reptiles * (1943) and * Gravitation * (1952). In Salvador Dalí's painting * The Sacrament of the Last Supper * (1955), the room is a hollow regular dodecahedron. Gerard Caris based his entire artistic oeuvre on the regular dodecahedron and the pentagon, which is presented as a new art movement coined as Pentagonism.

In modern role-playing games, the regular dodecahedron is often used as a twelve-sided die, one of the more common polyhedral dice.

Immersive Media Company, a former Canadian digital imaging company, made the Dodeca 2360 camera, the world's first 360° full-motion camera which captures high-resolution video from every direction simultaneously at more than 100 million pixels per second or 30 frames per second.^{[ promotion? ]} It is based on regular dodecahedron.^{[ citation needed ]}

The Megaminx twisty puzzle, alongside its larger and smaller order analogues, is in the shape of a regular dodecahedron.

In the children's novel * The Phantom Tollbooth *, the regular dodecahedron appears as a character in the land of Mathematics. Each of his faces wears a different expression – *e.g.* happy, angry, sad – which he swivels to the front as required to match his mood.

The fossil coccolithophore * Braarudosphaera bigelowii * (see figure), a unicellular coastal phytoplanktonic alga, has a calcium carbonate shell with a regular dodecahedral structure about 10 micrometers across.^{ [13] }

Some quasicrystals and cages have dodecahedral shape (see figure). Some regular crystals such as garnet and diamond are also said to exhibit "dodecahedral" habit, but this statement actually refers to the rhombic dodecahedron shape.^{ [14] }^{ [1] }

Various models have been proposed for the global geometry of the universe. In addition to the primitive geometries, these proposals include the Poincaré dodecahedral space, a positively curved space consisting of a regular dodecahedron whose opposite faces correspond (with a small twist). This was proposed by Jean-Pierre Luminet and colleagues in 2003,^{ [15] }^{ [16] } and an optimal orientation on the sky for the model was estimated in 2008.^{ [17] }

In Bertrand Russell's 1954 short story "The Mathematician's Nightmare: The Vision of Professor Squarepunt", the number 5 said: "I am the number of fingers on a hand. I make pentagons and pentagrams. And but for me dodecahedra could not exist; and, as everyone knows, the universe is a dodecahedron. So, but for me, there could be no universe."

Regular dodecahedra fill space with cubes and bilunabirotundas (Johnson solid 91), in the ratio of 1 to 1 to 3.^{ [18] }^{ [19] } The dodecahedra alone make a lattice of edge-to-edge pyritohedra. The bilunabirotundas fill the rhombic gaps. Each cube meets six bilunabirotundas in three orientations.

Block model | Lattice of dodecahedra | 6 bilunabirotundas around a cube |

The regular dodecahedron is topologically related to a series of tilings by vertex figure *n*^{3}.

*n32 symmetry mutation of regular tilings: {n,3} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Spherical | Euclidean | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||

{2,3} | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} | {12i,3} | {9i,3} | {6i,3} | {3i,3} |

The regular dodecahedron can be transformed by a truncation sequence into its dual, the icosahedron:

Family of uniform icosahedral polyhedra | |||||||
---|---|---|---|---|---|---|---|

Symmetry: [5,3], (*532) | [5,3]^{+}, (532) | ||||||

{5,3} | t{5,3} | r{5,3} | t{3,5} | {3,5} | rr{5,3} | tr{5,3} | sr{5,3} |

Duals to uniform polyhedra | |||||||

V5.5.5 | V3.10.10 | V3.5.3.5 | V5.6.6 | V3.3.3.3.3 | V3.4.5.4 | V4.6.10 | V3.3.3.3.5 |

Uniform octahedral polyhedra | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [4,3], (*432) | [4,3]^{+}(432) | [1^{+},4,3] = [3,3](*332) | [3^{+},4](3*2) | |||||||

{4,3} | t{4,3} | r{4,3} r{3 ^{1,1}} | t{3,4} t{3 ^{1,1}} | {3,4} {3 ^{1,1}} | rr{4,3} s _{2}{3,4} | tr{4,3} | sr{4,3} | h{4,3} {3,3} | h_{2}{4,3} t{3,3} | s{3,4} s{3 ^{1,1}} |

= | = | = | = or | = or | = | |||||

| | | | | ||||||

Duals to uniform polyhedra | ||||||||||

V4^{3} | V3.8^{2} | V(3.4)^{2} | V4.6^{2} | V3^{4} | V3.4^{3} | V4.6.8 | V3^{4}.4 | V3^{3} | V3.6^{2} | V3^{5} |

The regular dodecahedron is a member of a sequence of otherwise non-uniform polyhedra and tilings, composed of pentagons with face configurations (V3.3.3.3.*n*). (For *n* > 6, the sequence consists of tilings of the hyperbolic plane.) These face-transitive figures have (*n*32) rotational symmetry.

n32 symmetry mutations of snub tilings: 3.3.3.3.n | ||||||||
---|---|---|---|---|---|---|---|---|

Symmetryn32 | Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||

232 | 332 | 432 | 532 | 632 | 732 | 832 | ∞32 | |

Snub figures | ||||||||

Config. | 3.3.3.3.2 | 3.3.3.3.3 | 3.3.3.3.4 | 3.3.3.3.5 | 3.3.3.3.6 | 3.3.3.3.7 | 3.3.3.3.8 | 3.3.3.3.∞ |

Gyro figures | ||||||||

Config. | V3.3.3.3.2 | V3.3.3.3.3 | V3.3.3.3.4 | V3.3.3.3.5 | V3.3.3.3.6 | V3.3.3.3.7 | V3.3.3.3.8 | V3.3.3.3.∞ |

The regular dodecahedron shares its vertex arrangement with four nonconvex uniform polyhedra and three uniform polyhedron compounds.

Five cubes fit within, with their edges as diagonals of the regular dodecahedron's faces, and together these make up the regular polyhedral compound of five cubes. Since two tetrahedra can fit on alternate cube vertices, five and ten tetrahedra can also fit in a regular dodecahedron.

The 3 stellations of the regular dodecahedron are all regular (nonconvex) polyhedra: (Kepler–Poinsot polyhedra)

0 | 1 | 2 | 3 | |
---|---|---|---|---|

Stellation | Regular dodecahedron | Small stellated dodecahedron | Great dodecahedron | Great stellated dodecahedron |

Facet diagram |

Regular dodecahedron graph | |
---|---|

Vertices | 20 |

Edges | 30 |

Radius | 5 |

Diameter | 5 |

Girth | 5 |

Automorphisms | 120 (A_{5} × Z_{2})^{ [20] } |

Chromatic number | 3 |

Properties | Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph |

Table of graphs and parameters |

The skeleton of the dodecahedron (the vertices and edges) form a graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.

This graph can also be constructed as the generalized Petersen graph *G*(10,2) where the vertices of a decagon are connected to those of two pentagons, one pentagon connected to odd vertices of the decagon and the other pentagon connected to the even vertices. Geometrically, this can be visualized as the 10-vertex equatorial belt of the dodecahedron connected to the two 5-vertex polar regions, one on each side.

The high degree of symmetry of the polygon is replicated in the properties of this graph, which is distance-transitive, distance-regular, and symmetric. The automorphism group has order 120. The vertices can be colored with 3 colors, as can the edges, and the diameter is 5.^{ [21] }

The dodecahedral graph is Hamiltonian – there is a cycle containing all the vertices. Indeed, this name derives from a mathematical game invented in 1857 by William Rowan Hamilton, the icosian game. The game's object was to find a Hamiltonian cycle along the edges of a dodecahedron.

- 120-cell, a regular polychoron (4D polytope whose surface consists of 120 dodecahedral cells)
*Braarudosphaera bigelowii*− A dodecahedron shaped coccolithophore (a unicellular phytoplankton algae).- Dodecahedrane (molecule)
- Pentakis dodecahedron
- Snub dodecahedron
- Truncated dodecahedron

In geometry, a **dodecahedron** or **duodecahedron** is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

In geometry, the **regular 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.

In geometry, an **icosidodecahedron** is a polyhedron with twenty (*icosi*) triangular faces and twelve (*dodeca*) pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

In geometry, a **Kepler–Poinsot polyhedron** is any of four regular star polyhedra.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular octahedron**, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a **Platonic solid** is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

In geometry, the **truncated icosahedron** is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. In general usage, the degree of truncation is assumed to be uniform unless specified.

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.

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.

In geometry, the **truncated dodecahedron** is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

In geometry, the **rhombic dodecahedron** is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

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.

In geometry, the **120-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a **C _{120}**,

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.

In geometry, a **disdyakis triacontahedron**, **hexakis icosahedron**, **decakis dodecahedron** or **kisrhombic triacontahedron** is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

In geometry, a **pentagonal hexecontahedron** is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices.

In geometry, the **bilunabirotunda** is one of the Johnson solids. A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra. They were named by Norman Johnson, who first listed these polyhedra in 1966.

In geometry, the **complete** or **final stellation of the icosahedron** is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, simple, and star polyhedron.

In geometry, the **great icosahedron** is one of four Kepler–Poinsot polyhedra, with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

In geometry, the **medial rhombic triacontahedron** is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called **small stellated triacontahedron**. Its dual is the dodecadodecahedron.

- 1 2 Kai Wu, Jonathan Nitschke and co-workers "Systematic construction of progressively larger capsules from a fivefold linking pyrrole-based subcomponent",
*Nature Synthesis*,**2023**, DOI: https://doi.org/10.1038/s44160-023-00276-9. - ↑ Sutton, Daud (2002),
*Platonic & Archimedean Solids*, Wooden Books, Bloomsbury Publishing USA, p. 55, ISBN 9780802713865 . - ↑ Livio, Mario (2003) [2002].
*The Golden Ratio: The Story of Phi, the World's Most Astonishing Number*(First trade paperback ed.). New York City: Broadway Books. pp. 70–1. ISBN 0-7679-0816-3. - ↑ Weisstein, Eric W. "Icosahedral group".
*MathWorld*. - ↑ Coxeter, H.S.M. (1973) [1948]. "§1.8 Configurations".
*Regular Polytopes*(3rd ed.). New York: Dover. - ↑ Coxeter, H.S.M. (1991).
*Regular Complex Polytopes*(2nd ed.). Cambridge: Cambridge University Press. p. 117. - ↑ http://mathworld.wolfram.com/images/eps-gif/DodecahedronCube_700.gif
^{[ bare URL image file ]} - ↑ Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938).
*The Fifty-Nine Icosahedra*. Vol. 6. University of Toronto Studies (Mathematical Series). p. 4. - ↑ Knott, Ron (26 September 2016). "The Golden Geometry of Solids or Phi in 3 dimensions".
*Ron Knott's Mathematics Pages*. Retrieved 2022-03-19. - ↑ http://www.toshen.com/images/dodecahedronwithgoldrectang.gif
^{[ bare URL image file ]} - ↑ Florian Cajori,
*A History of Mathematics*(1893) - ↑ Plato,
*Timaeus*, Jowett translation [line 1317–8]; the Greek word translated as delineation is*diazographein*, painting in semblance of life. - ↑ Hagino, K., Onuma, R., Kawachi, M. and Horiguchi, T. (2013) "Discovery of an endosymbiotic nitrogen-fixing cyanobacterium UCYN-A in
*Braarudosphaera bigelowii*(Prymnesiophyceae)".*PLoS One*,**8**(12): e81749. doi : 10.1371/journal.pone.0081749. - ↑ Dodecahedral Crystal Habit Archived 12 April 2009 at the Wayback Machine
- ↑ Dumé, Belle (Oct 8, 2003). "Is The Universe A Dodecahedron?".
*PhysicsWorld*. Archived from the original on 2012-04-25. - ↑ Luminet, Jean-Pierre; Jeff Weeks; Alain Riazuelo; Roland Lehoucq; Jean-Phillipe Uzan (2003-10-09). "Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background".
*Nature*.**425**(6958): 593–5. arXiv: astro-ph/0310253 . Bibcode:2003Natur.425..593L. doi:10.1038/nature01944. PMID 14534579. S2CID 4380713. - ↑ Roukema, Boudewijn; Zbigniew Buliński; Agnieszka Szaniewska; Nicolas E. Gaudin (2008). "A test of the Poincaré dodecahedral space topology hypothesis with the WMAP CMB data".
*Astronomy and Astrophysics*.**482**(3): 747. arXiv: 0801.0006 . Bibcode:2008A&A...482..747L. doi:10.1051/0004-6361:20078777. S2CID 1616362. - ↑ "Dodecahedron and Bilunabirotunda – Wolfram Demonstrations Project".
- ↑ http://www.lcv.ne.jp/~hhase/memo/m09_08b.html
- ↑ Frucht, Roberto (1936–1937), "Die gruppe des Petersen'schen Graphen und der Kantensysteme der regulären Polyeder",
*Comment. Math. Helv.*,**9**: 217–223, doi:10.1007/bf01258190, S2CID 121791222 - ↑ Weisstein, Eric W. "Dodecahedral Graph".
*MathWorld*.

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