A regular dodecahedron or pentagonal dodecahedron is a dodecahedron (a polyhedron with 12 faces) composed of regular pentagonal faces, three meeting at each vertex. It is one of the Platonic solids, described in Plato's dialogues as the shape of the universe itself. Johannes Kepler used the dodecahedron in his 1596 model of the Solar System. However, the dodecahedron and other Platonic solids had already been described by other philosophers since antiquity. The regular dodecahedron is a truncated trapezohedron because it is the result of truncating axial vertices of a pentagonal trapezohedron. It is also a Goldberg polyhedron because it is the initial polyhedron to construct new polyhedra by the process of chamfering. It has a relation with other Platonic solids, one of them is the regular icosahedron as its dual polyhedron. Other new polyhedra can be constructed by using a regular dodecahedron. The regular dodecahedron's metric properties and construction are associated with the golden ratio. The regular dodecahedron is featured in some artistic and narrative works. Some toys and artifacts are also shaped like regular dodecahedra, including the Roman dodecahedron. Regular dodecahedra can also be found in nature and supramolecules, as well as the shape of the universe. The skeleton of a regular dodecahedron can be represented as the graph called the dodecahedral graph, a Platonic graph. Its property of the Hamiltonian, a path that visits all of its vertices exactly once, can be found in a toy called icosian game.
As a Platonic solid
Descriptions
The regular dodecahedron is a polyhedron with twelve pentagonal faces, thirty edges, and twenty vertices. It is one of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex. This set of polyhedrons is named after Plato. In Theaetetus, a dialogue of Plato, Plato hypothesized that the classical elements were made of the five uniform regular solids. Plato described the regular dodecahedron, obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven". Timaeus, as a personage of Plato's dialogue, associates the other four Platonic solids—regular tetrahedron, cube, regular octahedron, and regular icosahedron—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." Aristotle also postulated that the heavens were made of a fifth element, which he called aithêr (aether in Latin, ether in American English). Following its attribution with nature by Plato, Johannes Kepler in his Harmonices Mundi sketched each of the Platonic solids, one of them is a regular dodecahedron. In his Mysterium Cosmographicum, Kepler also proposed the Solar System by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron, regular icosahedron, regular dodecahedron, regular tetrahedron, and cube.
Many antiquity philosophers described the regular dodecahedron, including the rest of the Platonic solids. 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. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons". The regular dodecahedron, as the family of Platonic solids, is a regular polyhedron. It is isogonal, isohedral, and isotoxal: any two vertices, two faces, and two edges of a regular dodecahedron can be transformed by rotations and reflections under its symmetry orbit respectively, which preserves the appearance.
Relation to the regular icosahedron
The dual polyhedron of a dodecahedron is the regular icosahedron. One property of the dual polyhedron generally is that the original polyhedron and its dual share the same three-dimensional symmetry group. In the case of the regular dodecahedron, it has the same symmetry as the regular icosahedron, the icosahedral symmetry I h {\displaystyle \mathrm {I} _{\mathrm {h} }} . The regular dodecahedron has ten three-fold axes passing through pairs of opposite vertices, six five-fold axes passing through the opposite faces centers, and fifteen two-fold axes passing through the opposite sides midpoints. 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%). The result of both spheres' volumes initially began from the problem by ancient Greeks, 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 of Alexandria, Pappus of Alexandria, 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 are taken to different powers. Golden rectangle may also related to both regular icosahedron and regular dodecahedron. The regular icosahedron can be constructed by intersecting three golden rectangles perpendicularly, arranged in two-by-two orthogonal, and connecting each of the golden rectangle's vertices with a segment line. There are 12 regular icosahedron vertices, considered as the center of 12 regular dodecahedron faces.
Relation to the regular tetrahedron
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