Dodecahedron

A general dodecahedron is a polyhedron having 12 faces. Examples include the decagonal prism, elongated square dipyramid (Johnson solid ), hexagonal dipyramid, metabidiminished icosahedron (), pentagonal antiprism, pentagonal cupola (), (regular) dodecahedron, rhombic dodecahedron, snub disphenoid (), triakis tetrahedron, and undecagonal pyramid. Crystals of pyrite () resemble slightly distorted dodecahedra (Steinhaus 1999, pp. 207-208), and sphalerite (ZnS) crystals are irregular dodecahedra bounded by congruent deltoids (Steinhaus 1999, pp. 207 and 209). The hexagonal scalenohedron is another irregular dodecahedron.

The regular dodecahedron, often simply called "the dodecahedron," is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and the Wythoff symbol .

There are 43380 distinct nets for the dodecahedron, the same number as for the icosahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998). Questions of polyhedron coloring of the dodecahedron can be addressed using the Pólya enumeration theorem.

The image above shows an origami dodecahedron constructed using six dodecahedron units, each consisting of a single sheet of paper (Kasahara and Takahama 1987, pp. 86-87).

A dodecahedron appears as part of the staircase being ascending by alligator-like lizards in Escher's 1943 lithograph "Reptiles" (Bool et al. 1982, p. 284; Forty 2003, Plate 32). Two dodecahedra also appear as polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43). The IPV pod that transports Ellie Arroway (Jodi Foster) through a network of wormholes in the 1997 film Contact was enclosed in a dodecahedral framework.

A 40-foot high sculpture (Nath 1999) known as Eclipse is displayed in the Hyatt Regency Hotel in San Francisco. It was constructed by Charles Perry, and is composed of pieces of anodized aluminum tubes and assembled over a period of four months (Kraeuter 1999). The layered sculpture begins with a regular dodecahedron, but each face then rotates outward. At the midpoint of the rotation, it forms an icosidodecahedron. Then, as the 12 pentagons continue to rotate outward, it forms a small rhombicosidodecahedron.

Dodecahedra were known to the Greeks, and 90 models of dodecahedra with knobbed vertices have been found in a number of archaeological excavations in Europe dating from the Gallo-Roman period in locations ranging from military camps to public bath houses to treasure chests (Schuur).

The dodecahedron has the icosahedral group of symmetries. The connectivity of the vertices is given by the dodecahedral graph. There are three dodecahedron stellations.

The dual polyhedron of a dodecahedron with unit edge lengths is an icosahedron with edge lengths , where is the golden ratio. As a result, the centers of the faces of an icosahedron form a dodecahedron, and vice versa (Steinhaus 1999, pp. 199-201).

A plane perpendicular to a axis of a dodecahedron cuts the solid in a regular hexagonal cross section (Holden 1991, p. 27). A plane perpendicular to a axis of a dodecahedron cuts the solid in a regular decagonal cross section (Holden 1991, p. 24).

A cube can be constructed from the dodecahedron's vertices taken eight at a time (above left figure; Steinhaus 1999, pp. 198-199; Wells 1991). Five such cubes can be constructed, forming the cube 5-compound. In addition, joining the centers of the faces gives three mutually perpendicular golden rectangles (right figure; Wells 1991).

The short diagonals of the faces of the rhombic triacontahedron give the edges of a dodecahedron (Steinhaus 1999, pp. 209-210).

The following table gives polyhedra which can be constructed by cumulation of a dodecahedron by pyramids of given heights .

		result
		60-faced dimpled deltahedron
		pentakis dodecahedron
		60-faced star deltahedron
		small stellated dodecahedron

When the dodecahedron with edge length is oriented with two opposite faces parallel to the -plane, the vertices of the top and bottom faces lie at and the other polyhedron vertices lie at , where is the golden ratio. The explicit coordinates are

(1)

(2)

with , 1, ..., 4, where is the golden ratio.

Eight dodecahedra can be place in a closed ring, as illustrated above (Kabai 2002, pp. 177-178).

The polyhedron vertices of a dodecahedron can be given in a simple form for a dodecahedron of side length by (0, , ), (, 0, ), (, , 0), and (, , ).

For a dodecahedron of unit edge length , the circumradius and inradius of a pentagonal face are

			(3)
			(4)

The sagitta is then given by

(5)

Now consider the following figure.

Using the Pythagorean theorem on the figure then gives

			(6)
			(7)
			(8)

Equation (8) can be written

(9)

Solving (6), (7), and (9) simultaneously gives

			(10)
			(11)
			(12)

The inradius of the dodecahedron is then given by

(13)

so

(14)

and solving for gives

(15)

Now,

(16)

so the circumradius is

(17)

The midradius is given by

(18)

so

(19)

The dihedral angle is

(20)

The area of a single face is the area of a pentagon of unit edge length

(21)

so the surface area is 12 times this value, namely

(22)

The volume of the dodecahedron can be computed by summing the volume of the 12 constituent pentagonal pyramids,

(23)

Apollonius showed that for an icosahedron and a dodecahedron with the same inradius,

(24)

where is the volume and the surface area, with the actual ratio being

(25)

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