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Regular Dodecahedron


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The regular dodecahedron, often simply called "the" dodecahedron, is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, 12{5}. It is illustrated above together with a wireframe version and a net that can be used for its construction.

The regular dodecahedron is also the uniform polyhedron with Maeder index 23 (Maeder 1997), Wenninger index 5 (Wenninger 1989), Coxeter index 26 (Coxeter et al. 1954), and Har'El index 28 (Har'El 1993). It is given by the Schläfli symbol {5,3} and the Wythoff symbol 3|25.

DodecahedronProjections

A number of symmetric projections of the regular dodecahedron are illustrated above.

The regular dodecahedron is implemented in the Wolfram Language as Dodecahedron[] or UniformPolyhedron["Dodecahedron"], and precomputed properties are available as PolyhedronData["Dodecahedron"].

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

Origami dodecahedron

The image above shows an origami regular 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.

Charles Perry's Eclipse sculpture

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 1440 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).

DodecahedralGraph

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

DodecahedronConvexHulls

The regular dodecahedron is the convex hull of the cube 5-Compound, ditrigonal dodecadodecahedron, third dodecahedron stellation hull, great ditrigonal icosidodecahedron, great rhombic triacontahedron, great stellated dodecahedron, rhombic hexecontahedron, small ditrigonalI icosidodecahedron, tetrahedron 5-compound, and tetrahedron 10-compound.

DodecahedronAndDual
dodinico
icoindod

The dual polyhedron of a dodecahedron with unit edge lengths is an icosahedron with edge lengths phi, where phi 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).

DodecahedronHexagon
DodecahedronDecagon

A plane perpendicular to a C_3 axis of a dodecahedron cuts the solid in a regular hexagonal cross section (Holden 1991, p. 27). A plane perpendicular to a C_5 axis of a dodecahedron cuts the solid in a regular decagonal cross section (Holden 1991, p. 24).

DodecahedronCube
DodecahedronGolden

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).

RhombicTriacontDodec

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 augmentation of a dodecahedron by pyramids of given heights h.

When the dodecahedron with edge length sqrt(10-2sqrt(5)) is oriented with two opposite faces parallel to the xy-plane, the vertices of the top and bottom faces lie at z=+/-(phi+1) and the other polyhedron vertices lie at z=+/-(phi-1), where phi is the golden ratio. The explicit coordinates are

 +/-(2cos(2/5pii),2sin(2/5pii),phi+1)
(1)
 +/-(2phicos(2/5pii),2phisin(2/5pii),phi-1)
(2)

with i=0, 1, ..., 4, where phi is the golden ratio.

Dodecahedron8Projection
Dodecahedron8
Dodecahedron8Tilted

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 a=sqrt(5)-1 by (0, +/-phi^(-1), +/-phi), (+/-phi, 0, +/-phi^(-1)), (+/-phi^(-1), +/-phi, 0), and (+/-1, +/-1, +/-1).

PentagonApothem

For a dodecahedron of unit edge length a=1, the circumradius R^' and inradius r^' of a pentagonal face are

R^'=1/(10)sqrt(50+10sqrt(5))
(3)
r^'=1/(10)sqrt(25+10sqrt(5)).
(4)

The sagitta x is then given by

 x=R^'-r^'=1/(10)sqrt(125-10sqrt(5)).
(5)

Now consider the following figure.

DodecahedronTrig

Using the Pythagorean theorem on the figure then gives

z_1^2+m^2=(R^'+r^')^2
(6)
z_2^2+(m-x)^2=1
(7)
((z_1+z_2)/2)^2+R^('2)=((z_1-z_2)/2)^2+(m+r^')^2.
(8)

Equation (8) can be written

 z_1z_2+r^2=(m+r^')^2.
(9)

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

m=r^'=1/(10)sqrt(25+10sqrt(5))
(10)
z_1=2r^'=1/5sqrt(25+10sqrt(5))
(11)
z_2=R^'=1/(10)sqrt(50+10sqrt(5)).
(12)

The inradius of the regular dodecahedron is then given by

 r=1/2(z_1+z_2),
(13)

so

 r^2=1/(40)(25+11sqrt(5)),
(14)

and solving for r gives

 r=1/(20)sqrt(250+110sqrt(5))=1.11351....
(15)

Now,

 R^2=R^('2)+r^2=3/8(3+sqrt(5)),
(16)

so the circumradius is

 R=1/4(sqrt(15)+sqrt(3))=1.40125....
(17)

The midradius is given by

 rho^2=r^('2)+r^2=1/8(7+3sqrt(5)),
(18)

so

 rho=1/4(3+sqrt(5))=1.30901....
(19)

The dihedral angle is

 alpha=cos^(-1)(-1/5sqrt(5)) approx 116.57 degrees
(20)

and the Dehn invariant for a unit regular dodecahedron is

D=-30<5>_1
(21)
=-30tan^(-1)(2)
(22)
=-33.2144...,
(23)

(OEIS A377787), where the first expression uses the basis of Conway et al. (1999).

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

 A=1/4sqrt(25+10sqrt(5)),
(24)

so the surface area is 12 times this value, namely

 S=3sqrt(25+10sqrt(5)).
(25)

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

 V=12(1/3Ar)=1/4(15+7sqrt(5)).
(26)

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

 (V_(icosahedron))/(V_(dodecahedron))=(A_(icosahedron))/(A_(dodecahedron)),
(27)

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

 (V_(icosahedron))/(V_(dodecahedron))=(A_(icosahedron))/(A_(dodecahedron))=sqrt(3/(10)(5-sqrt(5))).
(28)

See also

Augmented Dodecahedron, Augmented Truncated Dodecahedron, Cairo Tessellation, Cuboctahedron, Deltoidal Hexecontahedron, Dodecagon, Dodecahedron 2-Compound, Dodecahedron 5-Compound, Dodecahedron 6-Compound, Dodecahedron-Icosahedron Compound, Dodecahedron-Small Triambic Icosahedron Compound, Dodecahedron Stellations, Elongated Dodecahedron, Great Dodecahedron, Great Stellated Dodecahedron, Hyperbolic Dodecahedron, Icosahedron, Metabiaugmented Dodecahedron, Metabiaugmented Truncated Dodecahedron, Parabiaugmented Dodecahedron, Parabiaugmented Truncated Dodecahedron, Polyhedron Coloring, Pyritohedron, Rhombic Dodecahedron, Rhombic Triacontahedron, Small Stellated Dodecahedron, Stellation, Triakis Tetrahedron, Triaugmented Dodecahedron, Triaugmented Truncated Dodecahedron, Trigonal Dodecahedron, Trigonometry Values--pi/5, Truncated Dodecahedron, Truncated Tetrahedron

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 228, 1987.Bool, F. H.; Kist, J. R.; Locher, J. L.; and Wierda, F. M. C. Escher: His Life and Complete Graphic Work. New York: Abrams, 1982.Bouzette, S. and Vandamme, F. "The Regular Dodecahedron and Icosahedron Unfold in 43380 Ways." Unpublished manuscript.Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <=4." Disc. Math. 186, 69-94, 1998.Conway, J. H.; Radin, C.; and Sadun, L. "On Angles Whose Squared Trigonometric Functions Are Rational." Discr. Computat. Geom. 22, 321-332, 1999.Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.Cundy, H. and Rollett, A. "Dodecahedron. 5^3." §3.5.4 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 87, 1989.Davie, T. "The Dodecahedron." http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/dodecahedron.html.Escher, M. C. "Reptiles." Lithograph. 1943. http://www.mcescher.com/Gallery/back-bmp/LW327.jpg.Escher, M. C. "Stars." Wood engraving. 1948. http://www.mcescher.com/Gallery/back-bmp/LW359.jpg.Forty, S. M.C. Escher. Cobham, England: TAJ Books, 2003.Geometry Technologies. "Dodecahedron." http://www.scienceu.com/geometry/facts/solids/dodeca.html.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Harris, J. W. and Stocker, H. "Dodecahedron." §4.4.5 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 101, 1998.Hippenmeyer, C. "Die Anzahl der inkongruenten ebenen Netze eines regulären Ikosaeders." Elem. Math. 34, 61-63, 1979.Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, 2002.Kasahara, K. "Regular-Pentagonal Flat Unit" and "From Regular to Semiregular Polyhedrons." Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, pp. 218-221 and 231, 1988.Kasahara, K. and Takahama, T. Origami for the Connoisseur. Tokyo: Japan Publications, 1987.Kraeuter, C. "Public Art: A Feast for the Eyes or an Ugly Eyesore?" San Francisco Business Times, Aug. 23, 1999. http://www.bizjournals.com/sanfrancisco/stories/1999/08/23/focus2.html.Maeder, R. E. "23: Dodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/23.html.Nath, A. "Potins d'Uranie: L'éclipse de Perry." Orion, 57/3, 22, 1999.Schuur, W. A. "Pentagonale Dodecaeder." http://home.wxs.nl/~wschuur/dcaeder.htm.Sloane, N. J. A. Sequence A377787 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 195-199, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 57-58, 1991.Wenninger, M. J. "The Dodecahedron." Model 5 in Polyhedron Models. Cambridge, England: Cambridge University Press, p. 19, 1989.

Cite this as:

Weisstein, Eric W. "Regular Dodecahedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RegularDodecahedron.html

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