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Regular Icosahedron


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The regular icosahedron, often simply called "the" icosahedron, is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces, 20{3}. It is illustrated above together with a wireframe version and a net that can be used for its construction.

The regular icosahedron is also the uniform polyhedron with Maeder index 22 (Maeder 1997), Wenninger index 4 (Wenninger 1989), Coxeter index 25 (Coxeter et al. 1954), and Har'El index 27 (Har'El 1993). It is described by the Schläfli symbol {3,5} and Wythoff symbol 5|23. Coxeter et al. (1999) have shown that there are 58 icosahedron stellations (giving a total of 59 solids when the icosahedron itself is included).

IcosahedronProjections

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

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

Origami icosahedron

Two icosahedra constructed in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper.

Two icosahedra appears as polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

There are 43380 distinct nets for the icosahedron, the same number as for the dodecahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998).

IcosahedralGraph

The icosahedron has the icosahedral group I_h of symmetries. The connectivity of the vertices is given by the icosahedral graph.

IcosahedronConvexHulls

The regular icosahedron is the convex hull of the first and second dodecahedron stellations, great dodecahedron great icosahedron, 11th icosahedron stellation, and small stellated dodecahedron.

IcosahedronAndDual
dodinico
icoindod

The dual polyhedron of an icosahedron with unit edge lengths is the dodecahedron with edge lengths 1/phi, where phi is the golden ratio. As a result, the centers of the faces of an icosahedron form a dodecahedron, and vice versa, illustrated above (Steinhaus 1999, pp. 199-201).

In particular, fifteen golden rectangles span the interior of the icosahedron. These rectangles have 30 edges, and each edge pairs up with its opposite edge to form a golden rectangle

There are 59 distinct icosahedra when each triangle is colored differently (Coxeter 1969). More general question of polyhedron coloring can be addressed using the Pólya enumeration theorem.

IcosahedronCube

Taken eight at a time, the centers of the faces of an icosahedron comprise the vertices of a cube. This leads to the beautiful cube 5-compound and is the basis for Jessen's orthogonal icosahedron.

IcosahedronDecagon

A plane perpendicular to a C_5 axis of an icosahedron cuts the solid in a regular decagonal cross section (Holden 1991, pp. 24-25).

RhombicTriacontIcos

The long diagonals of the faces of the rhombic triacontahedron give the edges of an icosahedron (Steinhaus 1999, pp. 209-210).

The following table gives polyhedra which can be constructed by augmentation of an icosahedron by pyramids of given heights h.

A construction for an icosahedron with side length a=sqrt(50-10sqrt(5))/5 places the end vertices at (0,0,+/-1) and the central vertices around two staggered circles of radii 2/5sqrt(5) and heights +/-1/5sqrt(5). By a suitable rotation, the polyhedron vertices of an icosahedron of side length 2 can also be placed at (0,+/-phi,+/-1), (+/-1,0,+/-phi), and (+/-phi,+/-1,0), where phi is the golden ratio. These points divide the polyhedron edges of an octahedron into segments with lengths in the ratio phi:1. Another orientation of the icosahedron places two opposite triangular faces in an orientation parallel to the xy-plane. In this orientation, the distance h^' from the top plane to the triangle T of vertices below it is h^'=sqrt(3)/3, equal to the circumradius of a face. The circumradius R_T of T is given by

 R_T=sqrt(1/6(3+sqrt(5))).
(1)
PentagonApothem

To derive the volume of an icosahedron having edge length a, consider the orientation so that two polyhedron vertices are oriented on top and bottom. The vertical distance between the top and bottom pentagonal dipyramids is then given by

 z=sqrt(l^2-x^2),
(2)

where

 l=1/2sqrt(3)a
(3)

is the height of an equilateral triangle, and the sagitta x=R^'-r^' of the pentagon is

 x=1/2a1/(10)sqrt(25-10sqrt(5))a,
(4)

giving

 x^2=1/(20)sqrt(5-2sqrt(5))a^2.
(5)

Plugging (3) and (5) into (2) gives

 z=1/(10)sqrt(50+10sqrt(5))a,
(6)

which is identical to the radius of a pentagon of side a. The circumradius is then

 R=h+1/2z,
(7)

where

 h=1/(10)sqrt(50-10sqrt(5))a
(8)

is the height of a pentagonal dipyramid. Therefore,

R^2=(h+1/2z)^2
(9)
=1/8(5+sqrt(5))a^2.
(10)

Taking the square root gives the circumradius

R=sqrt(1/8(5+sqrt(5)))a
(11)
=1/4sqrt(10+2sqrt(5))a
(12)
 approx 0.95105a.
(13)

The inradius is

 r=1/(12)(3sqrt(3)+sqrt(15))a approx 0.75576a.
(14)

The square of the midradius is

 rho^2=(1/2z)^2+x_l^2=1/8(3+sqrt(5))a^2,
(15)

so

 rho=sqrt(1/8(3+sqrt(5)))a=1/4(1+sqrt(5))a approx 0.80901a.
(16)

The dihedral angle is

 alpha=cos^(-1)(-1/3sqrt(5)) approx 138.19 degrees
(17)

and the Dehn invariant of a regular icosahedron is

D=60<3>_5
(18)
=30sin^(-1)((sqrt(5))/3),
(19)
=25.23206...
(20)

(OEIS A377698), where the first expression uses the basis of Conway et al. (1999).

The area of one face is the area of an equilateral triangle

 A=1/4a^2sqrt(3).
(21)

The volume can be computed by taking 20 pyramids of height r

 V=20[(1/3A)r]=5/(12)(3+sqrt(5))a^3.
(22)

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

 (V_(icosahedron))/(V_(dodecahedron))=(A_(icosahedron))/(A_(dodecahedron)),
(23)

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))).
(24)

See also

Augmented Tridiminished Icosahedron, Cube 5-Compound, Decagon, Dodecahedron, Great Icosahedron, Icosahedron, Icosahedron Stellations, Jessen's Orthogonal Icosahedron, Metabidiminished Icosahedron, Polyhedron Coloring, Regular Polyhedron, Rhombic Triacontahedron, Skew Icosahedron, Tridiminished Icosahedron, Trigonometry Values--pi/5

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 228, 1987.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.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Coxeter, H. S. M.; Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra. Stradbroke, England: Tarquin Publications, 1999.Cundy, H. and Rollett, A. "Icosahedron 3^5." §3.5.5 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 88, 1989.Davie, T. "The Icosahedron." http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/icosahedron.html.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. "Icosahedron." http://www.scienceu.com/geometry/facts/solids/icosa.html.Gurkewitz, R. and Arnstein, B. 3-D Geometric Origami: Modular Polyhedra. New York: Dover, 1995.Har'El, Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47, 57-110, 1993.Harris, J. W. and Stocker, H. "Icosahedron." §4.4.6 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.Kasahara, K. Origami Omnibus: Paper-Folding for Everyone. Tokyo: Japan Publications, p. 204, 1988.Klein, F. Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. New York: Dover, 1956.Maeder, R. E. "22: Icosahedron." 1997. https://www.mathconsult.ch/static/unipoly/22.html.Pappas, T. "The Icosahedron & the Golden Rectangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 115, 1989.Sloane, N. J. A. Sequence A377698 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 199-201 and 209-210, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 163, 1991.Wenninger, M. J. "The Icosahedron." Model 4 in Polyhedron Models. Cambridge, England: Cambridge University Press, pp. 17-18, 1989.

Cite this as:

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

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