In general, an icosidodecahedron is a 32-faced
A number of such solids are illustrated above.
(quasiregular) icosidodecahedron is the 32-faced Archimedean
solid with faces .
It is one of the two convex quasiregular polyhedra.
It is also the uniform polyhedron with Maeder
index 24 (Maeder 1997), Wenninger index 12 (Wenninger 1989), Coxeter index 28 (Coxeter
et al. 1954), and Har'El index 29 (Har'El 1993). It has Schläfli
and Wythoff symbol .
It is implemented in the
Wolfram Language as [ PolyhedronData "Icosidodecahedron"]
or [ UniformPolyhedron "Icosidodecahedron"].
Several symmetric projections of the icosidodecahedron are illustrated above.
polyhedron vertices of an icosidodecahedron of polyhedron edge length are , , , , , . The 30 polyhedron
vertices of an octahedron 5-compound
form an icosidodecahedron (Ball and Coxeter 1987). Faceted
versions include the small icosihemidodecahedron
and small dodecahemidodecahedron.
The regular icosidodecahedron is the
convex hull of the cube-octahedron 5-compound, dodecadodecahedron, great
dodecahemicosahedron, great dodecahemidodecahedron,
great icosidodecahedron, great
icosihemidodecahedron, third icosahedron
stellation hull, octahedron 5-compound,
small dodecahemicosahedron, small
dodecahemidodecahedron, and small
The icosidodecahedron constructed in
origami is illustrated above (Kasahara and Takahama 1987, pp. 48-49). This construction uses 120 sonobè
units, each made from a single sheet of origami paper.
The faces of the icosidodecahedron consist of 20 triangles and 12 pentagons. Furthermore, its 60 edges are bisected perpendicularly by those of the reciprocal
triacontahedron (Ball and Coxeter 1987).
Five octahedra of edge length
can be inscribed on the vertices of an icosidodecahedron
of unit edge length, resulting in the beautiful octahedron
dual polyhedron of the icosidodecahedron is the rhombic triacontahedron, both of which
are illustrated above together with their common midsphere.
inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are
surface area and volume
for an icosidodecahedron are given by
The distance to the centers of the triangular and pentagonal faces are
dihedral angle between triangular and pentagonal
The unit regular icosiedodecahedron has
where the first expression uses the basis of Conway
et al. (1999). It can be dissected into the pentagonal
orthobirotunda (E. Weisstein, Aug. 17, 2023).
See also Archimedean Solid
Explore with Wolfram|Alpha
References Baez, J. C. "The Icosidodecahedron." 26 Sep 2023. https://arxiv.org/abs/2309.15774. Ball,
W. W. R. and Coxeter, H. S. M. New York: Dover, p. 137, 1987. Mathematical
Recreations and Essays, 13th ed. 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. "Icosidodecahedron. ." §3.7.8 in Stradbroke, England: Tarquin Pub., p. 108, 1989. Mathematical
Models, 3rd ed. Har'El,
Z. "Uniform Solution for Uniform Polyhedra." Geometriae Dedicata 47,
57-110, 1993. Kasahara, K. "From Regular to Semiregular Polyhedrons."
Tokyo: Japan Publications, pp. 220-221,
Omnibus: Paper-Folding for Everyone. Kasahara, K. and Takahama, T. Tokyo: Japan Publications, 1987. Origami
for the Connoisseur. Geometry
Technologies. "Icosidodecahedron." http://www.scienceu.com/geometry/facts/solids/icosidodeca.html. Maeder,
R. E. "24: Icosidodecahedron." 1997. https://www.mathconsult.ch/static/unipoly/24.html. Wenninger,
M. J. "The Icosidodecahedron." Model 12 in
Cambridge, England: Cambridge University Press, pp. 26 and 73,
Models. Cite this as:
Weisstein, Eric W. "Icosidodecahedron."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/Icosidodecahedron.html