Disdyakis triacontahedron


Disdyakis triacontahedron
Disdyakis triacontahedron
Disdyakis triacontahedron
Click on picture for large version

spinning version

Type Catalan
Face polygon scalene triangle
Faces 120
Edges 180
Vertices 62 = 12 + 20 + 30
Face configuration V4.6.10
Symmetry group Ih, [5,3], *532
Dihedral angle 164° 53' 17"
Dual polyhedron truncated icosidodecahedron
Properties convex, face-transitive
Disdyakis triacontahedron
Net

In geometry, a disdyakis triacontahedron, or hexakis icosahedron is a Catalan solid and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons. It looks a bit like an inflated rhombic triacontahedron—if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It also has the most faces among the Archimedean and Catalan solids, with in second place the Snub dodecahedron, which is an enneacontadihedron.

The edges of the polyhedron projected onto a sphere form great circles, and represent all ten mirror planes of reflective Ih icosahedral symmetry, as shown in this image. Combining pairs of light and dark triangles define the fundamental domains of the nonreflective I icosahedral symmetry.

Icosahedral reflection domains.png Disdyakis triacontahedron.png

See also

References

  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
  • Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR730208  (The thirteen semiregular convex polyhedra and their duals, Page 25, Disdyakistriacontahedron )
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, kisRhombic triacontahedron )

External links