Truncated dodecadodecahedron

Truncated dodecadodecahedron
Type	Uniform star polyhedron
Elements	F = 54, E = 180 V = 120 (χ = −6)
Faces by sides	30{4}+12{10}+12{^10/3}
Wythoff symbol	2 5 ^5/3 |
Symmetry group	Ih, [5,3], *532
Index references	U59, C75, W98
Dual polyhedron	Medial disdyakis triacontahedron
Vertex figure	4.^10/9.^10/3
Bowers acronym	Quitdid

In geometry, the truncated dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U₅₉. It is given a Schläfli symbol t_0,1,2{5/3,5}. It has 120 vertices and 54 faces: 30 squares, 12 decagons, and 12 decagrams. The central region of the polyhedron is connected to the exterior via 20 small triangular holes.

The name truncated dodecadodecahedron is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by Coxeter, Longuet-Higgins & Miller (1954).^[1] For this reason, it is also known as the quasitruncated dodecadodecahedron.^[2] Coxeter et al. credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch.^[3]

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated dodecadodecahedron are all the triples of numbers obtained by circular shifts and sign changes from the following points (where $\varphi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio):

$(1,1,3);\quad (\frac{1}{\varphi}, \frac{1}{\varphi^2}, 2\varphi);\quad (\varphi, \frac{2}{\varphi}, \varphi^2);\quad (\varphi^2, \frac{1}{\varphi^2}, 2);\quad (2\varphi-1,1,2\varphi-1).$

Each of these five points has eight possible sign patterns and three possible circular shifts, giving a total of 120 different points.

As a Cayley graph

The truncated dodecadodecahedron forms a Cayley graph for the symmetric group on five elements, as generated by two group members: one that swaps the first two elements of a five-tuple, and one that performs a circular shift operation on the last four elements. That is, the 120 vertices of the polyhedron may be placed in one-to-one correspondence with the 5! permutations on five elements, in such a way that the three neighbors of each vertex are the three permutations formed from it by swapping the first two elements or circularly shifting (in either direction) the last four elements.^[4]

Related polyhedra

Medial disdyakis triacontahedron

Medial disdyakis triacontahedron
Type	Star polyhedron
Face
Elements	F = 120, E = 180 V = 54 (χ = −6)
Symmetry group	Ih, [5,3], *532
Index references	DU59
dual polyhedron	Truncated dodecadodecahedron

The medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron.

See also

• List of uniform polyhedra

References

1. Coxeter, H. S. M.; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 246: 401–450, doi:10.1098/rsta.1954.0003, JSTOR 91532, MR 0062446. See especially the description as a quasitruncation on p. 411 and the photograph of a model of its skeleton in Fig. 114, Plate IV.
2. Wenninger writes "quasitruncated dodecahedron", but this appear to be a mistake. Wenninger, Magnus J. (1971), "98 Quasitruncated dodecahedron", Polyhedron Models, Cambridge University Press, pp. 152–153.
3. Pitsch, Johann (1881), "Über halbreguläre Sternpolyeder", Zeitschrift für das Realschulwesen 6: 9–24, 72–89, 216. According to Coxeter, Longuet-Higgins & Miller (1954), the truncated dodecadodecahedron appears as no. XII on p.86.
4. Eppstein, David (2008), "The topology of bendless three-dimensional orthogonal graph drawing", in Tollis, Ioannis G.; Patrignani, Marizio, Proc. 16th Int. Symp. Graph Drawing, Lecture Notes in Computer Science 5417, Heraklion, Crete: Springer-Verlag, pp. 78–89, arXiv:0709.4087, doi:10.1007/978-3-642-00219-9_9.

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208

External links

• Weisstein, Eric W., "Truncated dodecadodecahedron", MathWorld.
• Weisstein, Eric W., "Medial disdyakis triacontahedron", MathWorld.