dodecadodecahedron

It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5 5/3, 2 | 5/2 5/4, 2 | 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: r{5/2,5}, r{5/3,5}, r{5/2,5/4}, and r{5/3,5/4} or as Coxeter-Dynkin diagrams: {{CDD|node|5|rat|d2|node_1|5|node}}, {{CDD|node|5|rat|d3|node_1|5|node}}, {{CDD|node|5|rat|d2|node_1|5|rat|d4|node}}, and {{CDD|node|5|rat|d3|node_1|5|rat|d4|node}}.

A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets:

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12 pentagrams and 20 rhombic clusters are necessary. However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal structure.

Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the small dodecahemicosahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the pentagonal faces in common).

File:Small stellated dodecahedron truncations.gif

This polyhedron can be considered a rectified great dodecahedron. It is center of a truncation sequence between a small stellated dodecahedron and great dodecahedron:

The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams). The truncation of the dodecadodecahedron itself is not uniform and attempting to make it uniform results in a degenerate polyhedron (that looks like a small rhombidodecahedron with {10/2} polygons filling up the dodecahedral set of holes), but it has a uniform quasitruncation, the truncated dodecadodecahedron.

It is topologically equivalent to a quotient space of the hyperbolic order-4 pentagonal tiling, by distorting the pentagrams back into regular pentagons. As such, it is topologically a regular polyhedron of index two:[http://homepages.wmich.edu/~drichter/regularpolyhedra.htm The Regular Polyhedra (of index two)] {{Webarchive|url=https://web.archive.org/web/20160304023021/http://homepages.wmich.edu/~drichter/regularpolyhedra.htm |date=2016-03-04 }}, David A. Richter[http://homepages.wmich.edu/~drichter/golay.htm The Golay Code on the Dodecadodecahedron] {{Webarchive|url=https://web.archive.org/web/20181018051918/http://homepages.wmich.edu/~drichter/golay.htm |date=2018-10-18 }}, David A. Richter

{{Uniform polyhedra db|Uniform dual polyhedron stat table|DD}}

The medial rhombic triacontahedron is the dual of the dodecadodecahedron. It has 30 intersecting rhombic faces.

It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares. As such, it is topologically a regular polyhedron of index two:[http://homepages.wmich.edu/~drichter/regularpolyhedra.htm The Regular Polyhedra (of index two)] {{Webarchive|url=https://web.archive.org/web/20160304023021/http://homepages.wmich.edu/~drichter/regularpolyhedra.htm |date=2016-03-04 }}, David A. Richter

Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.

• List of uniform polyhedra

{{reflist}}

• {{citation|last=Badoureau|year=1881|title=Mémoire sur les figures isoscèles|journal=Journal de l'École Polytechnique|volume=49|pages=47–172}}
• {{citation|last=Hess|first=Edmund|year=1878|publisher=Cassel. Th. Kay|jfm=10.0346.03|title=Vier archimedeische Polyeder höherer Art}}
• {{citation|last=Pitsch|title=Über halbreguläre Sternpolyheder|journal= Zeitschrift für das Realschulwesen|volume=7|year=1882|jfm=14.0448.01}}
• {{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983 | doi=10.1017/CBO9780511569371}}

• {{mathworld | urlname = Dodecadodecahedron| title =Dodecadodecahedron}}
• {{mathworld | urlname = MedialRhombicTriacontahedron| title =Medial Rhombic Triacontahedron}}
• [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]

{{Nonconvex polyhedron navigator}}

Category:Uniform polyhedra