ditrigonal dodecadodecahedron
{{Short description|Polyhedron with 24 faces}}
{{Uniform polyhedra db|Uniform polyhedron stat table|dDD}}
File:Ditrigonal dodecadodecahedron.stl
In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices.{{Cite web|url=https://www.mathconsult.ch/static/unipoly/41.html|title=41: ditrigonal dodecadodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|url-status=live|archive-url=https://web.archive.org/web/20150921043423/http://www.mathconsult.ch/static/unipoly/41.html |archive-date=2015-09-21 |access-date=}} It has extended Schläfli symbol b{5,{{Frac|5|2}}}, as a blended great dodecahedron, and Coxeter diagram {{CDD|node|5|node_h3|5-2|node}}. It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | {{Frac|5|3}} 5, and Coxeter diagram File:ditrigonal dodecadodecahedron cd.png.
Related polyhedra
Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron (having the pentagrammic faces in common), the great ditrigonal icosidodecahedron (having the pentagonal faces in common), and the regular compound of five cubes.
| class="wikitable" width="400" style="vertical-align:top;text-align:center"
!a{5,3} !a{{{Frac|5|2}},3} !b{5,{{Frac|5|2}}} |
| {{CDD|label5-2|branch_10ru|split2|node}} = {{CDD|node_h3|5|node|3|node}}
!{{CDD||label5-4|branch_10ru|split2|node}} = {{CDD|node_h3|5-2|node|3|node}} !File:ditrigonal dodecadodecahedron cd.png = {{CDD|node|5|node_h3|5-2|node}} |
|---|
| align=center|100px Small ditrigonal icosidodecahedron |align=center|100px |align=center|100px |
| align=center|100px Dodecahedron (convex hull) |align=center|100px |
Furthermore, it may be viewed as a facetted dodecahedron: the pentagrammic faces are inscribed in the dodecahedron's pentagons. Its dual, the medial triambic icosahedron, is a stellation of the icosahedron.
It is topologically equivalent to a quotient space of the hyperbolic order-6 pentagonal tiling, by distorting the pentagrams back into regular pentagons. As such, it is 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
See also
References
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External links
- {{mathworld | urlname = DitrigonalDodecadodecahedron| title = Ditrigonal dodecadodecahedron}}
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