great dodecicosahedron

{{Short description|Polyhedron with 32 faces}}

{{Uniform polyhedra db|Uniform polyhedron stat table|gDI}}

File:Great dodecicosahedron.stl

In geometry, the great dodecicosahedron (or great dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U₆₃. It has 32 faces (20 hexagons and 12 decagrams), 120 edges, and 60 vertices.{{Cite web|url=https://www.mathconsult.ch/static/unipoly/63.html|title=63: great dodecicosahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}} Its vertex figure is a crossed quadrilateral.

It has a composite Wythoff symbol, 3 {{frac|5|3}} ({{frac|3|2}} {{frac|5|2}}) |, requiring two different Schwarz triangles to generate it: (3 {{frac|5|3}} {{frac|3|2}}) and (3 {{frac|5|3}} {{frac|5|2}}). (3 {{frac|5|3}} {{frac|3|2}} | represents the great dodecicosahedron with an extra 12 {{mset|{{frac|10|2}}}} pentagons, and 3 {{frac|5|3}} {{frac|5|2}} | represents it with an extra 20 {{mset|{{frac|6|2}}}} triangles.){{cite book | first=Magnus | last=Wenninger | author-link=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }} pp. 9–10.

Its vertex figure 6.{{frac|10|3}}.{{frac|6|5}}.{{frac|10|7}} is also ambiguous, having two clockwise and two counterclockwise faces around each vertex.