Great disdyakis dodecahedron

Great disdyakis dodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Coxeter diagram	m4/3m3m
Elements
Faces	48 scalene triangles
Edges	24+24+24
Vertices	12+8+6
Vertex figures	12 squares
	8 hexagons
	6 octagrams
Measures (edge length 1)
Inradius	$3{\frac {\sqrt {194(15-8{\sqrt {2}})}}{97}}\approx 0.82708$
Volume	$144{\frac {{\sqrt {2}}-1}{7}}\approx 8.52096$
Dihedral angles	48 edges: $\arccos \left(-{\frac {71-12{\sqrt {2}}}{97}}\right)\approx 123.84889^{\circ }$
	24 edges: $180^{\circ }+\arccos \left({\frac {71-12{\sqrt {2}}}{97}}\right)\approx 236.15111^{\circ }$
Central density	–1
Number of external pieces	48
Related polytopes
Dual	Quasitruncated cuboctahedron
Abstract & topological properties
Flag count	288
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

The great disdyakis dodecahedron is a uniform dual polyhedron. It consists of 48 scalene triangles.

If its dual, the great cubicuboctahedron, has an edge length of 1, then the short edges of the triangles will measure $3{\frac {\sqrt {6\left(2-{\sqrt {2}}\right)}}{7}}\approx 0.80347$, the medium edges will be $2{\frac {\sqrt {3\left(10+{\sqrt {2}}\right)}}{7}}\approx 1.67192$, and the long edges will be $2{\frac {\sqrt {6\left(10-{\sqrt {2}}\right)}}{7}}\approx 2.05068$. The triangles have one interior angle of $\arccos \left({\frac {3}{4}}+{\frac {\sqrt {2}}{8}}\right)\approx 22.06219^{\circ }$, one of $\arccos \left(-{\frac {1}{6}}-{\frac {\sqrt {2}}{12}}\right)\approx 106.53003^{\circ }$, and one of $\arccos \left(-{\frac {1}{12}}+{\frac {\sqrt {2}}{2}}\right)\approx 51.40778^{\circ }$.

Vertex coordinates

A great disdyakis dodecahedron with dual edge length 1 has vertex coordinates given by all permutations of:

• $\left(\pm 3{\frac {3{\sqrt {2}}-2}{7}},\,0,\,0\right),$
• $\left(\pm 3{\frac {2{\sqrt {2}}-1}{7}},\,\pm 3{\frac {2{\sqrt {2}}-1}{7}},\,0\right),$
• $\left(\pm {\sqrt {2}},\,\pm {\sqrt {2}},\,\pm {\sqrt {2}}\right).$

External links

• Wikipedia Contributors. "Great disdyakis dodecahedron".
• McCooey, David. "Great Disdyakis Dodecahedron"