# Medial rhombic triacontahedron

Medial rhombic triacontahedron
Rank3
TypeUniform dual, Abstractly regular
Notation
Bowers style acronymMort
Coxeter diagramo5/2m5o ()
Schläfli symbol${\displaystyle \{4,5\}_{6}}$
Elements
Faces30 rhombi
Edges60
Vertices12+12
Vertex figure12 pentagons, 12 pentagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {3}{4}}=0.75}$
Dihedral angle120°
Central density3
Number of external pieces60
Related polytopes
HalvingGreat dodecahedron
ConjugateMedial rhombic triacontahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count240
Euler characteristic–6
Schläfli type{4,5}
SurfaceBring's surface
OrientableYes
Genus4
Properties
SymmetryH3, order 120
Flag orbits2
ConvexNo
NatureTame

The medial rhombic triacontahedron is a uniform dual polyhedron. It consists of 30 rhombi.

If its dual, the dodecadodecahedron, has an edge length of 1, then the edges of the rhombi will measure ${\displaystyle {\frac {3{\sqrt {3}}}{4}}\approx 1.29904}$. ​The rhombus faces will have length ${\displaystyle 3{\frac {1+{\sqrt {5}}}{4}}\approx 2.42705}$, and width ${\displaystyle 3{\frac {{\sqrt {5}}-1}{4}}\approx 0.92705}$. The rhombi have two interior angles of ${\displaystyle \arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }}$, and two of ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$.

## Vertex coordinates

A medial rhombic triacontahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(\pm 3{\frac {1+{\sqrt {5}}}{8}},\,\pm {\frac {3}{4}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {3}{4}},\,\pm 3{\frac {{\sqrt {5}}-1}{8}},\,0\right)}$.

## Related polytopes

This polyhedron is abstractly regular, being a quotient of the order-5 square tiling. As an abstract polytope, it is equivalent to a regular tessellation of Bring's surface. It has a symmetry group of order 240, which is the maximum possible for a tessellation of Bring's surface.

Its realization may also be considered regular if one also counts conjugacies as symmetries.