# Medial rhombic triacontahedron

Medial rhombic triacontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Bowers style acronymMort
Coxeter diagramo5/2m5o (       )
Schläfli symbol$\{4,5\}_6$ Elements
Faces30 rhombi
Edges60
Vertices12+12
Vertex figure12 pentagons, 12 pentagrams
Measures (edge length 1)
Inradius$\frac34 = 0.75$ Dihedral angle120°
Central density3
Number of external pieces60
Related polytopes
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count240
Euler characteristic–6
Schläfli type{4,5}
SurfaceBring's surface
OrientableYes
Genus4
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The medial rhombic triacontahedron, or mort, 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 $\frac{3\sqrt3}{4} ≈ 1.29904$ . ​The rhombus faces will have length $3\frac{1+\sqrt5}{4} ≈ 2.42705$ , and width $3\frac{\sqrt5-1}{4} ≈ 0.92705$ . The rhombi have two interior angles of $\arccos\left(\frac{\sqrt5}{3}\right) ≈ 41.81031°$ , and two of $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°$ .

## Vertex coordinates

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

• $\left(±3\frac{1+\sqrt5}{8},\,±\frac34,\,0\right),$ • $\left(±\frac34,\,±3\frac{\sqrt5-1}{8},\,0\right).$ ## 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.