Medial rhombic triacontahedron

Medial rhombic triacontahedron
Rank3
TypeUniform dual
SpaceSpherical
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${\displaystyle \frac34 = 0.75}$
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 ${\displaystyle \frac{3\sqrt3}{4} ≈ 1.29904}$. ​The rhombus faces will have length ${\displaystyle 3\frac{1+\sqrt5}{4} ≈ 2.42705}$, and width ${\displaystyle 3\frac{\sqrt5-1}{4} ≈ 0.92705}$. The rhombi have two interior angles of ${\displaystyle \arccos\left(\frac{\sqrt5}{3}\right) ≈ 41.81031°}$, and two of ${\displaystyle \arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°}$.

Vertex coordinates

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

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

Related polytopes

A fundamental domain of the great dodecahedron in {4,5}.

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.