Medial rhombic triacontahedron

Medial rhombic triacontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual, Abstractly regular
Notation
Bowers style acronym	Mort
Coxeter diagram	o5/2m5o ()
Schläfli symbol	$\{4,5\}_{6}$
Elements
Faces	30 rhombi
Edges	60
Vertices	12+12
Vertex figure	12 pentagons, 12 pentagrams
Measures (edge length 1)
Inradius	${\frac {3}{4}}=0.75$
Dihedral angle	120°
Central density	3
Number of external pieces	60
Related polytopes
Dual	Dodecadodecahedron
Halving	Great dodecahedron
Conjugate	Medial rhombic triacontahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	240
Euler characteristic	–6
Schläfli type	{4,5}
Surface	Bring's surface
Orientable	Yes
Genus	4
Properties
Symmetry	H3, order 120
Flag orbits	2
Convex	No
Nature	Tame

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 ${\frac {3{\sqrt {3}}}{4}}\approx 1.29904$. ​The rhombus faces will have length $3{\frac {1+{\sqrt {5}}}{4}}\approx 2.42705$, and width $3{\frac {{\sqrt {5}}-1}{4}}\approx 0.92705$. The rhombi have two interior angles of $\arccos \left({\frac {\sqrt {5}}{3}}\right)\approx 41.81031^{\circ }$, and two of $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$.

Vertex coordinates

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

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

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.

External links

• Hartley, Michael. "{4,5}*240".
• Klitzing, Richard. "did".
• Wikipedia contributors. "Medial rhombic triacontahedron".
• McCooey, David. "Medial Rhombic Triacontahedron"