Medial rhombic triacontahedron

{{Short description|Polyhedron with 30 faces}}

{{more footnotes|date=November 2021}}

{{Uniform polyhedra db|Uniform dual polyhedron stat table|DD}}

File:Medial rhombic triacontahedron.stl

In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.

Its 24 vertices are all on the 12 axes with 5-fold symmetry (i.e. each corresponds to one of the 12 vertices of the icosahedron). This means that on each axis there is an inner and an outer vertex. The ratio of outer to inner vertex radius is $\backslash varphi\; \backslash approx\; 1.618$, the golden ratio.

It has 30 intersecting rhombic faces, which correspond to the faces of the convex rhombic triacontahedron. The diagonals in the rhombs of the convex solid have a ratio of 1 to $\backslash varphi$. The medial solid can be generated from the convex one by stretching the shorter diagonal from length 1 to $\backslash varphi^3\; \backslash approx\; 4.236$. So the ratio of rhomb diagonals in the medial solid is 1 to $\backslash varphi^2\; \backslash approx\; 2.618$.

This solid is to the compound of small stellated dodecahedron and great dodecahedron what the convex one is to the compound of dodecahedron and icosahedron:

The crossing edges in the dual compound are the diagonals of the rhombs.

The faces have two angles of $\backslash arccos(\backslash frac\{1\}\{3\}\backslash sqrt\{5\})\backslash approx\; 41.810\backslash ,314\backslash ,895\backslash ,78^\{\backslash circ\}$, and two of $\backslash arccos(-\backslash frac\{1\}\{3\}\backslash sqrt\{5\})\backslash approx\; 138.189\backslash ,685\backslash ,104\backslash ,22^\{\backslash circ\}$. Its dihedral angles equal $\backslash arccos(-\backslash frac\{1\}\{2\})=\; 120^\{\backslash circ\}$. Part of each rhomb lies inside the solid, hence is invisible in solid models.

style="vertical-align: top;" | {{multiple image | align = left | total_width = 500 | image1 = Rhombic triacontahedron 1 (convex), size m (crop matching medial), pyritohedral.png | image2 = Rhombic triacontahedron 2 (medial), size m (crop), pyritohedral.png | image3 = Skeleton pair Gr12 and dual, size m (crop), thick.png | footer = Convex and medial rhombic triacontahedron (both shown with pyritohedral symmetry) and on the right the dual compound of Kepler–Poinsot solids }} | {{multiple image | align = left | total_width = 500 | image1 = Rhombic triacontahedron 2 (medial), size m, 2-fold.png | image2 = Rhombic triacontahedron 2 (medial), size m, 3-fold.png | image3 = Rhombic triacontahedron 2 (medial), size m, 5-fold.png | footer = Orthographic projections from 2-, 3- and 5-fold symmetry axes }}