Noble tetragonal tetracontoctahedron

Noble tetragonal tetracontoctahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Space	Spherical
Elements
Faces	48 irregular tetragons
Edges	24+24+48
Vertices	24
Vertex figure	Mirror-symmetric octagram
Number of external pieces	384
Level of complexity	48
Related polytopes
Army	Semi-uniform truncated octahedron
Dual	Noble faceting of great rhombicuboctahedron
Convex core	Non-Catalan disdyakis dodecahedron
Abstract & topological properties
Flag count	384
Euler characteristic	–24
Orientable	Yes
Genus	13
Properties
Symmetry	B3, order 48
Convex	No
Nature	Tame

The noble tetragonal tetracontoctahedron is a noble polyhedron. Its 48 congruent faces are convex irregular quadrilaterals meeting at congruent order-8 vertices. It is a faceting of a semi-uniform truncated octahedral convex hull.

The ratio between the longest and shortest edges is 1:a ≈ 1:1.57021, where a is the positive real root of $a^6-4a^4+5a^2-3$.

Vertex coordinates

The vertex coordinates are all permutations of $\left(±a,\,±b,\,0\right)$, where $\frac{a}{b} = \frac{1+\sqrt[3]{\frac{29-3\sqrt{93}}{2}}+\sqrt[3]{\frac{29+3\sqrt{93}}{2}}}{3} \approx 1.46557$ is the real root of $x^3-x^2-1$.