Noble tetragonal tetracontoctahedron

Noble tetragonal tetracontoctahedron
(3D model) (OFF file)
Rank	3
Type	Noble
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 octagrammic icositetrahedron
Convex core	Non-Catalan disdyakis dodecahedron
Abstract & topological properties
Flag count	384
Euler characteristic	–24
Schläfli type	{4,8}
Orientable	Yes
Genus	13
Properties
Symmetry	B3, order 48
Flag orbits	8
Convex	No
Nature	Tame
History
Discovered by	Plasmath
First discovered	2022

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 $\left(\pm a,\,\pm b,\,0\right)$, 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 $x^{3}-x^{2}-1$.