# Noble tetragonal tetracontoctahedron

Noble tetragonal tetracontoctahedron
Rank3
TypeNoble
Elements
Faces48 irregular tetragons
Edges24+24+48
Vertices24
Vertex figureMirror-symmetric octagram
Number of external pieces384
Level of complexity48
Related polytopes
ArmySemi-uniform truncated octahedron
DualNoble octagrammic icositetrahedron
Convex coreNon-Catalan disdyakis dodecahedron
Abstract & topological properties
Flag count384
Euler characteristic–24
Schläfli type{4,8}
OrientableYes
Genus13
Properties
SymmetryB3, order 48
Flag orbits8
ConvexNo
NatureTame
History
Discovered byPlasmath
First discovered2022

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 ${\displaystyle a^{6}-4a^{4}+5a^{2}-3}$.

## Vertex coordinates

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