Second noble kipiscoidal icositetrahedron

Second noble kipiscoidal icositetrahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Elements
Faces	24 irregular pentagons
Edges	12+12+12+12+12
Vertices	24
Vertex figure	Irregular pentagon
Related polytopes
Army	Nonuniform snub cube
Dual	Second noble kisombreroidal icositetrahedron
Convex core	Non-Catalan pentagonal icositetrahedron
Abstract & topological properties
Flag count	240
Euler characteristic	–12
Orientable	No
Genus	14
Properties
Symmetry	B3+, order 24
Convex	No
Nature	Tame

The second noble kipiscoidal icositetrahedron is a noble polyhedron. Its 24 congruent faces are irregular pentagons meeting at congruent order-5 vertices. It is a faceting of a non-uniform snub cubic hull.

The ratio between the longest and shortest edges is 1:2.68320.

Vertex coordinates

This polyhedron has coordinates given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes, of

• (1, a, b),

where

• $a={\sqrt[{3}]{1+{\sqrt {\frac {19}{27}}}}}+{\sqrt[{3}]{1-{\sqrt {\frac {19}{27}}}}}+1\approx 2.76929$

is the real root of $x^{3}-3x^{2}+x-1$, and

• $b={\sqrt[{3}]{{\frac {46}{27}}+{\sqrt {\frac {76}{27}}}}}+{\sqrt[{3}]{{\frac {46}{27}}-{\sqrt {\frac {76}{27}}}}}+{\frac {1}{3}}\approx 2.13039$

is the real root of $x^{3}-x^{2}-x-3$.

These are the same coordinates as the noble kipentagrammic icositetrahedron.