# Augmented sphenocorona

Augmented sphenocorona
Rank3
TypeCRF
Notation
Bowers style acronymAuwaco
Elements
Faces1+1+1+1+2+2+2+2+2+2 triangles, 1 square
Edges1+1+1+1+2+2+2+2+2+2+2+2+2+2+2
Vertices1+1+1+2+2+2+2
Vertex figures1+1+2+2 pentagons, edge length 1
2 trapezoids, edge lengths 1, 1, 1, 2
2 irregular pentagons, edge lengths 1, 1, 1, 1, 2
1 square, edge length 1
Measures (edge length 1)
Volume${\displaystyle {\frac {\sqrt {2}}{6}}+{\sqrt {\frac {2+3{\sqrt {6}}+2{\sqrt {13+3{\sqrt {6}}}}}{8}}}\approx 1.75105}$
Central density1
Number of external pieces17
Level of complexity52
Related polytopes
ArmyAuwaco
RegimentAuwaco
DualOrder-4 monotruncated order-5 truncated bi-apiculated tetrahedron
ConjugateAugmented sphenocorona
Abstract & topological properties
Flag count104
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA1×I×I, order 2
ConvexYes
NatureTame

The augmented sphenocorona is one of the 92 Johnson solids (J87). It consists of 1+1+1+1+2+2+2+2+2+2 triangles and 1 square. It can be constructed by attaching a square pyramid to one of the square faces of the sphenocorona.

## Coordinates

Coordinates for the vertices of an sphenocorona with unit edge length are given by:

• ${\displaystyle (0,\pm 1/2,{\sqrt {1-k^{2}}})}$,
• ${\displaystyle (\pm k,\pm 1/2,0)}$,
• ${\displaystyle \left(0,\pm \left({\frac {1}{2}}+{\frac {\sqrt {3-4k^{2}}}{2{\sqrt {1-k^{2}}}}}\right),{\frac {1-2k^{2}}{2{\sqrt {1-k^{2}}}}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},0,-{\sqrt {{\frac {1}{2}}+k-k^{2}}}\right)}$,
• ${\displaystyle \left({\frac {k+{\sqrt {2-2k^{2}}}}{2}},0,{\frac {k+{\sqrt {2-2k^{2}}}}{2}}\right)}$.

where k  ≈ 0.85273 is the smallest positive root of the quartic polynomial:

${\displaystyle 60x^{4}-48x^{3}-100x^{2}+56x+23}$
This root can also be given as:
${\displaystyle k={\frac {6+{\sqrt {6}}+2{\sqrt {213-57{\sqrt {6}}}}}{30}}}$