# Sphenomegacorona

Sphenomegacorona
Rank3
TypeCRF
Notation
Bowers style acronymWamco
Elements
Faces2+2+4+4+4 triangles, 2 squares
Edges1+1+2+2+2+4+4+4+4+4
Vertices2+2+2+2+4
Vertex figures2 kites, edge lengths 1 and 2
2 rhombi, edge length 1
4 irregular pentagons, edge lengths 1, 1, 1, 1, 2
2+2 pentagons, edge length 1
Measures (edge length 1)
Volume≈ 1.94811
Dihedral angles4–4: ≈ 72.97300°
3–3: ≈ 86.72683°
3–3: ≈ 117.35557°
3–3: ≈ 129.44457°
4–3: ≈ 137.24008°
3–3: ≈ 143.73833°
4–3: ≈ 154.72228°
3–3: ≈ 161.48285°
3–3: ≈ 171.64574°
Central density1
Number of external pieces18
Level of complexity28
Related polytopes
ConjugateSphenomegacorona
Abstract & topological properties
Flag count112
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryK2×I, order 4
ConvexYes
NatureTame

The sphenomegacorona or wamco is one of the 92 Johnson solids (J88). It consists of 2+2+4+4+4 triangles and 2 squares.

It is one of several polyhedra near the end of the list of Johnson solids with no obvious relation to any of the uniform polyhedra. The name is derived from "spheno", meaning a wedge-like arrangement of two "lunes", where each lune consists of a square attached to two triangles, and "megacorona", denoting a large crown-like structure composed of 12 triangles, as opposed to the smaller "corona" of 8 triangles found in the sphenocorona.

## Vertex coordinates

Let k  ≈ 0.59463 be the smallest positive root of the polynomial

{\displaystyle {\begin{aligned}&1680x^{16}-4800x^{15}-3712x^{14}+17216x^{13}+1568x^{12}-24576x^{11}+2464x^{10}+17248x^{9}\\{}&-3384x^{8}-5584x^{7}+2000x^{6}+240x^{5}-776x^{4}+304x^{3}+200x^{2}-56x-23.\end{aligned}}}

Then, coordinates for the vertices of a sphenomegacorona with edge length 1 are given by the points:[1]

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

## Measures

From the coordinates of the sphenomegacorona, one may calculate its volume for unit edge length as approximately 1.94811.[2] The exact value is the greatest real root of the polynomial

{\displaystyle {\begin{aligned}&521578814501447328359509917696x^{32}-985204427391622731345740955648x^{30}\\{}&-16645447351681991898880656015360x^{28}+79710816694053483249372512649216x^{26}\\{}&-152195045391070538203422101864448x^{24}+156280253448056209478031589244928x^{22}\\{}&-96188116617075838858708654227456x^{20}+30636368373570166303441645731840x^{18}\\{}&+5828527077458909552923002273792x^{16}-8060049780765551057159394951168x^{14}\\{}&+1018074792115156107372011716608x^{12}+35220131544370794950945931264x^{10}\\{}&+327511698517355918956755959808x^{8}-116978732884218191486738706432x^{6}\\{}&+10231563774949176791703149568x^{4}-366323949299263261553952192x^{2}\\{}&+3071435678740442112675625.\end{aligned}}}

The dihedral angles may also be calculated in terms of the constant k  given in § Vertex coordinates:

${\displaystyle {\text{4–4: }}2{\text{asin}}(k)\approx 72.97300^{\circ },}$
${\displaystyle {\text{3–3: }}2{\text{asin}}\left({\frac {2k}{\sqrt {3}}}\right)\approx 86.72683^{\circ },}$
${\displaystyle {\text{3–3: }}2{\text{asin}}\left({\sqrt {\frac {1+2k}{3}}}\right)\approx 117.35557^{\circ },}$
${\displaystyle {\text{3–3: }}2{\text{asin}}\left({\sqrt {\frac {-3+4k^{2}}{3\left(-1+k^{2}\right)}}}\right)\approx 129.44457^{\circ },}$
${\displaystyle {\text{4–3: acos}}\left({\frac {3-\left({\sqrt {4-4k^{2}}}+{\sqrt {2+4k-4k^{2}}}\right)^{2}}{2{\sqrt {3}}}}\right)\approx 137.24008^{\circ },}$
${\displaystyle {\text{3–3: }}2{\text{asin}}\left({\frac {\sqrt {2+2k^{4}+{\sqrt {3-7k^{2}+4k^{2}}}-2k^{2}\left(2+{\sqrt {3-7k^{2}+4k^{2}}}\right)}}{{\sqrt {3}}\left(1-k^{2}\right)}}\right)\approx 143.73833^{\circ },}$
${\displaystyle {\text{4–3: acos}}\left(-{\sqrt {\frac {-3+4k^{2}}{3\left(-1+k^{2}\right)}}}\right)\approx 154.72228^{\circ },}$
${\displaystyle {\text{3–3: }}2{\text{asin}}\left({\frac {{\sqrt {3-4k^{2}}}(2k^{2}-1)}{2(k^{2}-1){\sqrt {3(1-k^{2})}}}}+{\frac {1}{2{\sqrt {3}}}}\right)\approx 161.48285^{\circ },}$
${\displaystyle {\text{3–3: }}2{\text{asin}}\left({\frac {1}{2{\sqrt {3}}}}{\sqrt {1+\left({\sqrt {2+k-k^{2}}}+{\frac {2-4k^{2}}{\sqrt {1-k^{2}}}}\right)^{2}+\left(1+{\sqrt {\frac {-3+4k^{2}}{-1+k^{2}}}}\right)^{2}}}\right)\approx 171.64574^{\circ }.}$

## References

1. Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 720.
2. Sloane, N. J. A. (ed.). "Sequence A334114". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.