Sphenomegacorona Rank 3 Type CRF Notation Bowers style acronym W am co Elements Faces 2+2+4+4+4 triangles , 2 squares Edges 1+1+2+2+2+4+4+4+4+4 Vertices 2+2+2+2+4 Vertex figures 2 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 angles 4–4: ≈ 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 density 1 Number of external pieces 18 Level of complexity 28 Related polytopes Dual Parabitruncated dipentadeltotetratetragoditriangular decahedron Conjugate Sphenomegacorona Abstract & topological properties Flag count112 Euler characteristic 2 Surface Sphere Orientable Yes Genus 0 Properties Symmetry K2 ×I , order 4Convex Yes Nature Tame
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 .
Let k
1680
x
16
−
4800
x
15
−
3712
x
14
+
17216
x
13
+
1568
x
12
−
24576
x
11
+
2464
x
10
+
17248
x
9
−
3384
x
8
−
5584
x
7
+
2000
x
6
+
240
x
5
−
776
x
4
+
304
x
3
+
200
x
2
−
56
x
−
23.
{\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]
(
0
,
±
1
2
,
1
−
k
2
)
,
{\displaystyle \left(0,\pm {\frac {1}{2}},{\sqrt {1-k^{2}}}\right),}
(
±
k
,
±
1
2
,
0
)
,
{\displaystyle \left(\pm k,\pm {\frac {1}{2}},0\right),}
(
0
,
±
(
3
−
4
k
2
2
1
−
k
2
+
1
2
)
,
1
−
2
k
2
2
1
−
k
2
)
,
{\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),}
(
±
1
2
,
0
,
−
1
2
2
+
4
k
−
4
k
2
)
,
{\displaystyle \left(\pm {\frac {1}{2}},0,-{\frac {1}{2}}{\sqrt {2+4k-4k^{2}}}\right),}
(
0
,
±
(
3
−
4
k
2
(
2
k
2
−
1
)
2
(
k
2
−
1
)
1
−
k
2
+
1
2
)
,
2
k
4
−
1
2
(
1
−
k
2
)
3
2
)
.
{\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).}
From the coordinates of the sphenomegacorona, one may calculate its volume for unit edge length as approximately 1.94811.[2]
521578814501447328359509917696
x
32
−
985204427391622731345740955648
x
30
−
16645447351681991898880656015360
x
28
+
79710816694053483249372512649216
x
26
−
152195045391070538203422101864448
x
24
+
156280253448056209478031589244928
x
22
−
96188116617075838858708654227456
x
20
+
30636368373570166303441645731840
x
18
+
5828527077458909552923002273792
x
16
−
8060049780765551057159394951168
x
14
+
1018074792115156107372011716608
x
12
+
35220131544370794950945931264
x
10
+
327511698517355918956755959808
x
8
−
116978732884218191486738706432
x
6
+
10231563774949176791703149568
x
4
−
366323949299263261553952192
x
2
+
3071435678740442112675625.
{\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 § Vertex coordinates :
4–4:
2
asin
(
k
)
≈
72.97300
∘
,
{\displaystyle {\text{4–4: }}2{\text{asin}}(k)\approx 72.97300^{\circ },}
3–3:
2
asin
(
2
k
3
)
≈
86.72683
∘
,
{\displaystyle {\text{3–3: }}2{\text{asin}}\left({\frac {2k}{\sqrt {3}}}\right)\approx 86.72683^{\circ },}
3–3:
2
asin
(
1
+
2
k
3
)
≈
117.35557
∘
,
{\displaystyle {\text{3–3: }}2{\text{asin}}\left({\sqrt {\frac {1+2k}{3}}}\right)\approx 117.35557^{\circ },}
3–3:
2
asin
(
−
3
+
4
k
2
3
(
−
1
+
k
2
)
)
≈
129.44457
∘
,
{\displaystyle {\text{3–3: }}2{\text{asin}}\left({\sqrt {\frac {-3+4k^{2}}{3\left(-1+k^{2}\right)}}}\right)\approx 129.44457^{\circ },}
4–3: acos
(
3
−
(
4
−
4
k
2
+
2
+
4
k
−
4
k
2
)
2
2
3
)
≈
137.24008
∘
,
{\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 },}
3–3:
2
asin
(
2
+
2
k
4
+
3
−
7
k
2
+
4
k
2
−
2
k
2
(
2
+
3
−
7
k
2
+
4
k
2
)
3
(
1
−
k
2
)
)
≈
143.73833
∘
,
{\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 },}
4–3: acos
(
−
−
3
+
4
k
2
3
(
−
1
+
k
2
)
)
≈
154.72228
∘
,
{\displaystyle {\text{4–3: acos}}\left(-{\sqrt {\frac {-3+4k^{2}}{3\left(-1+k^{2}\right)}}}\right)\approx 154.72228^{\circ },}
3–3:
2
asin
(
3
−
4
k
2
(
2
k
2
−
1
)
2
(
k
2
−
1
)
3
(
1
−
k
2
)
+
1
2
3
)
≈
161.48285
∘
,
{\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 },}
3–3:
2
asin
(
1
2
3
1
+
(
2
+
k
−
k
2
+
2
−
4
k
2
1
−
k
2
)
2
+
(
1
+
−
3
+
4
k
2
−
1
+
k
2
)
2
)
≈
171.64574
∘
.
{\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 }.}
