Hebesphenomegacorona

Hebesphenomegacorona
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Hawmco
Elements
Faces	2+2+2+4+4+4 triangles, 1+2 squares
Edges	1+2+2+2+2+4+4+4+4+4+4
Vertices	2+2+2+4+4
Vertex figures	2+2+2 pentagons, edge length 1
	4 irregular pentagons, edge lengths 1, 1, 1, 1, √2
	4 kites, edge lengths 1 and √2
Measures (edge length 1)
Volume	$\approx 2.91291$[1]
Central density	1
Number of external pieces	21
Level of complexity	33
Related polytopes
Army	Hawmco
Regiment	Hawmco
Dual	Order-5 truncated tetratrigotetratetragodipentapentagonal dodecahedron
Abstract & topological properties
Flag count	132
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	K2×I, order 4
Convex	Yes
Nature	Tame

The hebesphenomegacorona is one of the 92 Johnson solids (J₈₉). It consists of 18 triangles and 3 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 "hebespheno" (meaning a wedge-like arrangement of three "lunes", where each lune consists of a square attached to two triangles) and "megacorona" denoting a crown-like structure composed of 12 triangles (as opposed to the smaller "corona" of 8 triangles found in the sphenocorona).

It has a weak relation to the icosahedron. If the middle square is contracted to an edge such that the neighboring squares become triangles and the neighboring triangles touch, the result is an icosahedron.

Vertex coordinates

Vertex coordinates for a hebesphenomegacorona with unit edge length are given by

• $\left(\pm {\frac {1}{2}},\pm {\frac {1}{2}},u\right)$,
• $\left(\pm \left({\frac {1}{2}}+k\right),\pm {\frac {1}{2}},0\right)$,
• $\left(\pm {\frac {1}{2}},0,-{\frac {w}{2}}\right)$,
• $\left(0,\pm \left({\frac {1}{2}}+{\frac {v}{2u}}\right),{\frac {v^{2}}{4u}}\right)$,
• $\left(0,\pm {\frac {wv+k+1}{4u^{2}}},{\frac {(2k-1)w}{4(1-k)}}-{\frac {v}{4u^{2}}}\right)$,

where $k\approx 0.21684$ is the second to smallest positive real root of

$26880x^{10}+35328x^{9}-25600x^{8}-39680x^{7}+6112x^{6}$

${}+13696x^{5}+2128x^{4}-1808x^{3}-1119x^{2}+494x-47$,

and with u , v , w  given by:

$u={\sqrt {1-k^{2}}},\ v={\sqrt {2-2k-4k^{2}}},\ w={\sqrt {3-4k^{2}}}$.

From these coordinates, the volume can be calculated by $\xi \approx 2.91291$, where ξ  is given as the greatest real root of

$2693461945329+132615435213216x^{2}-22211277300912896x^{4}$

${}-8337259437908852736x^{6}+366229890219212144640x^{8}$

${}-3065290664181478981632x^{10}+8973584611317745975296x^{12}$

${}-8432333285523990773760x^{14}+3596480447590271287296x^{16}$

${}-722445512980071186432x^{18}+47330370277129322496x^{20}$.

External links

• Klitzing, Richard. "hawmco".
• Quickfur. "The Hebesphenomegacorona".

• Wikipedia contributors. "Hebesphenomegacorona".
• McCooey, David. "Hebesphenomegacorona"