Sphenocorona (EntityTopic, 14)

From Hi.gher. Space

The sphenocorona is the 86th Johnson solid, J86.

Cartesian coordinates

The Cartesian coordinates of the sphenocorona, resting on the origin and with edge length 2, are:

(0, 0, ±1)

(±A, B, ±1)

(0, C, ±D)

(±1, E, 0)

where A, B, C, D, E are roots of the following polynomials:

92 + 112A - 100A² - 24A³ + 15A⁴	1 < A < 2
3600 - 96B² - 3176B⁴ - 24B⁶ + 225B⁸	1 < B < 1.5
3600 - 96C² - 3176C⁴ - 24C⁶ + 225C⁸	1.5 < C < 2
95 + 100D - 82D² - 36D³ + 15D⁴	1 < D < 2
-20 - 4E² + E⁴	1 < E < 2

Or, in closed form:

A = (1/15)(6 + √6 + 2√(213 - 57√6))

B = (2/5)√(1/6 + 6√6 - (1/3)√(538 + 18√6))

C = (2/5)√(1/6 + 6√6 + (1/3)√(538 + 18√6))

D = (1/15)(9 - √6 + 2√(213 - 57√6))

E = √(2(1 + √6))

Their numerical values are approximately:

A = 1.705453885692834

B = 1.044713857367277

C = 1.914399800381786

D = 1.578855253321743

E = 2.626590848527109

Images

Notable Trishapes
Regular:	tetrahedron • cube • octahedron • dodecahedron • icosahedron
Direct truncates:	tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate
Mesotruncates:	stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron
Snubs:	snub staurohedron • snub rhodohedron
Curved:	sphere • torus • cylinder • cone • frustum • crind