Sphenocorona (EntityTopic, 14)

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Revision as of 01:24, 2 February 2018

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

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