Spheritorus (EntityTopic, 11)

From Hi.gher. Space

Revision as of 20:41, 2 February 2014

Of all the four-dimensional torii, the spheritorus, previously known as the toracubinder, is the closest analog to the three-dimensional torus. It is formed by taking an uncapped spherinder and connecting its ends in a loop. Its toratopic dual is the torisphere. It has two possible cross-sections in coordinate planes through the origin: the torus, and two disjoint spheres.

Equations

• Variables:

R ⇒ major radius of the spheritorus
r ⇒ minor radius of the spheritorus
h ⇒ height of the spheritorus

• All points (x, y, z, w) that lie on the surcell of a spheritorus will satisfy the following equation:

(√(x² + y²) − R)² + z² + w² = r²

• The parametric equations are:

x = r cos a cos b cos c + R cos c
y = r cos a cos b sin c + R sin c
z = r cos a sin b
w = r sin a

• The hypervolumes of a spheritorus are given by:

total edge length = Unknown
total surface area = Unknown
surcell volume = 4π²Rr(r+h)
bulk = 2π²Rr²h

• The realmic cross-sections (n) of a spheritorus are:

Unknown

Cross-sections

Jonathan Bowers aka Polyhedron Dude created these two excellent cross-section renderings:
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ExPar: [#img] is obsolete, use [#embed] instead