Torisphere (EntityTopic, 11)

From Hi.gher. Space

Revision as of 20:41, 2 February 2014

The torisphere, previously known as the toraspherinder, is a four-dimensional torus formed by taking an uncapped spherinder and connecting its ends through its inside. Its toratopic dual is the spheritorus. It has two possible cross-sections in coordinate planes through the origin: the torus, and two concentric spheres.

Equations

• Variables:

r ⇒ minor radius of the torisphere
R ⇒ major radius of the torisphere

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

(√(x² + y² + z²) − R)² + w² = r²

• The parametric equations are:

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

• The hypervolumes of a torisphere are given by:

total edge length = 0
total surface area = 0
surcell volume = 8π²Rr²
bulk = 8π²Rr³3^-1

• The realmic cross-sections (n) of a torisphere 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ExPar: [#img] is obsolete, use [#embed] instead