# Torisphere (EntityTopic, 11)

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

## Cross-sections

Jonathan Bowers aka Polyhedron Dude created these two excellent cross-section renderings:

 Notable Tetrashapes Regular: pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron Powertopes: triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate Circular: glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder Torii: tiger • torisphere • spheritorus • torinder • ditorus

 6a. (II)(II)Duocylinder 6b. ((II)(II))Tiger 7a. (III)ISpherinder 7b. ((III)I)Torisphere 8a. ((II)I)ITorinder 8b. (((II)I)I)Ditorus List of toratopes