Torisphere (EntityTopic, 11)
From Hi.gher. Space
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:
(√(x^{2} + y^{2} + z^{2}) − R)^{2} + w^{2} = r^{2}
- 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 = 8π^{2}Rr^{2}
bulk = 8π^{2}Rr^{3}3^{-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:
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)I Spherinder | 7b. ((III)I) Torisphere | 8a. ((II)I)I Torinder | 8b. (((II)I)I) Ditorus |
List of toratopes |