Spheritorus (EntityTopic, 11)
From Hi.gher. Space
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*All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a toracubinder will satisfy the following equation: | *All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a toracubinder will satisfy the following equation: | ||
| - | <blockquote>(√(''x''<sup>2</sup> + ''y''<sup>2</sup>) | + | <blockquote>(√(''x''<sup>2</sup> + ''y''<sup>2</sup>) − ''R'')<sup>2</sup> + ''z''<sup>2</sup> + ''w''<sup>2</sup> = ''r''<sup>2</sup></blockquote> |
*The parametric equations are: | *The parametric equations are: | ||
Revision as of 10:59, 12 March 2011
Of all the four-dimensional torii, 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 toraspherinder.
Equations
- Variables:
R ⇒ major radius of the toracubinder
r ⇒ minor radius of the toracubinder
h ⇒ height of the toracubinder
- All points (x, y, z, w) that lie on the surcell of a toracubinder 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 toracubinder 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 toracubinder are:
Unknown
| 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 |
| 4a. IIII Tesseract | 4b. (IIII) Glome | 5a. (II)II Cubinder | 5b. ((II)II) Toracubinder | 6a. (II)(II) Duocylinder | 6b. ((II)(II)) Tiger |
| List of toratopes | |||||
