Torisphere (EntityTopic, 11)
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| - | {{Shape| | + | <[#ontology [kind topic] [cats 4D Curved Toratope]]> |
| + | {{STS Shape | ||
| + | | dim=4 | ||
| + | | elements=1, 0, 0, 0 | ||
| + | | genus=1 | ||
| + | | ssc=[(xyz)w]T | ||
| + | | ssc2=T((3)1) | ||
| + | | extra={{STS Toratope | ||
| + | | expand=[[Cylspherinder|32]] | ||
| + | | notation=((III)I) | ||
| + | | index=7b | ||
| + | }}}} | ||
| - | + | 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 [[sphere]]s. | |
| - | The toraspherinder is | + | |
| - | + | == Equations == | |
*Variables: | *Variables: | ||
| - | <blockquote>''r'' ⇒ minor radius of the | + | <blockquote>''r'' ⇒ minor radius of the torisphere<br> |
| - | ''R'' ⇒ major radius of the | + | ''R'' ⇒ major radius of the torisphere<br></blockquote> |
| - | *All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a | + | *All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a torisphere will satisfy the following equation: |
| - | <blockquote>( | + | <blockquote>(√(''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup>) − ''R'')<sup>2</sup> + ''w''<sup>2</sup> = ''r''<sup>2</sup></blockquote> |
*The parametric equations are: | *The parametric equations are: | ||
| Line 18: | Line 28: | ||
w = r sin a </blockquote> | w = r sin a </blockquote> | ||
| - | *The [[hypervolume]]s of a | + | *The [[hypervolume]]s of a torisphere are given by: |
<blockquote>total edge length = 0<br> | <blockquote>total edge length = 0<br> | ||
total surface area = 0<br> | total surface area = 0<br> | ||
| Line 24: | Line 34: | ||
bulk = 8π<sup>2</sup>''Rr''<sup>3</sup>3<sup>-1</sup></blockquote> | bulk = 8π<sup>2</sup>''Rr''<sup>3</sup>3<sup>-1</sup></blockquote> | ||
| - | *The [[realmic]] [[cross-section]]s (''n'') of a | + | *The [[realmic]] [[cross-section]]s (''n'') of a torisphere are: |
<blockquote>''Unknown''</blockquote> | <blockquote>''Unknown''</blockquote> | ||
| + | |||
| + | == Cross-sections == | ||
| + | [[User:Polyhedron Dude|Jonathan Bowers aka Polyhedron Dude]] created these two excellent cross-section renderings:<br/> | ||
| + | <[#embed [hash VNECTP4FCK6KRVHXZN8HC553GZ] [width 676]]><br/> | ||
| + | <[#embed [hash J16NA77JDZMTG938PXTKCGVQXX] [width 676]]> | ||
| + | |||
<br clear="all"><br> | <br clear="all"><br> | ||
| - | {{ | + | {{Tetrashapes}} |
| - | {{ | + | {{Toratope Nav B|6|7|8|(II)(II)<br>Duocylinder|((II)(II))<br>Tiger|(III)I<br>Spherinder|((III)I)<br>Torisphere|((II)I)I<br>Torinder|(((II)I)I)<br>Ditorus|chora}} |
Latest revision as of 20:46, 11 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:
| 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 | |||||
