Ditorus (EntityTopic, 11)
From Hi.gher. Space
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| + | <[#ontology [kind topic] [cats 4D Curved Toratope]]> | ||
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| + | The '''ditorus''' is a [[four-dimensional torus]] formed by taking an uncapped [[torinder]] and connecting its ends either in a loop or through its inside. Its [[toratopic dual]] is itself. | ||
== Equations == | == Equations == | ||
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*The parametric equations are: | *The parametric equations are: | ||
<blockquote> | <blockquote> | ||
| - | x = (R + (r + ρ cos | + | ''x'' = (''R'' + (''r'' + ''ρ'' cos ''θ''<sub>3</sub>) cos ''θ''<sub>2</sub>) cos ''θ''<sub>1</sub><br> |
| - | y = (R + (r + ρ cos | + | ''y'' = (''R'' + (''r'' + ''ρ'' cos ''θ''<sub>3</sub>) cos ''θ''<sub>2</sub>) sin ''θ''<sub>1</sub><br> |
| - | z = (r + ρ cos | + | ''z'' = (''r'' + ''ρ'' cos ''θ''<sub>3</sub>) sin ''θ''<sub>2</sub><br> |
| - | w = a sin | + | ''w'' = ''a'' sin ''θ''<sub>3</sub></blockquote> |
*The [[hypervolume]]s of a ditorus are given by: | *The [[hypervolume]]s of a ditorus are given by: | ||
| - | <blockquote>total surface area = 0<br> | + | <blockquote>total surface area = ''0''<br> |
| - | surcell volume = 8π<sup>3</sup>Rrρ<br> | + | surcell volume = 8π<sup>3</sup>''Rrρ''<br> |
| - | bulk = 4π<sup>3</sup>ρ<sup>2</sup>rR</blockquote> | + | bulk = 4π<sup>3</sup>''ρ''<sup>2</sup>''rR''</blockquote> |
*The [[realmic]] [[cross-section]]s (''n'') of a ditorus are: | *The [[realmic]] [[cross-section]]s (''n'') of a ditorus are: | ||
<blockquote>''Unknown''</blockquote> | <blockquote>''Unknown''</blockquote> | ||
| + | |||
| + | == Cross-sections == | ||
| + | [[User:Polyhedron Dude|Jonathan Bowers aka Polyhedron Dude]] created these three excellent cross-section renderings:<br/> | ||
| + | <[#embed [hash H6DXEQ7GQ5WXY6TJR4JYWJVEZA] [width 676]]><br/> | ||
| + | <[#embed [hash GGJMWW6SCSZ8ESAMECBKYNZAXP] [width 676]]><br/> | ||
| + | <[#embed [hash QVY0TNWN2PYSCXM0WF8TCZ9Z88] [width 676]]> | ||
| + | |||
{{Tetrashapes}} | {{Tetrashapes}} | ||
| - | {{Toratope Nav B|7|8|9|(III)I<br>Spherinder|((III)I)<br> | + | {{Toratope Nav B|7|8|9|(III)I<br>Spherinder|((III)I)<br>Torisphere|((II)I)I<br>Torinder|(((II)I)I)<br>Ditorus|IIIII<br>Penteract|(IIIII)<br>Pentasphere|chora}} |
Latest revision as of 20:49, 11 February 2014
The ditorus is a four-dimensional torus formed by taking an uncapped torinder and connecting its ends either in a loop or through its inside. Its toratopic dual is itself.
Equations
- Variables:
R ⇒ major-major radius of the ditorus
r ⇒ major-minor radius of the ditorus
ρ ⇒ minor-minor radius of the ditorus
- All points (x, y, z, w) that lie on the surcell of a ditorus will satisfy the following equation:
(√((√(x2 + y2) − ρ)2 + z2) − r)2 + w2 = R2
- The parametric equations are:
x = (R + (r + ρ cos θ3) cos θ2) cos θ1
y = (R + (r + ρ cos θ3) cos θ2) sin θ1
z = (r + ρ cos θ3) sin θ2
w = a sin θ3
- The hypervolumes of a ditorus are given by:
total surface area = 0
surcell volume = 8π3Rrρ
bulk = 4π3ρ2rR
- The realmic cross-sections (n) of a ditorus are:
Unknown
Cross-sections
Jonathan Bowers aka Polyhedron Dude created these three 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 |
| 7a. (III)I Spherinder | 7b. ((III)I) Torisphere | 8a. ((II)I)I Torinder | 8b. (((II)I)I) Ditorus | 9a. IIIII Penteract | 9b. (IIIII) Pentasphere |
| List of toratopes | |||||
