Ditorus (EntityTopic, 11)
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(Difference between revisions)
m (Tetratorus moved to Tritorus: switching to more systematic name) |
(tetratorus -> tritorus) |
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===Geometry=== | ===Geometry=== | ||
| - | The ''' | + | The '''tritorus''' is unique as it is the only [[rotope]] in four dimensions or less that has a [[pocket]]. |
===Equations=== | ===Equations=== | ||
*Variables: | *Variables: | ||
| - | <blockquote>''R'' ⇒ major radius of the | + | <blockquote>''R'' ⇒ major radius of the tritorus<br> |
| - | ''r'' ⇒ middle radius of the | + | ''r'' ⇒ middle radius of the tritorus<br> |
| - | ''a'' ⇒ minor radius of the | + | ''a'' ⇒ minor radius of the tritorus</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 tritorus will satisfy the following equation: |
<blockquote> | <blockquote> | ||
(sqrt((sqrt(x^2 + y^2) - a)^2 + z^2) - r)^2 + w^2 = R^2 | (sqrt((sqrt(x^2 + y^2) - a)^2 + z^2) - r)^2 + w^2 = R^2 | ||
| Line 22: | Line 22: | ||
w = a sin th<sub>3</sub> </blockquote> | w = a sin th<sub>3</sub> </blockquote> | ||
| - | *The [[hypervolume]]s of a | + | *The [[hypervolume]]s of a tritorus are given by: |
<blockquote>total surface area = 0<br> | <blockquote>total surface area = 0<br> | ||
surcell volume = 8π<sup>3</sup>Rra<br> | surcell volume = 8π<sup>3</sup>Rra<br> | ||
bulk = 4π<sup>3</sup>a<sup>2</sup>rR</blockquote> | bulk = 4π<sup>3</sup>a<sup>2</sup>rR</blockquote> | ||
| - | *The [[realmic]] [[cross-section]]s (''n'') of a | + | *The [[realmic]] [[cross-section]]s (''n'') of a tritorus are: |
<blockquote>''Unknown''</blockquote> | <blockquote>''Unknown''</blockquote> | ||
<br clear="all"><br> | <br clear="all"><br> | ||
{{Polychora}} | {{Polychora}} | ||
{{Rotopes}} | {{Rotopes}} | ||
Revision as of 02:22, 17 June 2007
Geometry
The tritorus is unique as it is the only rotope in four dimensions or less that has a pocket.
Equations
- Variables:
R ⇒ major radius of the tritorus
r ⇒ middle radius of the tritorus
a ⇒ minor radius of the tritorus
- All points (x, y, z, w) that lie on the surcell of a tritorus will satisfy the following equation:
(sqrt((sqrt(x^2 + y^2) - a)^2 + z^2) - r)^2 + w^2 = R^2
- The parametric equations are:
x = (R + (r + a cos th3) cos th2) cos th1
y = (R + (r + a cos th3) cos th2) sin th1
z = (r + a cos th3) sin th2
w = a sin th3
- The hypervolumes of a tritorus are given by:
total surface area = 0
surcell volume = 8π3Rra
bulk = 4π3a2rR
- The realmic cross-sections (n) of a tritorus are:
Unknown
