Ditorus (EntityTopic, 11)
From Hi.gher. Space
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| - | {{Shape| | + | {{Shape|Ditorus|''No image''|4|2, 0, 0, 0|1|N/A|N/A|[[Line (object)|E]][[Circle|L]][[Torus|Q]]Q|((21)1) (((x,y),z),w)|N/A|N/A|N/A|42}} |
===Geometry=== | ===Geometry=== | ||
| - | The ''' | + | The '''ditorus''' 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 ditorus<br> |
| - | ''r'' ⇒ middle radius of the | + | ''r'' ⇒ middle radius of the ditorus<br> |
| - | ''a'' ⇒ minor radius of the | + | ''a'' ⇒ minor radius of the ditorus</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 ditorus 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 | ||
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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 ditorus 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 ditorus are: |
<blockquote>''Unknown''</blockquote> | <blockquote>''Unknown''</blockquote> | ||
| - | |||
{{Polychora}} | {{Polychora}} | ||
| - | {{ | + | {{Rotope Nav|41|42|43|((II)I)'<br>Toric pyramid|(((II)I)I)<br>Ditorus|(II)(II)<br>Duocylinder}} |
Revision as of 13:04, 17 June 2007
Geometry
The ditorus is unique as it is the only rotope in four dimensions or less that has a pocket.
Equations
- Variables:
R ⇒ major radius of the ditorus
r ⇒ middle radius of the ditorus
a ⇒ minor radius of the ditorus
- All points (x, y, z, w) that lie on the surcell of a ditorus 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 ditorus are given by:
total surface area = 0
surcell volume = 8π3Rra
bulk = 4π3a2rR
- The realmic cross-sections (n) of a ditorus are:
Unknown
