Torus (EntityTopic, 11)
From Hi.gher. Space
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*The [[planar]] [[cross-section]]s (''n'') of a torus are: | *The [[planar]] [[cross-section]]s (''n'') of a torus are: | ||
<blockquote>''Unknown''</blockquote> | <blockquote>''Unknown''</blockquote> | ||
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| + | |||
| + | == Homology groups == | ||
| + | All homology groups are zero except where stated. Here X is the shape in the given frame, and nZ is the direct sum of n copies of the group of integers Z. | ||
| + | |||
| + | ;2-frame (torus):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = ℤ⊕ℤ, H<sub>2</sub>X = ℤ | ||
| + | ;3-frame (solid torus):H<sub>0</sub>X = ℤ,H<sub>1</sub>X = ℤ | ||
| + | |||
{{Trishapes}} | {{Trishapes}} | ||
{{Toratope Nav B|2|3|4|III<br>Cube|(III)<br>Sphere|(II)I<br>Cylinder|((II)I)<br>Torus|IIII<br>Tesseract|(IIII)<br>Glome|hedra}} | {{Toratope Nav B|2|3|4|III<br>Cube|(III)<br>Sphere|(II)I<br>Cylinder|((II)I)<br>Torus|IIII<br>Tesseract|(IIII)<br>Glome|hedra}} | ||
Revision as of 05:17, 25 November 2009
A torus is a special case of a surface of revolution where the base is a circle. The circle's radius is known as the minor radius and the distance from the center of the circle to the center of the torus is known as the major radius.
Equations
- Variables:
R ⇒ major radius of torus
r ⇒ minor radius of torus
- All points (x, y, z) that lie on the surface of a torus will satisfy the following equation:
(R-sqrt(x2+y2))2 + z2 = r2
- The hypervolumes of a torus are given by:
total edge length = 0
surface area = 4π2Rr
volume = 2π2Rr2
- The planar cross-sections (n) of a torus are:
Unknown
Homology groups
All homology groups are zero except where stated. Here X is the shape in the given frame, and nZ is the direct sum of n copies of the group of integers Z.
- 2-frame (torus)
- H0X = ℤ, H1X = ℤ⊕ℤ, H2X = ℤ
- 3-frame (solid torus)
- H0X = ℤ,H1X = ℤ
| Notable Trishapes | |
| Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
| Direct truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
| Mesotruncates: | stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |
| Snubs: | snub staurohedron • snub rhodohedron |
| Curved: | sphere • torus • cylinder • cone • frustum • crind |
| 2a. III Cube | 2b. (III) Sphere | 3a. (II)I Cylinder | 3b. ((II)I) Torus | 4a. IIII Tesseract | 4b. (IIII) Glome |
| List of toratopes | |||||
