Torus (EntityTopic, 11)
From Hi.gher. Space
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.
The expanded rotatope of the torus is the duocylinder.
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:
(√(x^{2}+y^{2}) − R)^{2} + z^{2} = r^{2}
- The hypervolumes of a torus are given by:
total edge length = 0
surface area = 4π^{2}Rr
volume = 2π^{2}Rr^{2}
- The planar cross-sections (n) of a torus are:
[!x,!y] ⇒ two separated circles
[!z] ⇒ two concentric circles
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_{0}X = ℤ, H_{1}X = 2ℤ, H_{2}X = ℤ
- 3-frame (solid torus)
- H_{0}X = ℤ,H_{1}X = 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 |