# Torus (EntityTopic, 11)

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:
(√(x2+y2) − R)2 + z2 = r2
total edge length = 0
surface area = 4π2Rr
volume = 2π2Rr2
[!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)
H0X = ℤ, H1X = 2ℤ, 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. IIICube 2b. (III)Sphere 3a. (II)ICylinder 3b. ((II)I)Torus 4a. IIIITesseract 4b. (IIII)Glome List of toratopes