Torus (EntityTopic, 11)

From Hi.gher. Space

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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.


  • 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: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind

2a. III
2b. (III)
3a. (II)I
3b. ((II)I)
4a. IIII
4b. (IIII)
List of toratopes