# Sphere (EntityTopic, 15)

A sphere refers to the surface of a perfectly symmetrical realmic object.

## Equations

• Variables:
• All points (x, y, z) that lie on the surface of a sphere will satisfy the following equation:
x2 + y2 + z2 = r2
total edge length = 0
surface area = 4π · r2
volume = 3 · r3
[!x,!y,!z] ⇒ circle of radius (rcos(πn/2))

## Homology groups

All homology groups are zero except where stated. Here X is the sphere in the given frame, and nZ is the direct sum of n copies of the group of integers Z.

2-frame (sphere)
H0X = ℤ, H1X = 0, H2X = ℤ
3-frame (ball)
H0X = ℤ

## Mapping

When the surface of a sphere is mapped onto a square centered at the origin with side length 2, the surface will be horizontally stretched such that the further away from the equator of the sphere a point is, the more it is stretched. The top and bottom edges of the rectangle will converge into a single point. Leaving the top edge of the rectangle will continue entering from the top edge at the position (x±1,1). Similarly, leaving the bottom edge of the rectangle will continue entering from the bottom edge at the position (x±1,-1).

 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

 4. 11Triangle 5. 3Sphere 6. 21Cylinder List of tapertopes

 1a. IISquare 1b. (II)Circle 2a. IIICube 2b. (III)Sphere 3a. (II)ICylinder 3b. ((II)I)Torus List of toratopes

 5. Octahedron 6. (III)Sphere 7. [(II)I]Cylinder List of bracketopes