Sphere (EntityTopic, 15)

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Revision as of 19:20, 9 March 2011 by Hayate (Talk | contribs)

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


  • Variables:
r ⇒ radius of sphere
  • 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 = 4πr2
volume = 4πr33-1
[!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


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

4. 11
5. 3
6. 21
List of tapertopes

1a. II
1b. (II)
2a. III
2b. (III)
3a. (II)I
3b. ((II)I)
List of toratopes

12. (z)
Narrow crind
13. (xyz)
14. [xyzw]
List of bracketopes