Sphere (EntityTopic, 15)

From Higher Dimensions Database


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

Equations

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


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


1a. II
Square
1b. (II)
Circle
2a. III
Cube
2b. (III)
Sphere
3a. (II)I
Cylinder
3b. ((II)I)
Torus
List of toratopes


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