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:
x^{2} + y^{2} + z^{2} = r^{2}
- The hypervolumes of a sphere are given by:
total edge length = 0
surface area = 4π · r^{2}
volume = ^{4π}∕_{3} · r^{3}
- The planar cross-sections (n) of a sphere are:
[!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)
- H_{0}X = ℤ, H_{1}X = 0, H_{2}X = ℤ
- 3-frame (ball)
- H_{0}X = 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. 1^{1} 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 |