Sphere (EntityTopic, 15)
From Hi.gher. Space
A sphere refers to the surface of a perfectly symmetrical realmic object.
Sometimes, the surface is called a sphere and the solid object is called a ball.
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
- The hypervolumes of a sphere are given by:
total edge length = 0
surface area = 4π · r2
volume = 4π∕3 · r3
[!x,!y,!z] ⇒ circle of radius (rcos(πn/2))
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).
|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|
|List of tapertopes|
|List of toratopes|
|List of bracketopes|