# Sphere (EntityTopic, 15)

(Difference between revisions)
 Revision as of 05:13, 25 November 2009 (view source)Pwrong (Talk | contribs)← Older edit Revision as of 05:30, 25 November 2009 (view source)Pwrong (Talk | contribs) (→Homology groups)Newer edit → Line 45: Line 45: ;2-frame (sphere):H0X = ℤ, H1X = 0, H2X = ℤ ;2-frame (sphere):H0X = ℤ, H1X = 0, H2X = ℤ ;3-frame (ball):H0X = ℤ ;3-frame (ball):H0X = ℤ - - == Mapping == == Mapping ==

## Revision as of 05:30, 25 November 2009

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 = 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

## 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

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