# Sphere (EntityTopic, 15)

(Difference between revisions)
 Revision as of 09:57, 10 November 2008 (view source)Hayate (Talk | contribs)m← Older edit Latest revision as of 16:29, 25 March 2017 (view source) (13 intermediate revisions not shown) Line 1: Line 1: + <[#ontology [kind topic] [cats 3D Hypersphere] [alt [[freebase:06y47]] [[wikipedia:Sphere]]]]> {{STS Shape {{STS Shape - | image=http://img457.imageshack.us/img457/787/sphere6jb.png + | image=<[#embed [hash EHJHWNKE23KPWCSG26N4W0CT14] [width 180]]> | dim=3 | dim=3 - | elements=1, 0, 0 + | elements=1 sphere surface, 0, 0 | genus=0 | genus=0 | ssc=(xyz) | ssc=(xyz) Line 11: Line 12: 0 0 0 0 0 0 0 0 - 3 1}}{{STS Rotope + 3 1}}{{STS Tapertope - | attrib=pure + | order=1, 0 - | notation=3 (xyz) + | notation=3 - | index=7 + | index=5 + }}{{STS Toratope + | expand=[[Sphere|3]] + | notation=(III) + | index=2b }}{{STS Bracketope }}{{STS Bracketope - | index=13 + | index=6 + | notation=(III) }}}} }}}} A '''sphere''' refers to the surface of a perfectly symmetrical [[realmic]] object. 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''. == Equations == == Equations == Line 30: Line 38: *The [[hypervolume]]s of a sphere are given by: *The [[hypervolume]]s of a sphere are given by:
total edge length = 0
total edge length = 0
- surface area = 4π''r''2
+ surface area = 4π {{DotHV|2|r}}
- volume = 4π''r''33-1
+ volume = {{Over|4π|3}} {{DotHV|3|r}}
*The [[planar]] [[cross-section]]s (''n'') of a sphere are: *The [[planar]] [[cross-section]]s (''n'') of a sphere are:
[!x,!y,!z] ⇒ [[circle]] of radius (''r''cos(π''n''/2))
[!x,!y,!z] ⇒ [[circle]] of radius (''r''cos(π''n''/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 == == 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). 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). + {{Trishapes}} {{Trishapes}} - {{Rotope Nav|6|7|8|II'
Square pyramid|(III)
Sphere|I'I
Triangular prism|hedra}} + {{Tapertope Nav|4|5|6|11
Triangle|3
Sphere|21
Cylinder|hedra}} - {{Bracketope Nav|12|13|14|(z)
Narrow crind|(xyz)
Sphere|[xyzw]
Tesseract|hedra}} + {{Toratope Nav B|1|2|3|II
Square|(II)
Circle|III
Cube|(III)
Sphere|(II)I
Cylinder|((II)I)
Torus|hedra}} + {{Bracketope Nav|5|6|7|
Octahedron|(III)
Sphere|[(II)I]
Cylinder|hedra}}

## Latest revision as of 16:29, 25 March 2017

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.

## 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 = 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: 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

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