Sphere (EntityTopic, 15)
From Hi.gher. Space
m (add) |
Username5243 (Talk | contribs) |
||
(27 intermediate revisions not shown) | |||
Line 1: | Line 1: | ||
- | {{Shape| | + | <[#ontology [kind topic] [cats 3D Hypersphere] [alt [[freebase:06y47]] [[wikipedia:Sphere]]]]> |
- | == | + | {{STS Shape |
+ | | image=<[#embed [hash EHJHWNKE23KPWCSG26N4W0CT14] [width 180]]> | ||
+ | | dim=3 | ||
+ | | elements=1 sphere surface, 0, 0 | ||
+ | | genus=0 | ||
+ | | ssc=(xyz) | ||
+ | | ssc2=T3 | ||
+ | | pv_circle=1 | ||
+ | | pv_square=~0.5236 | ||
+ | | extra={{STS Matrix| | ||
+ | 0 0 | ||
+ | 0 0 | ||
+ | 3 1}}{{STS Tapertope | ||
+ | | order=1, 0 | ||
+ | | notation=3 | ||
+ | | index=5 | ||
+ | }}{{STS Toratope | ||
+ | | expand=[[Sphere|3]] | ||
+ | | notation=(III) | ||
+ | | index=2b | ||
+ | }}{{STS Bracketope | ||
+ | | 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 == | ||
*Variables: | *Variables: | ||
<blockquote>''r'' ⇒ radius of sphere</blockquote> | <blockquote>''r'' ⇒ radius of sphere</blockquote> | ||
Line 13: | Line 38: | ||
*The [[hypervolume]]s of a sphere are given by: | *The [[hypervolume]]s of a sphere are given by: | ||
<blockquote>total edge length = 0<br> | <blockquote>total edge length = 0<br> | ||
- | surface area = 4π | + | surface area = 4π {{DotHV|2|r}}<br> |
- | volume = 4π | + | volume = {{Over|4π|3}} {{DotHV|3|r}}</blockquote> |
*The [[planar]] [[cross-section]]s (''n'') of a sphere are: | *The [[planar]] [[cross-section]]s (''n'') of a sphere are: | ||
<blockquote>[!x,!y,!z] ⇒ [[circle]] of radius (''r''cos(π''n''/2))</blockquote> | <blockquote>[!x,!y,!z] ⇒ [[circle]] of radius (''r''cos(π''n''/2))</blockquote> | ||
- | === | + | == Homology groups == |
- | When the surface of a sphere is mapped onto a [[ | + | 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<sub>0</sub>X = ℤ, H<sub>1</sub>X = 0, H<sub>2</sub>X = ℤ |
- | {{Bracketope Nav| | + | ;3-frame (ball):H<sub>0</sub>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). | ||
+ | |||
+ | {{Trishapes}} | ||
+ | {{Tapertope Nav|4|5|6|1<sup>1</sup><br>Triangle|3<br>Sphere|21<br>Cylinder|hedra}} | ||
+ | {{Toratope Nav B|1|2|3|II<br>Square|(II)<br>Circle|III<br>Cube|(III)<br>Sphere|(II)I<br>Cylinder|((II)I)<br>Torus|hedra}} | ||
+ | {{Bracketope Nav|5|6|7|<nowiki><III></nowiki><br>Octahedron|(III)<br>Sphere|[(II)I]<br>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:
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 |