Sphere (EntityTopic, 15)

From Hi.gher. Space

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<[#ontology [kind topic] [cats 3D Hypersphere] [alt [[freebase:06y47]] [[wikipedia:Sphere]]]]>
{{STS Shape
{{STS Shape
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| image=http://img457.imageshack.us/img457/787/sphere6jb.png
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| image=<[#embed [hash EHJHWNKE23KPWCSG26N4W0CT14] [width 180]]>
| dim=3
| dim=3
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| elements=1, 0, 0
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| elements=1 sphere surface, 0, 0
| genus=0
| genus=0
| ssc=(xyz)
| ssc=(xyz)
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| notation=3
| notation=3
| index=5
| index=5
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}}{{STS Tapertope
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}}{{STS Toratope
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| holeseq=[0, 1]
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| expand=[[Sphere|3]]
| notation=(III)
| notation=(III)
| index=2b
| index=2b
}}{{STS Bracketope
}}{{STS Bracketope
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| index=6
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| 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.
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Sometimes, the surface is called a sphere and the solid object is called a ''ball''.
== Equations ==
== Equations ==
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*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>
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surface area = 4π''r''<sup>2</sup><br>
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surface area = 4π {{DotHV|2|r}}<br>
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volume = 4π''r''<sup>3</sup>3<sup>-1</sup></blockquote>
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volume = {{Over||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>
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== Homology groups ==
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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.
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;2-frame (sphere):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 0, H<sub>2</sub>X = ℤ
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;3-frame (ball):H<sub>0</sub>X = ℤ
== Mapping ==
== Mapping ==
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{{Tapertope Nav|4|5|6|1<sup>1</sup><br>Triangle|3<br>Sphere|21<br>Cylinder|hedra}}
{{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}}
{{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}}
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{{Bracketope Nav|12|13|14|(<xy>z)<br>Narrow crind|(xyz)<br>Sphere|[xyzw]<br>Tesseract|hedra}}
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{{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:
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: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind


4. 11
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