Sphere (EntityTopic, 15)

From Hi.gher. Space

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<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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=== Mapping ===
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== 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'<br>Square pyramid|(III)<br>Sphere|I'I<br>Triangular prism|hedra}}
{{Rotope Nav|6|7|8|II'<br>Square pyramid|(III)<br>Sphere|I'I<br>Triangular prism|hedra}}
{{Bracketope Nav|12|13|14|(<xy>z)<br>Narrow crind|(xyz)<br>Sphere|[xyzw]<br>Tesseract|hedra}}
{{Bracketope Nav|12|13|14|(<xy>z)<br>Narrow crind|(xyz)<br>Sphere|[xyzw]<br>Tesseract|hedra}}

Revision as of 21:05, 22 September 2007

Template:Shape A sphere refers to the surface of a perfectly symmetrical realmic object.

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 = 4πr2
volume = 4πr33-1
[!x,!y,!z] ⇒ circle of radius (rcos(πn/2))

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

Template:Rotope Nav

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