Sphere (EntityTopic, 15)
From Hi.gher. Space
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== 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). | ||
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{{Trishapes}} | {{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|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 18:13, 24 November 2009
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
- The hypervolumes of a sphere are given by:
total edge length = 0
surface area = 4πr2
volume = 4πr33-1
- The planar cross-sections (n) of a sphere are:
[!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: | 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. 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 | |||||
| 12. ( Narrow crind | 13. (xyz) Sphere | 14. [xyzw] Tesseract |
| List of bracketopes | ||
