Sphere (EntityTopic, 15)
From Hi.gher. Space
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When the surface of a sphere is mapped onto a [[rectangle]] {(-1,-1),(1,-1),(1,1),(-1,1)}, 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 [[rectangle]] {(-1,-1),(1,-1),(1,1),(-1,1)}, 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). | ||
{{Polyhedra}} | {{Polyhedra}} | ||
| - | {{Rotope Nav|6|7|8|II'<br>Square pyramid|(III)<br>Sphere|I'I<br>Triangular prism}} | + | {{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}} | + | {{Bracketope Nav|12|13|14|(<xy>z)<br>Narrow crind|(xyz)<br>Sphere|[xyzw]<br>Tesseract|hedra}} |
Revision as of 07:15, 20 June 2007
Geometry
A sphere refers to the surface of a perfectly symmetrical realmic object.
Equations
- Assumption: Sphere is centered at the origin.
- 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 rectangle {(-1,-1),(1,-1),(1,1),(-1,1)}, 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 |
| 12. ( Narrow crind | 13. (xyz) Sphere | 14. [xyzw] Tesseract |
| List of bracketopes | ||
