Glome (EntityTopic, 15)
From Hi.gher. Space
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The glome, also known as the 3-sphere, is the 4-dimensional equivalent of a 3D sphere. It consists of a curved 3-manifold that forms circular intersections with planes, and spherical intersections with hyperplanes. The set of points midway between two antipodal points form a sphere, hence one may think of the glome as having a spherical "equator". Alternatively, one can think of the glome having two perpendicular circular equators - and no poles! | The glome, also known as the 3-sphere, is the 4-dimensional equivalent of a 3D sphere. It consists of a curved 3-manifold that forms circular intersections with planes, and spherical intersections with hyperplanes. The set of points midway between two antipodal points form a sphere, hence one may think of the glome as having a spherical "equator". Alternatively, one can think of the glome having two perpendicular circular equators - and no poles! | ||
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{{Tapertope Nav|11|12|13|1<sup>2</sup><br>Tetrahedron|4<br>Glome|31<br>Spherinder|chora}} | {{Tapertope Nav|11|12|13|1<sup>2</sup><br>Tetrahedron|4<br>Glome|31<br>Spherinder|chora}} | ||
{{Toratope Nav B|3|4|5|(II)I<br>Cylinder|((II)I)<br>Torus|IIII<br>Tesseract|(IIII)<br>Glome|(II)II<br>Cubinder|((II)II)<br>Toracubinder|chora}} | {{Toratope Nav B|3|4|5|(II)I<br>Cylinder|((II)I)<br>Torus|IIII<br>Tesseract|(IIII)<br>Glome|(II)II<br>Cubinder|((II)II)<br>Toracubinder|chora}} | ||
| - | {{Bracketope Nav| | + | {{Bracketope Nav|11|12|13|<nowiki><IIII></nowiki><br>Aerochoron|(IIII)<br>Glome|[(II)II]<br>Cubinder|chora}} |
Revision as of 17:25, 18 November 2011
The glome, also known as the 3-sphere, is the 4-dimensional equivalent of a 3D sphere. It consists of a curved 3-manifold that forms circular intersections with planes, and spherical intersections with hyperplanes. The set of points midway between two antipodal points form a sphere, hence one may think of the glome as having a spherical "equator". Alternatively, one can think of the glome having two perpendicular circular equators - and no poles!
Its projection to 3-space is a sphere — or, more properly, a ball: the image of its bounding manifold covers all points in a ball twice, once for each hemi-glome.
Equations
- Variables:
r ⇒ radius of the glome
- All points (x, y, z, w) that lie on the surcell of a glome will satisfy the following equation:
x2 + y2 + z2 + w2 = r2
- The hypervolumes of a glome are given by:
total edge length = 0
total surface area = 0
surcell volume = 2π2 · r3
bulk = π2∕2 · r4
- The realmic cross-sections (n) of a glome are:
[!x,!y,!z,!w] ⇒ sphere of radius (rcos(πn/2))
| Notable Tetrashapes | |
| Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
| Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
| Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
| Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
| 11. 12 Tetrahedron | 12. 4 Glome | 13. 31 Spherinder |
| List of tapertopes | ||
| 3a. (II)I Cylinder | 3b. ((II)I) Torus | 4a. IIII Tesseract | 4b. (IIII) Glome | 5a. (II)II Cubinder | 5b. ((II)II) Toracubinder |
| List of toratopes | |||||
| 11. <IIII> Aerochoron | 12. (IIII) Glome | 13. [(II)II] Cubinder |
| List of bracketopes | ||
