Glome (EntityTopic, 15)
From Hi.gher. Space
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 = 2π2r3
bulk = 2-1π2r4
- 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 |
39. (<xy>zw) Narrow dicrind | 40. (xyzw) Glome | 41. [<xy><zw>] Small tesseract |
List of bracketopes |