Glome (EntityTopic, 15)

From Hi.gher. Space

Revision as of 02:19, 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))