Manifold (ConceptTopic, 4)

From Hi.gher. Space

A manifold is a shape formed from a regular base shape, where various edges are connected either with or without twists.

Square manifolds

These are the best known manifolds. There are eight of them shown as follows:

To construct, first connect the red edges to each other, matching up the arrowheads, and then connect the blue arrows together in the same way. Edges without arrows are left unconnected.

The Klein figure 8 and Klein bottle are topologically equivalent, however they have been listed separately as they appear significantly different.

Cubic manifolds

There are 279 unique cubic manifolds out of 611 defined ones. Only sufficient examples and the most interesting are shown in the following table.

Construction is similar to that of the square manifolds: fold up each cubic net and attach the red, blue and green pairs of facets to each other in that order, making sure the triangles line up.

There are 3 more interesting group 0-3 cubic manifolds. These are SS0, S0S and S1S, and are currently unknown.

Tesseric manifolds

Here are the 4D p-toric q-hoses and p-spheric q-hoses along with the tesseract and möbial dihose:

To construct, first fold up the nets for each cube and attach the cubes into the net of a tesseract as shown below, making sure to preserve orientation. Solidify the tesseract net and fold that up too. Then, attach the red, blue, green and yellow pairs of facets to each other in that order, lining up the symbols.

http://teamikaria.com/dl/VT-jMGnNk4nWfj7Zdo9cYMhxSEWfHKo5Q1xWKmFM65sh4zwf.png

Since the number of defined tesseric manifolds is so large and the number of uniques has not been determined, these shall be summarized in the following table: