Manifold (ConceptTopic, 4)

From Hi.gher. Space

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Revision as of 21:54, 8 March 2011

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

Nullar and linear manifolds

There are only three of these, shown below:

Nullar group 0-0	Linear group 1-0	Linear group 0-1
Point	Line segment	Circle
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I	I	S

Square manifolds

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

Group 0-0	Group 1-0		Group 2-0				Group 0-2
Square	Hose (uncapped cylinder)	Möbius strip	Torus	Klein dalma	Klein bottle	Real projective plane	Sphere
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I	0	1	00	01	10	11	SS

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.

Group 0-0	Group 1-0	Group 2-0	Group 3-0	Group 0-2	Group 1-2	Group 0-3
1 defined	8 defined	64 defined	512 defined	1 defined	1 defined	24 defined
1 unique	6 unique	36 unique	216 unique	1 unique	1 unique	18 unique
1 shown	2 shown	2 shown	3 shown	1 shown	1 shown	3 shown
Cube	Dihose	Toric hose	Ditorus	Spherical hose	Glome	Toraspherinder	Toraspherindric bottle
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I	0	00	000	SS	SSS	0SS	1SS
	Möbial hose	Real projective planar hose	Toric bottle			Toraspherindric dalma
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	1	11	100			SS1
			Real projective realm
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			111

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:

Group 0-0	Group 1-0		Group 2-0	Group 3-0	Group 4-0	Group 0-2	Group 0-3	Group 0-4
Tesseract	Möbial dihose	Trihose	Toric dihose	Ditoric hose	Tritorus	Spherical dihose	Glomic hose	Pentasphere
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I	1	0	00	000	0000	SS	SSS	SSSS

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.

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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:

Group	Defined
0-0	1
1-0	128
2-0	16,384
3-0	2,097,152
4-0	268,435,456
Subtotal	270,548,736
0-2	1
1-2	384
2-2	98,304
Subtotal	98,689
0-3	1
1-3	8,388,608
Subtotal	8,388,609
0-4	1
Subtotal	1
Total	279,036,033

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 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
+<[#img [hash KV05CMH3C2Q7M8761HHSGX1GM6]]>
 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: