Manifold (ConceptTopic, 4)

From Hi.gher. Space

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A '''manifold''' is a [[shape]] formed from a [[regular]] base shape, where various edges are connected either with or without twists.
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A '''manifold''' is a topological object which locally resembles Euclidean space. Manifolds may or may not have boundaries and may or may not be orientable.
== Nullar and linear manifolds ==
== Nullar and linear manifolds ==
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The manifolds in 0D and 1D are relatively trivial, but are included for completeness:
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There are only three of these, shown below:
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{|
{|
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!style="background-color: #EEE;"|Nullar group 0-0
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!style="background-color: #EEE;"|nullar
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!style="background-color: #EEE;"|Linear group 1-0
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!style="background-color: #EEE;" colspan="2"|linear
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!style="background-color: #EEE;"|Linear group 0-1
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|-
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|width="33%" align="center"|[[Point]]
|width="33%" align="center"|[[Point]]
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|align="center"|<[#img [hash 76ARJ0KWJPGETCCXSNWQ813V9M]]>
|align="center"|<[#img [hash 76ARJ0KWJPGETCCXSNWQ813V9M]]>
|align="center"|<[#img [hash KBTKCV8P4Q8ETDE2WSJGTZKFRF]]>
|align="center"|<[#img [hash KBTKCV8P4Q8ETDE2WSJGTZKFRF]]>
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|-
 
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|align="center"|I
 
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|align="center"|I
 
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|align="center"|S
 
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== Square manifolds ==
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The colored boundaries (for the circle, the two red points) must be identified to form the desired manifold.
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These are the best known manifolds. There are eight of them shown as follows:
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== Planar manifolds ==
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These are the best known manifolds. There are seven "interesting" ones, shown below:
{|
{|
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!style="background-color: #EEE;"|Group 0-0
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!style="background-color: #EEE;"|Group 0
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!colspan="2" style="background-color: #EEE;"|Group 1-0
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!colspan="2" style="background-color: #EEE;"|Group 1
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!colspan="4" style="background-color: #EEE;"|Group 2-0
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!colspan="4" style="background-color: #EEE;"|Group 2
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!style="background-color: #EEE;"|Group 0-2
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|-
|-
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|width="12%" align="center"|[[Square]]
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|width="12%" align="center"|[[Disc]]
|width="12%" align="center"|[[Hose]] (uncapped [[cylinder]])
|width="12%" align="center"|[[Hose]] (uncapped [[cylinder]])
|width="12%" align="center"|[[Möbius strip]]
|width="12%" align="center"|[[Möbius strip]]
|width="12%" align="center"|[[Torus]]
|width="12%" align="center"|[[Torus]]
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|width="12%" align="center"|[[Klein dalma]]
 
|width="12%" align="center"|[[Klein bottle]]
|width="12%" align="center"|[[Klein bottle]]
|width="12%" align="center"|[[Real projective plane]]
|width="12%" align="center"|[[Real projective plane]]
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|align="center"|<[#img [hash GXSMG22779KTTFH16Q81TQXF36]]>
|align="center"|<[#img [hash GXSMG22779KTTFH16Q81TQXF36]]>
|align="center"|<[#img [hash EK4R8ABYEP6YCVZKM4QMJKC3S5]]>
|align="center"|<[#img [hash EK4R8ABYEP6YCVZKM4QMJKC3S5]]>
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|align="center"|<[#img [hash 80GKWC1VFQ28MFBSDM0HM86W7A]]>
 
|align="center"|<[#img [hash ZC0PJNBY5BZZEHD8Q6A3TZHBYX]]>
|align="center"|<[#img [hash ZC0PJNBY5BZZEHD8Q6A3TZHBYX]]>
|align="center"|<[#img [hash QY2NW8J9RCC9CK6GBPF3AZ01D0]]>
|align="center"|<[#img [hash QY2NW8J9RCC9CK6GBPF3AZ01D0]]>
|align="center"|<[#img [hash 62XZYY54QA05YXTE878CGPAPJ5]]>
|align="center"|<[#img [hash 62XZYY54QA05YXTE878CGPAPJ5]]>
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|align="center"|I
 
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|align="center"|0
 
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|align="center"|1
 
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|align="center"|00
 
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|align="center"|01
 
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|align="center"|10
 
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|align="center"|11
 
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|align="center"|SS
 
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|}
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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.
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To construct, connect up the colored edges so that the colors and arrowheads match.
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The Klein figure 8 and Klein bottle are topologically equivalent, however they have been listed separately as they appear significantly different.
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There are infinitely many more such manifolds in group 2, one each of the orientable and non-orientable varieties for each possible genus. Only the possibilities for genus 0 and 1 are shown above.
== Cubic manifolds ==
== Cubic manifolds ==
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There are 279 unique cubic manifolds out of 611 defined ones. Only sufficient examples and the most interesting are shown in the following table.
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Some examples are shown in the following table.
{|
{|
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!style="background-color: #EEE;"|Group 1-2
!style="background-color: #EEE;"|Group 1-2
!style="background-color: #EEE;" colspan="2"|Group 0-3
!style="background-color: #EEE;" colspan="2"|Group 0-3
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|-
 
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|align="center"|1 defined
 
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|align="center"|8 defined
 
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|align="center"|64 defined
 
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|align="center"|512 defined
 
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|align="center"|1 defined
 
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|align="center"|1 defined
 
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|align="center" colspan="2"|24 defined
 
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|-
 
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|align="center"|1 unique
 
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|align="center"|6 unique
 
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|align="center"|36 unique
 
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|align="center"|216 unique
 
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|align="center"|1 unique
 
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|align="center"|1 unique
 
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|align="center" colspan="2"|18 unique
 
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|align="center"|1 shown
 
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|align="center"|2 shown
 
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|align="center"|2 shown
 
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|align="center"|3 shown
 
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|align="center"|1 shown
 
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|align="center"|1 shown
 
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|align="center" colspan="2"|3 shown
 
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|align="center" width="12%"|[[Cube]]
|align="center" width="12%"|[[Cube]]
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|align="center"|<[#img [hash R2GEAJMABF01MCPMD04GPYAZV5]]>
|align="center"|<[#img [hash R2GEAJMABF01MCPMD04GPYAZV5]]>
|align="center"|<[#img [hash 83VKP6HCJ7350SJHQ3KY07HWA5]]>
|align="center"|<[#img [hash 83VKP6HCJ7350SJHQ3KY07HWA5]]>
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|-
 
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|align="center"|I
 
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|align="center"|0
 
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|align="center"|00
 
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|align="center"|000
 
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|align="center"|SS
 
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|align="center"|SSS
 
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|align="center"|0SS
 
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|align="center"|1SS
 
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|align="center"|
|align="center"|
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|align="center"|
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|align="center"|[[Toraspherindric dalma]]
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|align="center"|<[#img [hash B14HGCNWBQ6CY71WGJTZWYWT2D]]>
 
|align="center"|
|align="center"|
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|align="center"|
 
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|align="center"|1
 
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|align="center"|11
 
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|align="center"|100
 
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|align="center"|
 
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|align="center"|
 
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|align="center"|SS1
 
|align="center"|
|align="center"|
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|align="center"|
|align="center"|<[#img [hash 4HNZC40T26JRPN1DNB67WVWP9X]]>
|align="center"|<[#img [hash 4HNZC40T26JRPN1DNB67WVWP9X]]>
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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.
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.
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There are 3 more interesting group 0-3 cubic manifolds. These are SS0, S0S and S1S, and are currently unknown.
 
== Tesseric manifolds ==
== Tesseric manifolds ==
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<[#img [hash KV05CMH3C2Q7M8761HHSGX1GM6]]>
<[#img [hash KV05CMH3C2Q7M8761HHSGX1GM6]]>
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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:
 
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{|
 
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!width="50%" style="background-color:#EEE;"|Group
 
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!width="50%" style="background-color:#EEE;"|Defined
 
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|-
 
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|0-0
 
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|align="right"|1
 
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|-
 
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|1-0
 
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|align="right"|128
 
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|-
 
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|2-0
 
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|align="right"|16,384
 
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|-
 
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|3-0
 
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|align="right"|2,097,152
 
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|-
 
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|4-0
 
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|align="right"|268,435,456
 
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|-
 
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|Subtotal
 
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|align="right"|270,548,736
 
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|0-2
 
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|align="right"|1
 
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|1-2
 
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|align="right"|384
 
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|2-2
 
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|align="right"|98,304
 
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|-
 
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|Subtotal
 
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|align="right"|98,689
 
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|-
 
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|0-3
 
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|align="right"|1
 
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|-
 
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|1-3
 
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|align="right"|8,388,608
 
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|-
 
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|Subtotal
 
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|align="right"|8,388,609
 
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|-
 
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|0-4
 
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|align="right"|1
 
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|-
 
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|Subtotal
 
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|align="right"|1
 
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|-
 
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|Total
 
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|align="right"|279,036,033
 
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|}
 
== See also ==
== See also ==

Revision as of 21:57, 25 March 2011

A manifold is a topological object which locally resembles Euclidean space. Manifolds may or may not have boundaries and may or may not be orientable.

Nullar and linear manifolds

The manifolds in 0D and 1D are relatively trivial, but are included for completeness:

nullar linear
Point Line segment Circle
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The colored boundaries (for the circle, the two red points) must be identified to form the desired manifold.

Planar manifolds

These are the best known manifolds. There are seven "interesting" ones, shown below:

Group 0 Group 1 Group 2
Disc Hose (uncapped cylinder) Möbius strip Torus Klein bottle Real projective plane Sphere
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To construct, connect up the colored edges so that the colors and arrowheads match.

There are infinitely many more such manifolds in group 2, one each of the orientable and non-orientable varieties for each possible genus. Only the possibilities for genus 0 and 1 are shown above.

Cubic manifolds

Some examples 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
Cube Dihose Toric hose Ditorus Spherical hose Glome Toraspherinder Toraspherindric bottle
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Möbial hose Real projective planar hose Toric bottle
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Real projective realm
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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.

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.

ExPar: [#img] is obsolete, use [#embed] instead

See also

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