Manifold (ConceptTopic, 4)

From Hi.gher. Space

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{|
{|
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!style="background-color: #EEE;"|Group I0
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!style="background-color: #EEE;"|Group 0-0
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!colspan="2" style="background-color: #EEE;"|Group I1
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!colspan="2" style="background-color: #EEE;"|Group 1-0
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!colspan="4" style="background-color: #EEE;"|Group I2
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!colspan="4" style="background-color: #EEE;"|Group 2-0
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!style="background-color: #EEE;"|Group S2
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!style="background-color: #EEE;"|Group 0-2
|-
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|width="12%" align="center"|[[Square]]
|width="12%" align="center"|[[Square]]
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{|
{|
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!style="background-color: #EEE;"|Group I0
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!style="background-color: #EEE;"|Group 0-0
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!style="background-color: #EEE;"|Group I1
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!style="background-color: #EEE;"|Group 1-0
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!style="background-color: #EEE;"|Group I2
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!style="background-color: #EEE;"|Group 2-0
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!style="background-color: #EEE;" colspan="2"|Group I3
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!style="background-color: #EEE;"|Group 3-0
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!style="background-color: #EEE;"|Group S2
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!style="background-color: #EEE;"|Group 0-2
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!style="background-color: #EEE;" colspan="2"|Group S3
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!style="background-color: #EEE;"|Group 1-2
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!style="background-color: #EEE;" colspan="2"|Group 0-3
|-
|-
|align="center"|1 defined
|align="center"|1 defined
|align="center"|8 defined
|align="center"|8 defined
|align="center"|64 defined
|align="center"|64 defined
-
|align="center" colspan="2"|512 defined
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|align="center"|512 defined
|align="center"|1 defined
|align="center"|1 defined
-
|align="center" colspan="2"|25 defined
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|align="center"|1 defined
 +
|align="center" colspan="2"|24 defined
|-
|-
|align="center"|1 unique
|align="center"|1 unique
|align="center"|6 unique
|align="center"|6 unique
|align="center"|36 unique
|align="center"|36 unique
-
|align="center" colspan="2"|216 unique
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|align="center"|216 unique
 +
|align="center"|1 unique
|align="center"|1 unique
|align="center"|1 unique
-
|align="center" colspan="2"|19 unique
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|align="center" colspan="2"|18 unique
|-
|-
|align="center"|1 shown
|align="center"|1 shown
|align="center"|2 shown
|align="center"|2 shown
|align="center"|2 shown
|align="center"|2 shown
-
|align="center" colspan="2"|3 shown
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|align="center"|3 shown
|align="center"|1 shown
|align="center"|1 shown
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|align="center" colspan="2"|4 shown
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|align="center"|1 shown
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|align="center" colspan="2"|3 shown
|-
|-
|align="center" width="12%"|[[Cube]]
|align="center" width="12%"|[[Cube]]
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|align="center" width="12%"|[[Toric hose]]
|align="center" width="12%"|[[Toric hose]]
|align="center" width="12%"|[[Ditorus]]
|align="center" width="12%"|[[Ditorus]]
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|align="center" width="12%"|[[Toric bottle]]
 
|align="center" width="12%"|[[Spherical hose]]
|align="center" width="12%"|[[Spherical hose]]
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|align="center" width="12%"|[[Toraspherinder]]
 
|align="center" width="12%"|[[Glome]]
|align="center" width="12%"|[[Glome]]
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|align="center" width="12%"|[[Toraspherinder]]
 +
|align="center" width="12%"|[[Toraspherindric bottle]]
|-
|-
|align="center"|http://teamikaria.com/dl/jGzBHc8mcwh4Dmf4wKU0l6L-RHMGonZJ-UFLSz1lqBY4JywN.png
|align="center"|http://teamikaria.com/dl/jGzBHc8mcwh4Dmf4wKU0l6L-RHMGonZJ-UFLSz1lqBY4JywN.png
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|align="center"|http://teamikaria.com/dl/JpQKfGN7vea3nHKcWCIJhh5Q3n9P1868bafu31nJl-sBmEM2.png
|align="center"|http://teamikaria.com/dl/JpQKfGN7vea3nHKcWCIJhh5Q3n9P1868bafu31nJl-sBmEM2.png
|align="center"|http://teamikaria.com/dl/NJ5W30kaUY8xQlTsXh2tizXABsTdwYmId9nd0CtcwpqNskcf.png
|align="center"|http://teamikaria.com/dl/NJ5W30kaUY8xQlTsXh2tizXABsTdwYmId9nd0CtcwpqNskcf.png
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|align="center"|http://teamikaria.com/dl/ujCnRAl3fVM5XJtQmyne3A_1QD_qq7ys_wijaGiYp8ara-hz.png
 
|align="center"|http://teamikaria.com/dl/2hQD0jvviin64DzXw6AW52mh7VmrSL_6Cz1Er82_vrXEHe2x.png
|align="center"|http://teamikaria.com/dl/2hQD0jvviin64DzXw6AW52mh7VmrSL_6Cz1Er82_vrXEHe2x.png
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|align="center"|http://teamikaria.com/dl/W6Za757XuFgeg1uEqs_cVjf-miNfwZfHVcDAvzaAN3XHaUX8.png
 
|align="center"|http://teamikaria.com/dl/vpPTy_3DZg86sfyXmHEMWFBq3nKgUC-kxj3v5akICautHBGF.png
|align="center"|http://teamikaria.com/dl/vpPTy_3DZg86sfyXmHEMWFBq3nKgUC-kxj3v5akICautHBGF.png
 +
|align="center"|http://teamikaria.com/dl/W6Za757XuFgeg1uEqs_cVjf-miNfwZfHVcDAvzaAN3XHaUX8.png
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|align="center"|http://teamikaria.com/dl/T3PaGuR2qrVGOFTcYIZKQH5EjygmYUN4HNtjSHMmYir_CsMJ.png
|-
|-
|align="center"|I
|align="center"|I
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|align="center"|00
|align="center"|00
|align="center"|000
|align="center"|000
-
|align="center"|100
 
|align="center"|SS
|align="center"|SS
-
|align="center"|0SS
 
|align="center"|SSS
|align="center"|SSS
 +
|align="center"|0SS
 +
|align="center"|1SS
|-
|-
|align="center"|
|align="center"|
|align="center"|[[Möbial hose]]
|align="center"|[[Möbial hose]]
|align="center"|[[Real projective planar hose]]
|align="center"|[[Real projective planar hose]]
-
|align="center"|[[Real projective realm]]
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|align="center"|[[Toric bottle]]
|align="center"|
|align="center"|
|align="center"|
|align="center"|
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|align="center"|[[Toraspherindric bottle]]
 
|align="center"|[[Toraspherindric dalma]]
|align="center"|[[Toraspherindric dalma]]
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|align="center"|
|-
|-
|align="center"|
|align="center"|
|align="center"|http://teamikaria.com/dl/6gAZrn9Z77MrrUPMXNA6h-RSrAaGZ3JbHp-gcowPgo0GjeOr.png
|align="center"|http://teamikaria.com/dl/6gAZrn9Z77MrrUPMXNA6h-RSrAaGZ3JbHp-gcowPgo0GjeOr.png
|align="center"|http://teamikaria.com/dl/rmkny1xj0N2nxBzjd53RA3brepA4EH97KwSxvAsXp2eQV08p.png
|align="center"|http://teamikaria.com/dl/rmkny1xj0N2nxBzjd53RA3brepA4EH97KwSxvAsXp2eQV08p.png
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|align="center"|http://teamikaria.com/dl/1oBEeFhW91X-0NnU9eTfsNjroY7VTNDTwLQGbQflaCvwIAre.png
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|align="center"|http://teamikaria.com/dl/ujCnRAl3fVM5XJtQmyne3A_1QD_qq7ys_wijaGiYp8ara-hz.png
|align="center"|
|align="center"|
|align="center"|
|align="center"|
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|align="center"|http://teamikaria.com/dl/T3PaGuR2qrVGOFTcYIZKQH5EjygmYUN4HNtjSHMmYir_CsMJ.png
 
|align="center"|http://teamikaria.com/dl/jzaElp9Oo2w0udQTST58jKWMrOMTuWhYu8SzgpzUBfOJtJDZ.png
|align="center"|http://teamikaria.com/dl/jzaElp9Oo2w0udQTST58jKWMrOMTuWhYu8SzgpzUBfOJtJDZ.png
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|align="center"|
|-
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|align="center"|
|align="center"|
|align="center"|1
|align="center"|1
|align="center"|11
|align="center"|11
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|align="center"|111
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|align="center"|100
|align="center"|
|align="center"|
|align="center"|
|align="center"|
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|align="center"|1SS
 
|align="center"|SS1
|align="center"|SS1
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|align="center"|
 +
|-
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|align="center"|
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|align="center"|
 +
|align="center"|
 +
|align="center"|[[Real projective realm]]
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|align="center"|
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|align="center"|
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|align="center"|
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|align="center"|
 +
|-
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|align="center"|
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|align="center"|
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|align="center"|
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|align="center"|http://teamikaria.com/dl/1oBEeFhW91X-0NnU9eTfsNjroY7VTNDTwLQGbQflaCvwIAre.png
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|align="center"|
 +
|align="center"|
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|align="center"|
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|align="center"|
 +
|-
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|align="center"|
 +
|align="center"|
 +
|align="center"|
 +
|align="center"|111
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|align="center"|
 +
|align="center"|
 +
|align="center"|
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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.
-
There are 3 more interesting group S3 cubic manifolds. These are SS0, S0S and S1S, and are currently unknown.
+
There are 3 more interesting group 0-3 cubic manifolds. These are SS0, S0S and S1S, and are currently unknown.
[[Category:Topology]]
[[Category:Topology]]

Revision as of 13:43, 31 August 2008

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:

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
http://teamikaria.com/dl/zebnuw8TVYvi5oa7PHJFVXjeWIzl6j5b_OqpV9YRZnQ1HR5q.png http://teamikaria.com/dl/HGJdxDSgKdFFmJwMx3NmaeeTiQYuIdQTfVqyhFLWxV8c60WH.png http://teamikaria.com/dl/iCJxkx0R_t4XE_yO1bF8QpBRH-XMi5nDn-ELIknrjyQDa9X8.png http://teamikaria.com/dl/K0bGhfOG3hetb1QT2ev6gCwfdT_JioJJVxSOe65WDn2pyrQW.png http://teamikaria.com/dl/LpJNtb-p34CMW2XuXAEHxPWu_Yo2bx-gXbyxr8eOQU8xdJAr.png http://teamikaria.com/dl/GonihCjgciwA83T8wiya7qsB93p4mpXIi2_t-EWe11UwKv71.png http://teamikaria.com/dl/4h4Ag96nOfc3GRzA8r_rwcqoGURpR8NFiLFWJ9VogyAGV2az.png http://teamikaria.com/dl/6DfUWpRUCTvIcPzgXCfjaVzgQueSMNorAmgm8o87JArm5GhR.png
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
http://teamikaria.com/dl/jGzBHc8mcwh4Dmf4wKU0l6L-RHMGonZJ-UFLSz1lqBY4JywN.png http://teamikaria.com/dl/Z5maD1K5HvjYhHkveqYdTTOP2T39flttEhuhK0e2uFVuKSpu.png http://teamikaria.com/dl/JpQKfGN7vea3nHKcWCIJhh5Q3n9P1868bafu31nJl-sBmEM2.png http://teamikaria.com/dl/NJ5W30kaUY8xQlTsXh2tizXABsTdwYmId9nd0CtcwpqNskcf.png http://teamikaria.com/dl/2hQD0jvviin64DzXw6AW52mh7VmrSL_6Cz1Er82_vrXEHe2x.png http://teamikaria.com/dl/vpPTy_3DZg86sfyXmHEMWFBq3nKgUC-kxj3v5akICautHBGF.png http://teamikaria.com/dl/W6Za757XuFgeg1uEqs_cVjf-miNfwZfHVcDAvzaAN3XHaUX8.png http://teamikaria.com/dl/T3PaGuR2qrVGOFTcYIZKQH5EjygmYUN4HNtjSHMmYir_CsMJ.png
I 0 00 000 SS SSS 0SS 1SS
Möbial hose Real projective planar hose Toric bottle Toraspherindric dalma
http://teamikaria.com/dl/6gAZrn9Z77MrrUPMXNA6h-RSrAaGZ3JbHp-gcowPgo0GjeOr.png http://teamikaria.com/dl/rmkny1xj0N2nxBzjd53RA3brepA4EH97KwSxvAsXp2eQV08p.png http://teamikaria.com/dl/ujCnRAl3fVM5XJtQmyne3A_1QD_qq7ys_wijaGiYp8ara-hz.png http://teamikaria.com/dl/jzaElp9Oo2w0udQTST58jKWMrOMTuWhYu8SzgpzUBfOJtJDZ.png
1 11 100 SS1
Real projective realm
http://teamikaria.com/dl/1oBEeFhW91X-0NnU9eTfsNjroY7VTNDTwLQGbQflaCvwIAre.png
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.

Retrieved from "http://hi.gher.space/wiki/Manifold"