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.
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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== [[Square]] manifolds ==
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== Square manifolds ==
These are the best known manifolds. There are eight of them shown as follows:
These are the best known manifolds. There are eight of them shown as follows:
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|width="12%" align="center"|[[Square]]
|width="12%" align="center"|[[Square]]
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|width="12%" align="center"|Hose (uncapped [[cylinder]])
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|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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The Klein figure 8 and Klein bottle are topologically equivalent, however they have been listed separately as they appear significantly different.
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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== Cubic manifolds ==
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There are 278 unique cubic manifolds out of 610 defined ones. Only sufficient examples and the most interesting are shown in the following table.
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{|
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!style="background-color: #EEE;"|Group 0
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!style="background-color: #EEE;"|Group 1
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!style="background-color: #EEE;"|Group 2
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!style="background-color: #EEE;" colspan="2"|Group 3
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!style="background-color: #EEE;" colspan="2"|Group S
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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" colspan="2"|512 defined
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|align="center" colspan="2"|25 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" colspan="2"|216 unique
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|align="center" colspan="2"|19 unique
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|-
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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" colspan="2"|3 shown
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|align="center" colspan="2"|4 shown
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|-
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|align="center" width="14%"|[[Cube]]
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|align="center" width="14%"|[[Dihose]]
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|align="center" width="14%"|[[Toric hose]]
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|align="center" width="14%"|[[Ditorus]]
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|align="center" width="14%"|[[Toric bottle]]
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|align="center" width="14%"|[[Glome]]
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|align="center" width="14%"|[[Toraspherinder]]
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|-
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|align="center"|
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|align="center"|[[Möbial hose]]
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|align="center"|[[Real projective planar hose]]
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|align="center"|[[Real projective realm]]
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|align="center"|
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|align="center"|[[Toraspherindric bottle]]
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|align="center"|[[Toraspherindric dalma]]
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|}
[[Category:Topology]]
[[Category:Topology]]

Revision as of 01:47, 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 Group 1 Group 2 Group S
Square Hose (uncapped cylinder) Möbius strip Torus Klein figure 8 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 278 unique cubic manifolds out of 610 defined ones. Only sufficient examples and the most interesting are shown in the following table.

Group 0 Group 1 Group 2 Group 3 Group S
1 defined 8 defined 64 defined 512 defined 25 defined
1 unique 6 unique 36 unique 216 unique 19 unique
1 shown 2 shown 2 shown 3 shown 4 shown
Cube Dihose Toric hose Ditorus Toric bottle Glome Toraspherinder
Möbial hose Real projective planar hose Real projective realm Toraspherindric bottle Toraspherindric dalma
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