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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== Representation ==
 
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The edges that shall be connected are marked by arrows, and the direction of the arrow indicates the orientation of the connection. Edges without arrows are left unconnected.
 
== [[Square]] manifolds ==
== [[Square]] manifolds ==
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These are the best known manifolds.
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These are the best known manifolds. There are eight of them shown as follows:
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=== [[Cylinder]] ===
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<blockquote>http://teamikaria.com/dl/4f5UTQm7QPcUbsnKRt7lxQ98mV2_tVAaD1LfkWhF4s44855N.png</blockquote>
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*Note that the cylinder formed this way is actually an [[uncapped]] cylinder.
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=== [[Möbius strip]] ===
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<blockquote>http://teamikaria.com/dl/XVm9N4bZiV6O_SjicHHgeqAa42JKtOA2gc52LZlWllVEYSqy.png</blockquote>
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*The Möbius strip is the only [[nonorientable]] surface that can be embedded in 3D.
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=== [[Torus]] ===
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<blockquote>http://teamikaria.com/dl/a713qY7uBrzErdtgJTSt7Hl2ukV5p_dP58rOFNeMzwYSjXxJ.png</blockquote>
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=== [[Klein bottle]] ===
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<blockquote>http://teamikaria.com/dl/b1DlbJwwy1gB1TB_tBplkefvyyKbWlXRC1cUN4TomJawMmDC.png</blockquote>
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*Note that there are two forms of Klein bottle: the Figure-8 shape, and the "ordinary" bottle shape.
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This is because there are two ways to "fold up" the shape: you can either make the cylinder first and then get the bottle shape, or you can make the Möbius strip first and then get the Figure-8 shape.
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*When [[immersed]] in three dimensions, the Klein bottle is [[self-intersecting]].
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=== [[Real projective plane]] ===
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<blockquote>http://teamikaria.com/dl/0aavLBCz3scuR_PywCLhKq0A_sEiOWeWgnhzF2qLVb2GAq-i.png</blockquote>
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*When immersed in three dimensions, the real projective plane is self-intersecting.
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{|
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This immersion is a combination of the two forms of Klein bottle: you take the figure-8 shape, split it open and insert one end through the side of the other, attaching it on the inside.{{hmm}}
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|width="12%" align="center"|[[Square]]
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|width="12%" align="center"|Uncapped [[cylinder]]
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|width="12%" align="center"|[[Möbius strip]]
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|width="12%" align="center"|[[Torus]]
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|width="12%" align="center"|[[Klein figure 8]]
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|width="12%" align="center"|[[Klein bottle]]
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|width="12%" align="center"|[[Real projective plane]]
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|width="12%" align="center"|[[Sphere]]
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|-
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|align="center"|http://teamikaria.com/dl/zebnuw8TVYvi5oa7PHJFVXjeWIzl6j5b_OqpV9YRZnQ1HR5q.png
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|align="center"|http://teamikaria.com/dl/HGJdxDSgKdFFmJwMx3NmaeeTiQYuIdQTfVqyhFLWxV8c60WH.png
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|align="center"|http://teamikaria.com/dl/iCJxkx0R_t4XE_yO1bF8QpBRH-XMi5nDn-ELIknrjyQDa9X8.png
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|align="center"|http://teamikaria.com/dl/K0bGhfOG3hetb1QT2ev6gCwfdT_JioJJVxSOe65WDn2pyrQW.png
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|align="center"|http://teamikaria.com/dl/LpJNtb-p34CMW2XuXAEHxPWu_Yo2bx-gXbyxr8eOQU8xdJAr.png
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|align="center"|http://teamikaria.com/dl/GonihCjgciwA83T8wiya7qsB93p4mpXIi2_t-EWe11UwKv71.png
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|align="center"|http://teamikaria.com/dl/4h4Ag96nOfc3GRzA8r_rwcqoGURpR8NFiLFWJ9VogyAGV2az.png
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|align="center"|http://teamikaria.com/dl/6DfUWpRUCTvIcPzgXCfjaVzgQueSMNorAmgm8o87JArm5GhR.png
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|}
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== Notes ==
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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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*There is no manifold for a [[sphere]]. This is because a sphere has a point of [[convergance]], and if you go off the top of a sphere, you end up going down it again, which cannot be defined by the manifold representations. Similarly, there is no manifold for any 3D shape with a [[genus]] of zero.
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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.
[[Category:Topology]]
[[Category:Topology]]

Revision as of 22:24, 30 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:

Square 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

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.

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