Manifold (ConceptTopic, 4)

From Hi.gher. Space

Revision as of 13:43, 31 August 2008 by Hayate (Talk | contribs)

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.

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