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Four different views of the Morin surface (with "passage barriers" cut out).

What is meant here by "passage barrier"? It is an informal designation for parts of a self-intersecting inmersion in 3-space of a surface: which parts of such surface are bounded by closed loops of double points. Examples: the Klein bottle has one passage barrier, and the Boy's surface has three passage barriers. The Morin surface has four passage barriers.

The Klein bottle in upright position has a "cave entrance" at its bottom, which narrows into an upward tubular passageway which then intersects the wall of the "inner chamber". The intersection is a nearly circular loop of double points, and the surface of the main chamber's wall bounded by this loop is what is being referred to as a "passage barrier". (After passing through the barrier, the passageway continues upward then starts doing a U-turn until it faces downwards and begins to widen, eventually becoming the bottle's main chamber. Before reaching the bottom, though, the main chamber begins to narrow before turning itself back into a cave entrance.)

Designed by AugPi in Texas.
Rendered with Mathematica.
©AugPi, MMV.

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• (del) (cur) 03:06, 15 June 2005 . . en:User:AugPi AugPi ( en:User_talk:AugPi Talk) . . 902x743 (115714 bytes) (Four different views of the Morin surface (with "passage barriers" cut out).)