Subject: Re: triangulation of projective plane
Date: Mon, 16 Sep 2002 11:01:40 +0200 (MEST)
From: David J Green
Dear John Shu,
If I understand your question correctly you are asking why the
obvious triangle in the centre of the diagram has been dissected up
into four smaller triangles (the ones labelled 1,2,3,4).
The reason has to do with the fact that there are precisely three
different ways in which two triangles are allowed to intersect with each
other: (see the lower half of Alexandroff, page 6)
1) The intersection is empty.
2) The intersection consists of one point. This point is a vertex of
both triangles.
3) The intersection consists of a line segment. This line segment is an
edge of each triangle. (And the whole edge, not just part of it.)
Now let's look at the diagram. How does triangle 5 intersect with
that big central triangle? Answer: the intersection consists of top left
hand vertex of the big central triangle, together with half of the
opposite edge. That's dissallowed on two counts: Firstly, only whole edges
are allowed; secondly, you can have an edge or a vertex but not both.
Note that merging triangles 5 and 10 would overcome the half edge
difficulty, but would do nothing about the edge and vertex problem.
I hope this helps.
Yours sincerely,
David Green.
On Sat 14 Sep 2002 John Shu asked:
> Could anyone help me figure out the triangulation of the projective
> plane as presented on page 16 of Paul Alexandroff's book "Elementary
> Concepts of Topology" translated by Alan Farley. Why should there be
> three triangles in the middle of the graph?
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Subject: Re: triangulation of projective plane
Date: Mon, 16 Sep 2002 12:31:26 +0100
From: Ronnie Brown
There is a picture for a part of the Brehm model of the Projective plane on
http://www.cpm.informatics.bangor.ac.uk/sculmath/BMbrehm.htm
The reference for the model is
Brehm, Ulrich How to build minimal polyhedral models of the Boy surface.
Math. Intelligencer 12 (1990), no. 4, 51--56. (Reviewer: T. Banchoff)
Ronnie Brown