david@cvl.UUCP (David Harwood) (06/17/85)
Re Euler formula: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ There is an intuitive constructive 'proof': Verify that the formula is true for a trivial 'object'; supposing that any object may be constructed from this with the addition of vertices, we observe that if the added vertex splits an edge, then F + (V+1) = (E+1) + 2, else if it splits a face, then (F+2) + (V+1) = (E+3). (Of course, I am ignoring the holomogy of these 'objects'.) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Sorry about the typo for the last formula (F+2) + (V+1) = (E+3) + 2.
gjerawlins@watdaisy.UUCP (Gregory J.E. Rawlins) (06/23/85)
In article <555@cvl.UUCP> david@cvl.UUCP (David Harwood) writes: >Re Euler formula: >[.......] >if the added vertex splits an edge, then F + (V+1) = (E+1) + 2, else >if it splits a face, then (F+2) + (V+1) = (E+3). > (Of course, I am ignoring the holomogy of these 'objects'.) >~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > Sorry about the typo for the last formula (F+2) + (V+1) = >(E+3) + 2. Shouldn't that be (F+1) + (V+2) = (E+3) + 2 ? (assuming that the "added vertex" is an edge (in the normal sense) which splits a face). -- Gregory J.E. Rawlins, Department of Computer Science, U. Waterloo {allegra|clyde|linus|inhp4|decvax}!watmath!watdaisy!gjerawlins