Spoiler alert: a solution is included beneath the quoted text.
----- Original Message ----- From: Dan Asimov Sent: 04/20/14 11:11 PM To: math-fun Subject: [math-fun] Surface puzzle
Puzzle:
Start with a vertically flattish spheroidal surface, and make one Y-shaped cut in the top surface and an identical Y-shaped cut in the bottom surface. Each leg of each Y, after being cut, gives rise to two edges that we'll call "neighbors".
Now identify each of the 12 edges created this way with one other of these 12 edges, as follows: Given an edge on top, identify it with the neighbor of its corresponding edge on the bottom.
The resulting depiction of a surface in 3-space can be thought of as crossing itself along three segments sharing a common endpoint, making a Y in space:
* * * | * | * | / \ * / \ * / \ * *
* * *
QUESTION: Identify which topological surface this depicts among all the compact connected surfaces without boundary.*
--Dan _____________________________________ n * Which are the sphere S^2, the connected sum of n tori # T^2 for any n >= 1, k=1, n and the connected sum of n projective planes # P^2 for any n >= 1. k=1 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It's evident that if you walk through a cut and over the sphere, you end up changing your chirality; consequently, the surface is non- orientable so is a connected sum of n projective planes. Now we wish to determine the value of n. When you make the cut followed by the identification, you don't alter the number of faces or edges (in a hypothetical triangulation), but you end up with four fewer vertices and thus have an Euler characteristic of -2. Hence, I claim this is the connected sum of 4 cross-caps. Sincerely, Adam P. Goucher