{"id":425,"date":"2019-06-27T17:39:43","date_gmt":"2019-06-27T17:39:43","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=425"},"modified":"2019-11-18T22:41:25","modified_gmt":"2019-11-18T22:41:25","slug":"connected-sums","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/connected-sums\/","title":{"rendered":"Connected Sums"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle, a torus, a 2-holed torus are all examples of surfaces. We do not distinguish between a sphere and a deformed sphere… we say they are “topologically equivalent”. <\/p>\n\n\n\n

You know how to add numbers. But did you know that there is a way to add surfaces? It’s called the “connect sum”. To connect sum two surfaces you pull out a disc from each, creating “holes”, and then sew the two surfaces together along the boundaries of the holes. This gives another surface! Connect sum a 1-holed torus to a 2-holed torus, and you get a 3-holed torus. Connect sum a projective plane with a projective plane, and you get a Klein+bottle! And, it can be shown that if you connect sum three projective planes it is the same surface as the connect sum of a torus and one projective plane!<\/p>\n\n\n\n

The operation is commutative, associative and there is even an identity element: just add a sphere to any surface and you get back that surface!<\/p>\n\n\n\n

But there is no “inverse” operation: you cannot connect sum a torus to anything and hope to get a sphere…<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Draw some fun pictures to illustrate.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
This belongs to a field of mathematics known as topology, which, loosely speaking, is the study of continuous functions and properties of objects which do not change under continuous deformations.<\/p>\n\n\n\n

How to Cite this Page:<\/strong>\u00a0
Su, Francis E., et al. “Connected Sums.”\u00a0Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"

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