{"id":196,"date":"2019-06-26T22:58:53","date_gmt":"2019-06-26T22:58:53","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=196"},"modified":"2019-11-20T21:13:47","modified_gmt":"2019-11-20T21:13:47","slug":"euler-characteristic","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/euler-characteristic\/","title":{"rendered":"Euler Characteristic"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Take out a sheet of paper. Pop quiz! (just kidding).<\/p>\n\n\n\n

Draw any number of dots on your page. Now connect the dots with lines, subject to the following rules: lines may not cross each other as they move from dot to dot, and every dot on your page must be connected to every other dot through a sequence of lines.<\/p>\n\n\n\n

Now count the number dots (D), lines (L), and regions separated by lines (R). (Don’t forget to count the outside as a region too.) Compute D-L+R. What do you get?<\/p>\n\n\n\n

No matter how you started, the number you will always get is 2!<\/p>\n\n\n\n

In the figure, D=9, L=12, R=5, and indeed, D-L+R=2.<\/p>\n\n\n\n

If the lines represent fences, and the dots fenceposts, then the regions separated by the fenceposts are the pastures. So, if you are a farmer who wants to fence off 4 pastures together with 55 sections of fence, you can calculate exactly how many fenceposts you need, no matter how you arrange the fences<\/em>! 
(L=55, R=5=4+outside, so D=2+L-R=2+55-5=52 fenceposts.)<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
You may wish to have everyone shout out their answer at the same time… students will be surprised they all get the same answer.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
The number D-L+R is called the Euler characteristic<\/em> of a surface. It is an invariant<\/em> of a surface, meaning that while it looks like it may depend on the system of fences you draw, it really does not (as long as every pasture, including the outside, is topologically a disk with no holes). Thus the number only depends on the topology of the surface that you are on! For planes and spheres, this number is always 2.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Euler Characteristic.” Math 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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