{"id":359,"date":"2019-06-26T23:44:44","date_gmt":"2019-06-26T23:44:44","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=359"},"modified":"2019-12-20T23:28:26","modified_gmt":"2019-12-20T23:28:26","slug":"sperners-lemma","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/sperners-lemma\/","title":{"rendered":"Sperner’s Lemma"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Divide a triangle T<\/em> into lots of baby triangles, so that baby triangles only meet at a common edge or a common vertex. Label each main vertex of the whole triangle by 1, 2, or 3; then label vertices on the (12) side by either 1 or 2, on the (23) side by either 2 or 3, and the (13) side by either 1 or 3. Label the points in the interior by any of 1, 2, or 3. For instance, see Figure 1.<\/p>\n\n\n\n

Fun Fact: any such labelling must contain an baby (123) triangle! (In fact, there must be an odd<\/em> number!)<\/p>\n\n\n\n

Actually, a version of Sperner’s Lemma holds in all dimensions. Can you figure out how it generalizes?<\/p>\n\n\n\n

Sperner’s Lemma is equivalent to the\u00a0Brouwer fixed point theorem.<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Have everyone make their own labelled triangle and see how many (123) triangles they have in their picture.<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
There are many proofs of this fact. Some short non-constructive proofs rely on parity arguments. Constructive proofs are the key to many fixed point algorithms as well as\u00a0fair division\u00a0procedures. See the reference.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Sperner’s Lemma.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
F.E. Su, “Rental harmony: Sperner’s lemma in fair division”,
Amer. Math. Monthly, 1999.<\/p>\n\n\n\n

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

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