Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 3 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 4 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 6 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 6 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 8 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 8 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/header.inc on line 12 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 3 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 4 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 8 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/category.php on line 8 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 12 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 16 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/category.php on line 16 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 20 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/category.php on line 20 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 20 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/category.php on line 20 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 23 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 23 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 27 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/category.php on line 27 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 28 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 36 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/category.php on line 36 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 36 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/category.php on line 36 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 37 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/category.php on line 37 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 37 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/category.php on line 37 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 37 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/category.php on line 37 Minimal surfaces and soap films
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Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 91 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 91 The Möbius band
built out of an unknot
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This twisted unknot bounds a minimal Möbius band.
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Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 107 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 107 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/category.php on line 107 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/category.php on line 107
When you pull the loop from the soap, there is normally an extra disk in the center. You need to pop it to see the Möbius band
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