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 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 catenoid
made from two unlinked loops
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If you pull the two circles far enough apart, the topological type changes. You end up with a pair of flat disks.
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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
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