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/browse.php on line 3 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 4 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 8 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 8 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 12 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 12 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 16 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 16 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 20 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 20 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 20 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 20 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 23 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 23 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 26 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 27 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 29 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 32 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 39 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 39 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 43 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 43 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 47 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 47 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 51 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 54 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 54 Notice: Use of undefined constant img - assumed 'img' in /home/pat/website/public/math/fun/bubbles/browse.php on line 56 Notice: Use of undefined constant img - assumed 'img' in /home/pat/website/public/math/fun/bubbles/browse.php on line 57 Notice: Use of undefined constant img - assumed 'img' in /home/pat/website/public/math/fun/bubbles/browse.php on line 59 Notice: Use of undefined constant img - assumed 'img' in /home/pat/website/public/math/fun/bubbles/browse.php on line 62 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 68 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 68 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 70 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 70 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 70 Notice: Use of undefined constant subcats - assumed 'subcats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 70 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 72 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 72 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 72 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 72 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 75 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 75 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 75 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 75 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 75 Notice: Use of undefined constant cats - assumed 'cats' in /home/pat/website/public/math/fun/bubbles/browse.php on line 75 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 86 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 86 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 99 Notice: Use of undefined constant imgs - assumed 'imgs' in /home/pat/website/public/math/fun/bubbles/browse.php on line 99 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 100 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 100 Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 101 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 101 Minimal surfaces and soap films
previous Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 157 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 157 upnext
Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 164 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 164 An interesting unknot
Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 172 Notice: Use of undefined constant subcat - assumed 'subcat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 172 This the classical example of an unknot which bounds a minimal surface with topology. The surface pictured is a once punctured torus.
previous Notice: Use of undefined constant bub - assumed 'bub' in /home/pat/website/public/math/fun/bubbles/browse.php on line 179 Notice: Use of undefined constant cat - assumed 'cat' in /home/pat/website/public/math/fun/bubbles/browse.php on line 179 upnext