{"id":950,"date":"2023-01-09T00:13:08","date_gmt":"2023-01-09T08:13:08","guid":{"rendered":"https:\/\/puzzlezapper.com\/blog\/?p=950"},"modified":"2023-01-09T00:13:08","modified_gmt":"2023-01-09T08:13:08","slug":"rebels-of-flexible-polyforms","status":"publish","type":"post","link":"https:\/\/puzzlezapper.com\/blog\/2023\/01\/rebels-of-flexible-polyforms\/","title":{"rendered":"Rebels of Flexible Polyforms"},"content":{"rendered":"\n

After the last two flexible polyform posts, I had a couple of misfit tilings left over, rebels that didn’t want to fit into a larger theme. But they reminded me that creativity is itself a rebellion; to create something novel, we must set ourselves apart from the paths followed by others.<\/p>\n\n\n\n

A lot of creativity is mixing existing ideas in ways nobody else has yet thought to mix them. That’s much of what I try to do in recreational mathematics in general, but in polyform tilings there is a more literal sense of mixing: combining different types of shapes within the same tiling. Previously I tried mixing different symmetry variants of polyominoes<\/a>. With flexible polyforms, we can mix polyforms with entirely different base cells, since distorting angles can allow them to become compatible. Here is a tiling of flexible pentominoes together with the hexiamonds:<\/p>\n\n\n

\n
\"\"<\/a><\/figure><\/div>\n\n\n

Another creative tool is tweaking magnitudes of qualities. We can turn one negative, and ask what it’s opposite might mean, or what happens if we reverse a process. Or we can tweak a knob the other way toward an extreme. We already do that with flexible polyforms when we ask, “Can we get more symmetry by squeezing more repeated segments around the center?” We can also ask, “What would it mean to make flexible polyominoes even more flexible?” Here’s one answer:<\/p>\n\n\n

\n
\"\"<\/a><\/figure><\/div>\n\n\n

As it happens, my motivation for finding this was seeking a different extreme. I wanted to find the “best” possible shape to tile with flexible polyominoes. There is no clear definition for “best” here. More symmetry is nice, but so is convexity and a smooth border. Regular polygons seem good if you can pull them off. (Previously, we managed to squeeze the hexiamonds into an octagon<\/a>.) So I started with a regular pentagon, and looked for a good way to subdivide it into 60 cells, and ended up with the scheme used above.<\/p>\n\n\n\n

And with that coda, I conclude my series on flexible polyominoes. I’m sure there is much more out there to be found, but for now I’ll be searching elsewhere for new and fun ideas.<\/p>\n","protected":false},"excerpt":{"rendered":"

After the last two flexible polyform posts, I had a couple of misfit tilings left over, rebels that didn’t want to fit into a larger theme. But they reminded me that creativity is itself a rebellion; to create something novel, we must set ourselves apart from the paths followed by others. A lot of creativity … Continue reading Rebels of Flexible Polyforms<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[66,40,10],"class_list":["post-950","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-flexible-polyforms","tag-hexiamonds","tag-pentominoes"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/950","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/comments?post=950"}],"version-history":[{"count":1,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/950\/revisions"}],"predecessor-version":[{"id":954,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/950\/revisions\/954"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=950"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=950"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=950"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}