{"id":658,"date":"2017-05-12T16:31:58","date_gmt":"2017-05-12T23:31:58","guid":{"rendered":"http:\/\/puzzlezapper.com\/blog\/?p=658"},"modified":"2017-05-12T17:58:47","modified_gmt":"2017-05-13T00:58:47","slug":"a-pocketful-of-pentapennies","status":"publish","type":"post","link":"https:\/\/puzzlezapper.com\/blog\/2017\/05\/a-pocketful-of-pentapennies\/","title":{"rendered":"A Pocketful of Pentapennies"},"content":{"rendered":"

We can think of two connected unit coin configurations (or polypennies) as being equivalent if we can transform one into the other by reflection and\/or sliding coins without changing which coins are adjacent. (Coins may not overlap.)<\/p>\n

There are 13 pentapennies. A tiling with fivefold rotational symmetry may be possible, but I haven’t been able to find one.<\/a> (This is problem #27<\/strong>.) However, I recently found a way to tile a figure with fourfold rotational symmetry with them:<\/p>\n

\"\"<\/p>\n

Since I’ve had trouble with five symmetries, you’d think ten would be out of the question. But I found a repeating pattern on the plane with ten symmetries that can be tiled with the pentapennies:<\/p>\n

\"\"<\/p>\n

Notice that there are five translation symmetries. Reflecting the pattern on a vertical axis gives five more symmetries. This pattern uses the wallpaper group<\/a> cm<\/em>. (Conway orbifold symbol: *\u00d7) We could also try to find a tilable pattern with the same amount of symmetry using the wallpaper group p2<\/em>. (Conway orbifold symbol: 2222) <\/p>\n

Problem #45<\/strong>: Find a tiling of the pentapennies on a repeating pattern on the plane that has at least as many symmetries as the one above, but a different wallpaper group. I don’t think going above 10 symmetries is possible, but I’d love to be surprised.<\/p>\n","protected":false},"excerpt":{"rendered":"

We can think of two connected unit coin configurations (or polypennies) as being equivalent if we can transform one into the other by reflection and\/or sliding coins without changing which coins are adjacent. (Coins may not overlap.) There are 13 pentapennies. A tiling with fivefold rotational symmetry may be possible, but I haven’t been able … Continue reading A Pocketful of Pentapennies<\/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":[69,67,71,187],"class_list":["post-658","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-pentapennies","tag-polypennies","tag-symmetry","tag-wallpaper-groups"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/658","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=658"}],"version-history":[{"count":9,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/658\/revisions"}],"predecessor-version":[{"id":669,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/658\/revisions\/669"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=658"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=658"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=658"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}