![\"\"](\"https:\/\/puzzlezapper.com\/blog\/wp-content\/uploads\/2024\/10\/3-plus-2-om.png\")
<\/a><\/figure><\/div>\n\n\nThis use of a plus sign seems natural enough, but we might want to think a bit more about what it implies. We have established an operation on polyform sets, and a notation for that operation. This raises some questions: what other operations might we want to use? How should we notate them? And finally, can we design a notation system that readably describes a wide variety of polyform sets? (And should we?)<\/p>\n\n\n\n
After addition, a notation for multiplication would be handy. We’ve recently looked at di-triamonds\u2009and t<\/a>ri-diamonds<\/a>. We can call these 2\u00b73-iamonds and 3\u00b72-iamonds respectively. Notice that this multiplication, unlike the addition, doesn’t commute. But it does decompose into addition in the natural way; the 3\u00b72-iamonds are the same as the 2+2+2-iamonds. <\/p>\n\n\n In a way, we were already using polyform multiplication to define n-forms in the first place. The pentominoes are essentially the 5\u00b7monominoes. In the interest of brevity, we can use symbols for common base monoforms:\u25b2, \u25a0, \u2b23, and \u25e3 for the moniamond, monomino, monohex and monotan respectively. If we are consistent with the above examples, a n<\/em>\u25a0 has subdivisions for the individual cells. That may seem a little weird, but it can be useful; a 2\u00d71 rectangle could be either a domino or a tetratan, and we’d like to be able to know which. I won’t show these subdivisions in my graphics unless it aids with clarity.<\/p>\n\n\n\n We would also like to combine sets together into a larger one. This is multiset addition rather than set union, because we could want to work with multiple copies of the same polyform. I’ll use circled operator symbols for multiset operations, even though that’s a little nonstandard. They’re nicely readable, and the circle will be our mnemonic that we’re doing multiset things. The tetrominoes and pentominoes together would be 4\u22955\u25a0. We can read the ‘\u2295’ as “and”, so 4\u22955\u25a0 is read as “the four and five -ominoes”. Making a set from multiple copies of the same set is the same as scalar multiset multiplication. So five copies of the tetrominoes is 5\u22994\u25a0. As before, this is non-commutative left multiplication; the dot is our mnemonic for that. And it decomposes as expected into multiset addition: 5\u22994\u25a0 = 4\u22954\u22954\u22954\u22954\u25a0. I can’t think of any reason I would ever want to do element-wise multiset multiplication with polyforms, but \u2297 is there if I ever need it.<\/p>\n\n\n\n Now that we have multiset operations and polyform connection operations, we can start to combine them. There are 22 4+1\u25a0. I hope to share more problems involving them soon, but one thing I noticed was that with some smaller pieces included I could get an area of 144, and make a square. With my notation system, I can call these 2+1\u22953+1\u22954+1\u25a0. Or I could write that as (2\u22953\u22954)+1\u25a0. Polyform addition distributes over multiset operations!<\/p>\n\n\n (Well, I could have<\/em> made a square. I’m showing this shape instead because PolySolver<\/a> wasn’t finding solutions for the square with separated monominoes. Thanks to Bryce Herdt for showing me a technique for getting PolySolver to find solutions with this property.)<\/p>\n\n\n\n Finally, I must address the final question from the start of this post. Is a notation system for polyform sets actually a reasonable thing to develop, given that I am a lone crank and nobody else is likely to use this stuff? And I think that I am finding, for my own explorations with polyforms, that the answer is yes. With algebraic notation, the concepts behind the notation can be expressed with words, and were for a long time. But symbols are easier to mentally manipulate, and formulas that could not fit into working memory as a paragraph can do so as a modest number of symbols. I am already finding it easier to think about polyform sets because I have symbolic notation for them. As I hinted in my fuzzy polyominoes post<\/a>, I’m working on notation for related concepts, so more posts on polyform notation are sure to follow.<\/p>\n","protected":false},"excerpt":{"rendered":" Here’s a tiling of the 3+2-ominoes: This use of a plus sign seems natural enough, but we might want to think a bit more about what it implies. We have established an operation on polyform sets, and a notation for that operation. This raises some questions: what other operations might we want to use? How … Continue reading Notation Notions: Operations on Ominoes<\/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":[283,282,278,248,279,11,218],"class_list":["post-1073","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-283","tag-282","tag-278","tag-di-triamonds","tag-notation","tag-polyominoes","tag-tri-diamonds"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1073","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=1073"}],"version-history":[{"count":6,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1073\/revisions"}],"predecessor-version":[{"id":1218,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1073\/revisions\/1218"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=1073"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=1073"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=1073"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}<\/a><\/figure><\/div>\n\n\n<\/a><\/figure><\/div>\n\n\n