{"id":1265,"date":"2026-07-06T07:09:10","date_gmt":"2026-07-06T14:09:10","guid":{"rendered":"https:\/\/puzzlezapper.com\/blog\/?p=1265"},"modified":"2026-07-06T07:10:03","modified_gmt":"2026-07-06T14:10:03","slug":"notation-notions-actions-on-fractions","status":"publish","type":"post","link":"https:\/\/puzzlezapper.com\/blog\/2026\/07\/notation-notions-actions-on-fractions\/","title":{"rendered":"Notation Notions: Actions on Fractions"},"content":{"rendered":"\n

Previously,<\/a> we looked at this set of dihexes and trihexes with half-weight hexes affixed, which we used in a generalized type of tiling where fractional cells may overlap if the weights on those cells sum to one.<\/p>\n\n\n\n

This notion of generalized tiling turns out to be powerful enough to describe all kinds of problems involving tilings with overlap, or “pilings” as I have called them.Because of this, I would like to extend my notation system for polyform sets to encompass the kinds of sets that could appear in such tilings.<\/p>\n\n\n\n

Conveniently, I’m not already using fractions, so I can use them unambiguously to denote a fractional weight cell. The set used in the tiling from that previous post can then be notated as (2\u22953)+\u00bd\u2b23. We may also choose to use a fractional weight cell as a base cell of a polyform. There are 12 5\u00b7\u00bd\u25b2. Here’s a piling:<\/p>\n\n\n

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I’m representing the half-weight cells here with a pinwheel of smaller triangles. Note that when agglomerating half-weight cells, we may stack them in place to make a full-weight cell in addition to placing them in adjacent positions.<\/p>\n\n\n\n

Pilings of entire polyforms of fractional weight are possible. The \u2153\u00b75\u25a0 can tile a 4\u00d75 rectangle, as shown here. (If the pentominoes could tile three copies of a 4\u00d75 rectangle, you could use this as the basis for such a piling, but they cannot.)<\/p>\n\n\n

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Since we have two different kinds of multiplication with the same notation, we’ll want to be careful about the details of how they work. It makes sense for fractional weight multiplication and agglomeration multiplication to both be right associative. Writing out the implicit multiplication and parentheses for the above examples, 5\u00b7\u00bd\u25b2 = 5\u00b7(\u00bd\u00b7\u25a0), and \u2153\u00b75\u25a0 = \u2153\u00b7(5\u00b7\u25a0). Consecutive instances of fractional weight multiplication work like arithmetic multiplication; the \u00bd\u00b7\u00bd\u00b75\u25a0 should be identical to the \u00bc\u00b75\u25a0.<\/p>\n\n\n\n

As a final example, here is a set using the “with<\/a>” (colon) operator. These are the trominoes with two half-weight monominoes, or 3:(2\u2299\u00bd)\u25a0. There are 22 of these. Here the overlaps form a single loop.<\/p>\n\n\n

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There are certainly more things we can do with fractional weights, but some of them will have to wait until we have other operations to work with, subtraction in particular. You can look forward to more posts on my polyform set notation system, but don’t hold your breath; I don’t appear to be a very prolific blogger these days.<\/p>\n","protected":false},"excerpt":{"rendered":"

Previously, we looked at this set of dihexes and trihexes with half-weight hexes affixed, which we used in a generalized type of tiling where fractional cells may overlap if the weights on those cells sum to one. This notion of generalized tiling turns out to be powerful enough to describe all kinds of problems involving … Continue reading Notation Notions: Actions on Fractions<\/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":[279,241,17,11],"class_list":["post-1265","post","type-post","status-publish","format-standard","hentry","category-recreational-mathematics","tag-notation","tag-pilings","tag-polyiamonds","tag-polyominoes"],"_links":{"self":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1265","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=1265"}],"version-history":[{"count":3,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1265\/revisions"}],"predecessor-version":[{"id":1360,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/posts\/1265\/revisions\/1360"}],"wp:attachment":[{"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/media?parent=1265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/categories?post=1265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/puzzlezapper.com\/blog\/wp-json\/wp\/v2\/tags?post=1265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}