{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T03:04:01Z","timestamp":1765422241858},"reference-count":8,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T00:00:00Z","timestamp":1620604800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T00:00:00Z","timestamp":1620604800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Hungarian National Foundation for Scientific Research","award":["K124749"],"award-info":[{"award-number":["K124749"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2021,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We say that a triangle <jats:italic>T<\/jats:italic> tiles a polygon <jats:italic>A<\/jats:italic>, if <jats:italic>A<\/jats:italic> can be dissected into finitely many nonoverlapping triangles similar to\u00a0<jats:italic>T<\/jats:italic>. We show that if <jats:inline-formula><jats:alternatives><jats:tex-math>$$N&gt;42$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>N<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>42<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then there are at most three nonsimilar triangles <jats:italic>T<\/jats:italic> such that the angles of <jats:italic>T<\/jats:italic> are rational multiples of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\pi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c0<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:italic>T<\/jats:italic> tiles the regular <jats:italic>N<\/jats:italic>-gon. A tiling into similar triangles is called regular, if the pieces have two angles, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b2<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, such that at each vertex of the tiling the number of angles <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the same as that of\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b2<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Otherwise the tiling is irregular. It is known that for every regular polygon <jats:italic>A<\/jats:italic> there are infinitely many triangles that tile <jats:italic>A<\/jats:italic> regularly. We show that if <jats:inline-formula><jats:alternatives><jats:tex-math>$$N&gt;10$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>N<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>10<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then a triangle <jats:italic>T<\/jats:italic> tiles the regular <jats:italic>N<\/jats:italic>-gon irregularly only if the angles of <jats:italic>T<\/jats:italic> are rational multiples of\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\pi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c0<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Therefore, the number of triangles tiling the regular <jats:italic>N<\/jats:italic>-gon irregularly is at most three for every <jats:inline-formula><jats:alternatives><jats:tex-math>$$N&gt;42$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>N<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>42<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00454-021-00297-1","type":"journal-article","created":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T14:04:17Z","timestamp":1620655457000},"page":"1239-1261","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Irregular Tilings of Regular Polygons with Similar Triangles"],"prefix":"10.1007","volume":"66","author":[{"given":"Miklos","family":"Laczkovich","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,10]]},"reference":[{"key":"297_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511574917","volume-title":"Dissections. 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