The Greek architect Kostas Vittas published in 2006 a beautiful theorem ([1]) on the cyclic quadrilateral as follows:Theorem 1 (Kostas Vittas, 2006): If ABCD is a cyclic quadrilateral with P being the intersection of two diagonals AC and BD, then the four Euler lines of the triangles PAB, PBC, PCD and PDA are concurrent.
{"title":"Extensions of Vittas’ Theorem","authors":"N. Dergiades, Quang Hung Tran","doi":"10.1017/mag.2024.9","DOIUrl":"https://doi.org/10.1017/mag.2024.9","url":null,"abstract":"The Greek architect Kostas Vittas published in 2006 a beautiful theorem ([1]) on the cyclic quadrilateral as follows:Theorem 1 (Kostas Vittas, 2006): If ABCD is a cyclic quadrilateral with P being the intersection of two diagonals AC and BD, then the four Euler lines of the triangles PAB, PBC, PCD and PDA are concurrent.","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"2 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139774346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The 2 × 2 identity matrix, $${I_2} = left( begin{gathered}{rm{1 ,,,0}} hfill {rm{0 ,,,1}} hfill end{gathered} right)$$, has an infinite number of square roots. The purpose of this paper is to show some interesting patterns that appear among these square roots. In the process, we will take a brief tour of some topics in number theory, including Pythagorean triples, Eisenstein triples, Fibonacci numbers, Pell numbers and Diophantine triples.
{"title":"Patterns among square roots of the 2 × 2 identity matrix","authors":"H. Sporn","doi":"10.1017/mag.2024.14","DOIUrl":"https://doi.org/10.1017/mag.2024.14","url":null,"abstract":"The 2 × 2 identity matrix, $${I_2} = left( begin{gathered}{rm{1 ,,,0}} hfill {rm{0 ,,,1}} hfill end{gathered} right)$$, has an infinite number of square roots. The purpose of this paper is to show some interesting patterns that appear among these square roots. In the process, we will take a brief tour of some topics in number theory, including Pythagorean triples, Eisenstein triples, Fibonacci numbers, Pell numbers and Diophantine triples.","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"58 24","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139775414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A cautionary tale about the pole of polar coordinates","authors":"Nick Lord","doi":"10.1017/mag.2024.37","DOIUrl":"https://doi.org/10.1017/mag.2024.37","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"17 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139775423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The polyhedrists: art and geometry in the long sixteenth century by Noam Andrews , pp. 300, $44.95, ISBN 978-0-26204-664-0, MIT Press (2022)","authors":"G. Leversha","doi":"10.1017/mag.2024.43","DOIUrl":"https://doi.org/10.1017/mag.2024.43","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"553 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139835009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Euler’s polynomial f (n) = n2 + n + 41 is famous for producing 40 different prime numbers when the consecutive values 0, 1, …, 39 are substituted: see Table 1. Some authors, including Euler, prefer the polynomial f (n − 1) = n2 − n + 41 with prime values for n = 1, …, 40. Since f (−n) = f (n − 1), f (n) actually takes prime values (with each value repeated once) for n = −40, −39, …, 39; equivalently the polynomial f (n − 40) = n2 − 79n + 1601 takes (repeated) prime values for n = 0, 1, …, 79.
欧拉的多项式 f (n) = n2 + n + 41 以连续替换 0、1、...、39 的值时产生 40 个不同的质数而闻名：见表 1。包括欧拉在内的一些学者更倾向于使用多项式 f (n - 1) = n2 - n + 41，其中 n = 1, ..., 40 为质数。由于 f (-n) = f (n - 1)，f (n) 在 n = -40，-39，...，39 时实际上取质数值（每个值重复一次）；等价多项式 f (n - 40) = n2 - 79n + 1601 在 n = 0，1，...，79 时取（重复）质数值。
{"title":"Euler’s prime-producing polynomial revisited","authors":"R. Heffernan, Nick Lord, Des MacHale","doi":"10.1017/mag.2024.11","DOIUrl":"https://doi.org/10.1017/mag.2024.11","url":null,"abstract":"Euler’s polynomial f (n) = n2 + n + 41 is famous for producing 40 different prime numbers when the consecutive values 0, 1, …, 39 are substituted: see Table 1. Some authors, including Euler, prefer the polynomial f (n − 1) = n2 − n + 41 with prime values for n = 1, …, 40. Since f (−n) = f (n − 1), f (n) actually takes prime values (with each value repeated once) for n = −40, −39, …, 39; equivalently the polynomial f (n − 40) = n2 − 79n + 1601 takes (repeated) prime values for n = 0, 1, …, 79.","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"39 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139775806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A student's guide to Laplace transforms by Daniel Fleisch , pp. 218, £17.99, (paper), ISBN 978-1-00909-629-4, Cambridge University Press (2022)","authors":"Sue Colwell","doi":"10.1017/mag.2024.49","DOIUrl":"https://doi.org/10.1017/mag.2024.49","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"13 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139776585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lost in the Math Museum by Colin Adams, pp 209, $35 (paperback), ISBN 978-1-47046-858-3, American Mathematical Society (2022)","authors":"Mark Hunacek","doi":"10.1017/mag.2024.53","DOIUrl":"https://doi.org/10.1017/mag.2024.53","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"387 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139834395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"108.14 A triangle number identity","authors":"Paul Stephenson","doi":"10.1017/mag.2024.31","DOIUrl":"https://doi.org/10.1017/mag.2024.31","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"382 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139835420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The one true logic: a monist manifesto, by Owen Griffiths and A. C. Paseau, pp 232, ISBN 978-0-19-882971-3, Oxford University Press (2022).","authors":"Alan Slomson","doi":"10.1017/mag.2024.51","DOIUrl":"https://doi.org/10.1017/mag.2024.51","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"363 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139835530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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