Pub Date : 2022-07-03DOI: 10.1080/17513472.2022.2116745
Christopher Adler, J. Allouche
We partly decipher a family of finite integer sequences used in a musical composition of the first author, by showing in particular that they relate to arithmetic classical problems (counting cycles in a permutation, primitive roots modulo a prime number, Wieferich primes, etc.), and also to the art of shuffling cards and to the art of juggling. GRAPHICAL ABSTRACT
{"title":"Finite self-similar sequences, permutation cycles, and music composition","authors":"Christopher Adler, J. Allouche","doi":"10.1080/17513472.2022.2116745","DOIUrl":"https://doi.org/10.1080/17513472.2022.2116745","url":null,"abstract":"We partly decipher a family of finite integer sequences used in a musical composition of the first author, by showing in particular that they relate to arithmetic classical problems (counting cycles in a permutation, primitive roots modulo a prime number, Wieferich primes, etc.), and also to the art of shuffling cards and to the art of juggling. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"14 1","pages":"244 - 261"},"PeriodicalIF":0.2,"publicationDate":"2022-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90104276","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}
Pub Date : 2022-05-30DOI: 10.1080/17513472.2022.2079941
Martin Skrodzki, Milena Damrau
Recent years saw a rapid increase in conference formats that take place either fully online or in a hybrid fashion with some people on-site and others online. While these formats brought new challenges, they also opened up new opportunities. In the present article, we first outline advantages and disadvantages of different conference formats as discussed in the literature. We then share our own experiences based on two mathematics and art events that occurred during the respective annual meetings of the German Mathematical Society in 2020 and 2021. This is to illustrate the main benefits of online formats, in particular for the MathArt community. We conclude by highlighting two specific aspects – the facilitated presentation of large artworks and the availability of talk recordings – and give a brief outlook on hybrid events.
{"title":"Benefits of online meetings for the MathArt community: experiences from two events","authors":"Martin Skrodzki, Milena Damrau","doi":"10.1080/17513472.2022.2079941","DOIUrl":"https://doi.org/10.1080/17513472.2022.2079941","url":null,"abstract":"Recent years saw a rapid increase in conference formats that take place either fully online or in a hybrid fashion with some people on-site and others online. While these formats brought new challenges, they also opened up new opportunities. In the present article, we first outline advantages and disadvantages of different conference formats as discussed in the literature. We then share our own experiences based on two mathematics and art events that occurred during the respective annual meetings of the German Mathematical Society in 2020 and 2021. This is to illustrate the main benefits of online formats, in particular for the MathArt community. We conclude by highlighting two specific aspects – the facilitated presentation of large artworks and the availability of talk recordings – and give a brief outlook on hybrid events.","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"43 1","pages":"262 - 269"},"PeriodicalIF":0.2,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74203099","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}
Pub Date : 2022-04-09DOI: 10.1080/17513472.2022.2058819
S. Goldstine, C. Yackel
Mosaic knitting is a method of two-colour knitting that has become popular in recent decades. Our analysis begins with the mathematical rules that govern stitch patterns in mosaic knitting. Through this characterization, we find the total number of mosaic patterns possible in a given size of fabric and bound the number of patterns that are practical to knit. We proceed to a classification of the symmetry types that are compatible with mosaic designs, including theorems that enumerate which one- and two-colour frieze and wallpaper groups are and are not attainable in mosaic knitting. Our discussion includes practical information for knitwear designers and a multitude of sample patterns. GRAPHICAL ABSTRACT
{"title":"A mathematical analysis of mosaic knitting: constraints, combinatorics, and colour-swapping symmetries","authors":"S. Goldstine, C. Yackel","doi":"10.1080/17513472.2022.2058819","DOIUrl":"https://doi.org/10.1080/17513472.2022.2058819","url":null,"abstract":"Mosaic knitting is a method of two-colour knitting that has become popular in recent decades. Our analysis begins with the mathematical rules that govern stitch patterns in mosaic knitting. Through this characterization, we find the total number of mosaic patterns possible in a given size of fabric and bound the number of patterns that are practical to knit. We proceed to a classification of the symmetry types that are compatible with mosaic designs, including theorems that enumerate which one- and two-colour frieze and wallpaper groups are and are not attainable in mosaic knitting. Our discussion includes practical information for knitwear designers and a multitude of sample patterns. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"29 1","pages":"183 - 217"},"PeriodicalIF":0.2,"publicationDate":"2022-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81699605","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}
Pub Date : 2022-04-03DOI: 10.1080/17513472.2022.2082003
Rodrigo Treviño
In this paper, I try to explain how, by using concepts and ideas from the mathematical theory of tilings, we can approach metre in music through a geometric and algebraic point of view, being pinned down by a subgroup of with the hierarchical structure, leading to an abstract approach to rhythm, tempo and time signatures. I will also describe an algorithmic approach to write down sound using this structure which gives a way in which music can be written in an irrational metre.
{"title":"Quasimusic: tilings and metre","authors":"Rodrigo Treviño","doi":"10.1080/17513472.2022.2082003","DOIUrl":"https://doi.org/10.1080/17513472.2022.2082003","url":null,"abstract":"In this paper, I try to explain how, by using concepts and ideas from the mathematical theory of tilings, we can approach metre in music through a geometric and algebraic point of view, being pinned down by a subgroup of with the hierarchical structure, leading to an abstract approach to rhythm, tempo and time signatures. I will also describe an algorithmic approach to write down sound using this structure which gives a way in which music can be written in an irrational metre.","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"123 1","pages":"162 - 181"},"PeriodicalIF":0.2,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77072014","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}
Pub Date : 2022-04-03DOI: 10.1080/17513472.2022.2058865
Taneli Luotoniemi
Although projective geometry is an elegant and enlightening domain of spatial thinking and doing, it remains largely unknown to the general audience. This shortcoming can be mended with the aid of figures consisting of points, lines, and planes, that illustrate various projective phenomena. In practice, these configurations can be assembled physically from sticks tied together at their crossings. As an example, I discuss a set of five configurations and some of the projective topics connected to them. The activity of building the stick models offers an instructive, simple, and sculpturally engaging approach to projective geometry. GRAPHICAL ABSTRACT
{"title":"Stick models of projective configurations","authors":"Taneli Luotoniemi","doi":"10.1080/17513472.2022.2058865","DOIUrl":"https://doi.org/10.1080/17513472.2022.2058865","url":null,"abstract":"Although projective geometry is an elegant and enlightening domain of spatial thinking and doing, it remains largely unknown to the general audience. This shortcoming can be mended with the aid of figures consisting of points, lines, and planes, that illustrate various projective phenomena. In practice, these configurations can be assembled physically from sticks tied together at their crossings. As an example, I discuss a set of five configurations and some of the projective topics connected to them. The activity of building the stick models offers an instructive, simple, and sculpturally engaging approach to projective geometry. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"27 1","pages":"104 - 120"},"PeriodicalIF":0.2,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73996006","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}
Pub Date : 2022-04-03DOI: 10.1080/17513472.2022.2081047
S. Lucas, Laura Taalman
Graphs are typically represented in published research literature as two-dimensional images, for obvious reasons. With the increased accessibility of 3D rendering software and 3D printing hardware, we can now represent graphs in three dimensions more easily. Years of published work in the field have led to certain ‘standard’ two-dimensional configurations of well-known graphs such as the Petersen graph or , but there is no such standard for illustrations of graphs in three-dimensional space. Ideally, a spatial graph configuration should highlight the primary properties and features of the graph, as well as be aesthetically pleasing to view. In this paper, we will suggest and realize standard ideal spatial configurations for a variety of well-known graphs and families of graphs. These configurations can help provide fresh three-dimensional intuition about certain families of graphs, in particular the relationships between graphs in the Petersen family. GRAPHICAL ABSTRACT
{"title":"Ideal spatial graph configurations","authors":"S. Lucas, Laura Taalman","doi":"10.1080/17513472.2022.2081047","DOIUrl":"https://doi.org/10.1080/17513472.2022.2081047","url":null,"abstract":"Graphs are typically represented in published research literature as two-dimensional images, for obvious reasons. With the increased accessibility of 3D rendering software and 3D printing hardware, we can now represent graphs in three dimensions more easily. Years of published work in the field have led to certain ‘standard’ two-dimensional configurations of well-known graphs such as the Petersen graph or , but there is no such standard for illustrations of graphs in three-dimensional space. Ideally, a spatial graph configuration should highlight the primary properties and features of the graph, as well as be aesthetically pleasing to view. In this paper, we will suggest and realize standard ideal spatial configurations for a variety of well-known graphs and families of graphs. These configurations can help provide fresh three-dimensional intuition about certain families of graphs, in particular the relationships between graphs in the Petersen family. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"364 1","pages":"121 - 132"},"PeriodicalIF":0.2,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78977003","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}
Pub Date : 2022-04-03DOI: 10.1080/17513472.2022.2085977
E. Harriss, Henry Segerman
Edmund viscerally remembers, during his PhD, Simon Donaldson describing the hopf fibration, sketching it on the board and discussing it. Those words and images triggered something, and a fundamental intuition of the hopf fibration was created in his mind. The experience was so intense that he could still picture the room, down to the people sitting in it. Edmund created Figure 1 shortly after. An act of resonance, as in Gromov’s words above, had occurred, and it did so without the transcription of logical symbols. This story highlights the intriguing mixture of the personal and objective that good illustration enables. The articles in this special issue,many inspired by the 2019 semester on Illustrating Mathematics that took place at the ICERM,1 show this idea in many different ways. We begin, however, by considering the role of illustration in mathematics and its relationship to art.
{"title":"The art of illustrating mathematics","authors":"E. Harriss, Henry Segerman","doi":"10.1080/17513472.2022.2085977","DOIUrl":"https://doi.org/10.1080/17513472.2022.2085977","url":null,"abstract":"Edmund viscerally remembers, during his PhD, Simon Donaldson describing the hopf fibration, sketching it on the board and discussing it. Those words and images triggered something, and a fundamental intuition of the hopf fibration was created in his mind. The experience was so intense that he could still picture the room, down to the people sitting in it. Edmund created Figure 1 shortly after. An act of resonance, as in Gromov’s words above, had occurred, and it did so without the transcription of logical symbols. This story highlights the intriguing mixture of the personal and objective that good illustration enables. The articles in this special issue,many inspired by the 2019 semester on Illustrating Mathematics that took place at the ICERM,1 show this idea in many different ways. We begin, however, by considering the role of illustration in mathematics and its relationship to art.","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"12 1","pages":"1 - 10"},"PeriodicalIF":0.2,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76328735","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}
Pub Date : 2022-04-03DOI: 10.1080/17513472.2022.2059645
Enric Cosme Llópez, Raúl Ruiz Mora, Núria Tamarit
Given a homomorphism between algebras, there exists an isomorphism between the quotient of the domain by its kernel and the subalgebra in the codomain given by its image. This theorem, commonly known as the first isomorphism theorem, is a fundamental algebraic result. Different problems have been identified in its instruction, mainly related to the abstraction inherent to its content and to the lack of conceptual models to improve its understanding. In response to this situation, in this paper, we present an illustration that explores the narrative and graphical resources of comics with the aim of describing the set-theoretic elements that are involved in the proof of this theorem. GRAPHICAL ABSTRACT
{"title":"A comic page for the first isomorphism theorem","authors":"Enric Cosme Llópez, Raúl Ruiz Mora, Núria Tamarit","doi":"10.1080/17513472.2022.2059645","DOIUrl":"https://doi.org/10.1080/17513472.2022.2059645","url":null,"abstract":"Given a homomorphism between algebras, there exists an isomorphism between the quotient of the domain by its kernel and the subalgebra in the codomain given by its image. This theorem, commonly known as the first isomorphism theorem, is a fundamental algebraic result. Different problems have been identified in its instruction, mainly related to the abstraction inherent to its content and to the lack of conceptual models to improve its understanding. In response to this situation, in this paper, we present an illustration that explores the narrative and graphical resources of comics with the aim of describing the set-theoretic elements that are involved in the proof of this theorem. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"9 1","pages":"29 - 56"},"PeriodicalIF":0.2,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87233936","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}
Pub Date : 2022-03-04DOI: 10.1080/17513472.2022.2045047
K. Sugihara
The geometric principle of a new 3D optical illusion, which we refer to as the ‘rising object illusion,’ is presented. In this illusion, a horizontally lying columnar object rises vertically in a mirror, although the mirror stands vertically, and consequently the horizontal directions in the real world remain horizontal in the mirror. Actually, the illusion object is a picture of the original columnar object expanded by 1.41 (square root of two) in the direction of the axis and placed horizontally. This visual effect occurs only when the axis of the object is directed toward the viewer, and the viewer sees the object with a -downward orientation. GRAPHICAL ABSTRACT
{"title":"Rising object illusion","authors":"K. Sugihara","doi":"10.1080/17513472.2022.2045047","DOIUrl":"https://doi.org/10.1080/17513472.2022.2045047","url":null,"abstract":"The geometric principle of a new 3D optical illusion, which we refer to as the ‘rising object illusion,’ is presented. In this illusion, a horizontally lying columnar object rises vertically in a mirror, although the mirror stands vertically, and consequently the horizontal directions in the real world remain horizontal in the mirror. Actually, the illusion object is a picture of the original columnar object expanded by 1.41 (square root of two) in the direction of the axis and placed horizontally. This visual effect occurs only when the axis of the object is directed toward the viewer, and the viewer sees the object with a -downward orientation. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"9 1","pages":"232 - 243"},"PeriodicalIF":0.2,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85250900","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}
Pub Date : 2021-12-30DOI: 10.1080/17513472.2022.2069417
Henriette Lipschütz, Ulrich Reitebuch, Martin Skrodzki, K. Polthier
Various computer simulations regarding, e.g. the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual. GRAPHICAL ABSTRACT
{"title":"Weaving patterns inspired by the pentagon snub subdivision scheme","authors":"Henriette Lipschütz, Ulrich Reitebuch, Martin Skrodzki, K. Polthier","doi":"10.1080/17513472.2022.2069417","DOIUrl":"https://doi.org/10.1080/17513472.2022.2069417","url":null,"abstract":"Various computer simulations regarding, e.g. the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"94 1","pages":"75 - 103"},"PeriodicalIF":0.2,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76413474","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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