Pub Date : 2023-11-01DOI: 10.1080/17513472.2023.2275489
Daniel A. Griffith
AbstractA spectral geometry utility awareness, with specific reference to isospectralisation and art painting analytics, is permeating the academy today, with special interest in its ability to foster interfaces between a range of analytical quantitative disciplines and art, exhibiting popularity in, for example, computer engineering/image processing and GIScience/spatial statistics, among other subject areas. This paper contributes to the emerging literature about such mathematized interdisciplinarities and synergies. It more specifically extends the matrix algebra based 2-D Graph Moranian operator that dominates spatial statistics/econometrics to the 3-D Riemannian manifold sphere whose analysis the general Graph Laplacian (i.e. Laplace-Beltrami) operator monopolizes today. One conclusion is that harmonizing the use of these two operators offers a way to expand knowledge and comprehension. Another is a continuing demonstration that the understanding and analysis of art sculptures dovetails with mathematics-art studies.KEYWORDS: Geary ratioMoran coefficientRiemannian manifoldspatial autocorrelationspectral geometry AcknowledgementsDaniel A. Griffith is an Ashbel Smith Professor of Geospatial Information Sciences.Disclosure statementNo potential conflict of interest was reported by the author(s).Statements and declarationsThe author did not receive support from any organization for this work. The author further certifies that he has no affiliations with or involvement in any organization or entity with any financial or non-financial interest in the subject matter or materials discussed in this paper. Finally, the datasets generated and/or analyzed during the study summarized in this paper are available from the corresponding author by reasonable request; a number of the source datasets also are retrievable from online depositories cited in this paper.Notes1 The spatial statistics/econometrics literature notation almost universally symbolizes this matrix with C, and its row-standardized Laplacian companion with W.2 The Laplace-Beltrami operator based upon an unbounded equilateral triangle mesh essentially is a scalar multiple of this Laplacian matrix (Xu, Citation2004; Wu et al., Citation2010).3 They disagree about the spatial autocorrelation portrayal of numerous regular square tessellation eigenvectors, based upon either a rook or a queen definition of adjacency, both of which extract exactly the same eigenvectors.4 See https://github.com/alecjacobson/common-3d-test-models, https://github.com/MPI-IS/mesh/blob/master/data/unittest/sphere.obj, https://cims.nyu.edu/gcl/datasets.html, https://www.cs.cornell.edu/courses/cs4620/2015fa/assignments/a1/a1mesh.html, https://people.sc.fsu.edu/~jburkardt/data/obj/obj.html, http://graphics.stanford.edu/data/3Dscanrep/, https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/, https://github.com/yig/graphics101-meshes/tree/master/examples, and https://archive.lib.msu.edu/crcmath/math/math/t/t200.htm.5 Many d
{"title":"Spectral geometry and Riemannian manifold mesh approximations: some autocorrelation lessons from spatial statistics","authors":"Daniel A. Griffith","doi":"10.1080/17513472.2023.2275489","DOIUrl":"https://doi.org/10.1080/17513472.2023.2275489","url":null,"abstract":"AbstractA spectral geometry utility awareness, with specific reference to isospectralisation and art painting analytics, is permeating the academy today, with special interest in its ability to foster interfaces between a range of analytical quantitative disciplines and art, exhibiting popularity in, for example, computer engineering/image processing and GIScience/spatial statistics, among other subject areas. This paper contributes to the emerging literature about such mathematized interdisciplinarities and synergies. It more specifically extends the matrix algebra based 2-D Graph Moranian operator that dominates spatial statistics/econometrics to the 3-D Riemannian manifold sphere whose analysis the general Graph Laplacian (i.e. Laplace-Beltrami) operator monopolizes today. One conclusion is that harmonizing the use of these two operators offers a way to expand knowledge and comprehension. Another is a continuing demonstration that the understanding and analysis of art sculptures dovetails with mathematics-art studies.KEYWORDS: Geary ratioMoran coefficientRiemannian manifoldspatial autocorrelationspectral geometry AcknowledgementsDaniel A. Griffith is an Ashbel Smith Professor of Geospatial Information Sciences.Disclosure statementNo potential conflict of interest was reported by the author(s).Statements and declarationsThe author did not receive support from any organization for this work. The author further certifies that he has no affiliations with or involvement in any organization or entity with any financial or non-financial interest in the subject matter or materials discussed in this paper. Finally, the datasets generated and/or analyzed during the study summarized in this paper are available from the corresponding author by reasonable request; a number of the source datasets also are retrievable from online depositories cited in this paper.Notes1 The spatial statistics/econometrics literature notation almost universally symbolizes this matrix with C, and its row-standardized Laplacian companion with W.2 The Laplace-Beltrami operator based upon an unbounded equilateral triangle mesh essentially is a scalar multiple of this Laplacian matrix (Xu, Citation2004; Wu et al., Citation2010).3 They disagree about the spatial autocorrelation portrayal of numerous regular square tessellation eigenvectors, based upon either a rook or a queen definition of adjacency, both of which extract exactly the same eigenvectors.4 See https://github.com/alecjacobson/common-3d-test-models, https://github.com/MPI-IS/mesh/blob/master/data/unittest/sphere.obj, https://cims.nyu.edu/gcl/datasets.html, https://www.cs.cornell.edu/courses/cs4620/2015fa/assignments/a1/a1mesh.html, https://people.sc.fsu.edu/~jburkardt/data/obj/obj.html, http://graphics.stanford.edu/data/3Dscanrep/, https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/, https://github.com/yig/graphics101-meshes/tree/master/examples, and https://archive.lib.msu.edu/crcmath/math/math/t/t200.htm.5 Many d","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"21 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135270769","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 : 2023-10-27DOI: 10.1080/17513472.2023.2272328
Payam Seraji
AbstractAfter a short review of Pythagorean theory of harmonic ratios and musical scales as it is described in Plato’s Timaeus treatise, the concept of ‘optimality of a sequence of (real) numbers with respect to Pythagorean ratios’ is defined and main theorem of this article proves that there are only three optimal sequences of length 6, which correspond to three well-known pentatonic scales which are used in many musical traditions (including Chinese, Japanese and others). It is also noted that a definition similar to our optimal scales has appeared in a treatise by Sadi-al-Din Urmavi, a thirteenth century Iranian musicologist.KEYWORDS: Optimal scalePythagorean ratiosTimaeusPentatonic scaleSafi-al-Din al-Urmavi Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 It may be thought that optimal scales can be constructed by simply choosing first notes in the circle of fifths but it is not the case: the first seven notes in the circle of fifths are Do, Sol, Re, La, Mi, Si, Fa# and it can be easily checked that the corresponding scale is not optimal.
摘要简要回顾了柏拉图《提梅乌斯》中毕达哥拉斯和声比与音阶的理论，定义了“毕达哥拉斯数列(实数)相对于毕达哥拉斯数列的最优性”的概念，并证明了只有三个长度为6的最优数列，它们对应于许多音乐传统(包括中国、日本等)中使用的三个著名的五声音阶。值得注意的是，类似于我们的最佳音阶的定义出现在13世纪伊朗音乐学家Sadi-al-Din Urmavi的一篇论文中。关键词:最优尺度;毕达古比例;时间尺度;五声尺度;注1人们可能认为，最优音阶可以通过简单地选择五度圈中的第一个音符来构建，但事实并非如此:五度圈中的前七个音符是Do, Sol, Re, La, Mi, Si, Fa#，很容易检查出相应的音阶不是最优的。
{"title":"Plato’s Timaeus and optimal pentatonic scales","authors":"Payam Seraji","doi":"10.1080/17513472.2023.2272328","DOIUrl":"https://doi.org/10.1080/17513472.2023.2272328","url":null,"abstract":"AbstractAfter a short review of Pythagorean theory of harmonic ratios and musical scales as it is described in Plato’s Timaeus treatise, the concept of ‘optimality of a sequence of (real) numbers with respect to Pythagorean ratios’ is defined and main theorem of this article proves that there are only three optimal sequences of length 6, which correspond to three well-known pentatonic scales which are used in many musical traditions (including Chinese, Japanese and others). It is also noted that a definition similar to our optimal scales has appeared in a treatise by Sadi-al-Din Urmavi, a thirteenth century Iranian musicologist.KEYWORDS: Optimal scalePythagorean ratiosTimaeusPentatonic scaleSafi-al-Din al-Urmavi Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 It may be thought that optimal scales can be constructed by simply choosing first notes in the circle of fifths but it is not the case: the first seven notes in the circle of fifths are Do, Sol, Re, La, Mi, Si, Fa# and it can be easily checked that the corresponding scale is not optimal.","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234891","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 : 2023-10-02DOI: 10.1080/17513472.2023.2281895
Peter beim Graben
After reviewing the physicalistic or metaphorical accounts to musical and visual forces by Arnheim and Larson, respectively, which were inspired by the basic tenets of gestalt psychology, I present a novel, naturalistic, mathematical framework, based on symmetry principles and gauge theory. In musicology, this approach has already been applied to the phenomenon of tonal attraction, leading to a deformation of the circle of fifths. The underlying gauge symmetry turns out as the SO(2) Lie group of a musical quantum model. Here, I present an alternative description in terms of Riemannian geometry. Its essential constraint of invariance of the infinitesimal line element leads to a deformation of the circle of fifths into a heart of fifths. In vision, the same approach is applied to Fraser's twisted cord illusion where concentric circles are deformed to squircle objects by means of an optical gauge field induced through a checkerboard background. GRAPHICAL ABSTRACT
{"title":"Gauge symmetries of musical and visual forces","authors":"Peter beim Graben","doi":"10.1080/17513472.2023.2281895","DOIUrl":"https://doi.org/10.1080/17513472.2023.2281895","url":null,"abstract":"After reviewing the physicalistic or metaphorical accounts to musical and visual forces by Arnheim and Larson, respectively, which were inspired by the basic tenets of gestalt psychology, I present a novel, naturalistic, mathematical framework, based on symmetry principles and gauge theory. In musicology, this approach has already been applied to the phenomenon of tonal attraction, leading to a deformation of the circle of fifths. The underlying gauge symmetry turns out as the SO(2) Lie group of a musical quantum model. Here, I present an alternative description in terms of Riemannian geometry. Its essential constraint of invariance of the infinitesimal line element leads to a deformation of the circle of fifths into a heart of fifths. In vision, the same approach is applied to Fraser's twisted cord illusion where concentric circles are deformed to squircle objects by means of an optical gauge field induced through a checkerboard background. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"10 1","pages":"347 - 382"},"PeriodicalIF":0.2,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139324860","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 : 2023-08-04DOI: 10.1080/17513472.2023.2236005
Pedro Freitas
{"title":"The mathematics of Almada Negreiros","authors":"Pedro Freitas","doi":"10.1080/17513472.2023.2236005","DOIUrl":"https://doi.org/10.1080/17513472.2023.2236005","url":null,"abstract":"","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"83 14","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72404677","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 : 2023-07-20DOI: 10.1080/17513472.2023.2233883
R. Ajlouni
{"title":"Derived from the traditional principles of Islamic geometry, a methodology for generating non-periodic long-range sequences in one-dimension for 8-fold, 10-fold, and 12-fold rotational symmetries","authors":"R. Ajlouni","doi":"10.1080/17513472.2023.2233883","DOIUrl":"https://doi.org/10.1080/17513472.2023.2233883","url":null,"abstract":"","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"11 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82695413","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 : 2023-04-03DOI: 10.1080/17513472.2023.2197831
Kimberly A. Roth
The Hue Shift afghan consists of 100 squares knit with 10 colours in a manner determined by a diagram. How many ways can you knit a Hue Shift afghan? What makes an afghan a Hue Shift is defined. Then the number of different afghans is determined up to symmetry considering colour order, stripe order, and direction of knitting for each square. GRAPHICAL ABSTRACT
{"title":"A mathematician knits an afghan and counts the number of possible patterns","authors":"Kimberly A. Roth","doi":"10.1080/17513472.2023.2197831","DOIUrl":"https://doi.org/10.1080/17513472.2023.2197831","url":null,"abstract":"The Hue Shift afghan consists of 100 squares knit with 10 colours in a manner determined by a diagram. How many ways can you knit a Hue Shift afghan? What makes an afghan a Hue Shift is defined. Then the number of different afghans is determined up to symmetry considering colour order, stripe order, and direction of knitting for each square. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"83 1","pages":"85 - 98"},"PeriodicalIF":0.2,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75836887","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 : 2023-04-03DOI: 10.1080/17513472.2023.2200897
E. Baker, C. Wampler, Daniel R. Baker
We provide relevant math and detailed sewing instructions for constructing a toroidal scarf that reverses three ways and whose design uses the unique inversion properties of a particular torus geometry and particular 3-component link. We explain how the scarf’s sewing instructions are guided by the mathematics underlying its construction. GRAPHICAL ABSTRACT
{"title":"Triply invertible scarf sewing adventures (and instructions)","authors":"E. Baker, C. Wampler, Daniel R. Baker","doi":"10.1080/17513472.2023.2200897","DOIUrl":"https://doi.org/10.1080/17513472.2023.2200897","url":null,"abstract":"We provide relevant math and detailed sewing instructions for constructing a toroidal scarf that reverses three ways and whose design uses the unique inversion properties of a particular torus geometry and particular 3-component link. We explain how the scarf’s sewing instructions are guided by the mathematics underlying its construction. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"15 1","pages":"22 - 49"},"PeriodicalIF":0.2,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76565495","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 : 2023-04-03DOI: 10.1080/17513472.2023.2197832
Berit Nilsen Givens
We investigate a variation on Pascal's triangle and approximations to Sierpinski's triangle, by considering the coefficients in the trinomial expansion . These trinomial coefficients have many properties similar to those of the binomial coefficients. We illustrate the triangle of numbers with a knitted shawl. GRAPHICAL ABSTRACT
{"title":"The trinomial triangle knitted shawl","authors":"Berit Nilsen Givens","doi":"10.1080/17513472.2023.2197832","DOIUrl":"https://doi.org/10.1080/17513472.2023.2197832","url":null,"abstract":"We investigate a variation on Pascal's triangle and approximations to Sierpinski's triangle, by considering the coefficients in the trinomial expansion . These trinomial coefficients have many properties similar to those of the binomial coefficients. We illustrate the triangle of numbers with a knitted shawl. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"193 1","pages":"178 - 193"},"PeriodicalIF":0.2,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78588445","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 : 2023-04-03DOI: 10.1080/17513472.2023.2183805
M. Fernandez-Guasti
{"title":"The cusphere","authors":"M. Fernandez-Guasti","doi":"10.1080/17513472.2023.2183805","DOIUrl":"https://doi.org/10.1080/17513472.2023.2183805","url":null,"abstract":"","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"18 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84241922","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 : 2023-04-03DOI: 10.1080/17513472.2023.2197829
Mary D. Shepherd
The Drunkard's Path quilt block is a basic square quilt block consisting of a quarter circle in one corner on a square of some contrasting fabric. In this paper, we use symmetry to organize a library of quilting patterns using the Drunkard's Path quilt block. The organizational strategy begins by arranging the basic quilt blocks into squares that we call arrangements. We categorize these arrangements by symmetry type. We also act upon the arrangements by rotations, reflections, and colour exchanges, using the results to produce squares that we call tiles. These tiles are subsequently considered as tiles for quilt tops, thereby giving fodder for analysis of the underlying wallpaper symmetry groups and sometimes even two-colour symmetry patterns. Over 90 of the tiles are shown representing just a small number of the possible quilt patterns. GRAPHICAL ABSTRACT
{"title":"Categorizing Drunkard's Path type quilting patterns","authors":"Mary D. Shepherd","doi":"10.1080/17513472.2023.2197829","DOIUrl":"https://doi.org/10.1080/17513472.2023.2197829","url":null,"abstract":"The Drunkard's Path quilt block is a basic square quilt block consisting of a quarter circle in one corner on a square of some contrasting fabric. In this paper, we use symmetry to organize a library of quilting patterns using the Drunkard's Path quilt block. The organizational strategy begins by arranging the basic quilt blocks into squares that we call arrangements. We categorize these arrangements by symmetry type. We also act upon the arrangements by rotations, reflections, and colour exchanges, using the results to produce squares that we call tiles. These tiles are subsequently considered as tiles for quilt tops, thereby giving fodder for analysis of the underlying wallpaper symmetry groups and sometimes even two-colour symmetry patterns. Over 90 of the tiles are shown representing just a small number of the possible quilt patterns. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"386 1","pages":"62 - 84"},"PeriodicalIF":0.2,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87550307","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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