Pub Date : 2021-10-02DOI: 10.1080/17513472.2021.2008764
Sujan Shrestha
The 24th annual Bridges Conference 2021 amalgamates a series of events, including invited and contributed paper presentations, a juried exhibition of mathematical art, hands-on workshops, a short film festival, a poetry reading, an informal music night, and art performance events. Since 1988, the conference has provided a notable interdisciplinary model as one of the largest conferences on the mathematical connections with art, music, architecture, and culture. GRAPHICAL ABSTRACT
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Pub Date : 2021-10-02DOI: 10.1080/17513472.2021.2011687
Mark D. Tomenes, M. D. L. De Las Peñas
A tiling is isotoxal if its edges form orbits or transitivity classes under the action of its symmetry group. In this article, a method is presented that facilitates the systematic derivation of planar edge-to-edge isotoxal tilings from isohedral tilings. Two well-known subgroups of triangle groups will be used to create and determine classes of isotoxal tilings in the Euclidean, hyperbolic and spherical planes which will be described in terms of their symmetry groups and symbols. The symmetry properties of isotoxal tilings make these appropriate tools to create geometrically influenced artwork such as Escher-like patterns or aesthetically pleasing designs in the three classical geometries. GRAPHICAL ABSTRACT
{"title":"k–isotoxal tilings from [pn ] tilings","authors":"Mark D. Tomenes, M. D. L. De Las Peñas","doi":"10.1080/17513472.2021.2011687","DOIUrl":"https://doi.org/10.1080/17513472.2021.2011687","url":null,"abstract":"A tiling is isotoxal if its edges form orbits or transitivity classes under the action of its symmetry group. In this article, a method is presented that facilitates the systematic derivation of planar edge-to-edge isotoxal tilings from isohedral tilings. Two well-known subgroups of triangle groups will be used to create and determine classes of isotoxal tilings in the Euclidean, hyperbolic and spherical planes which will be described in terms of their symmetry groups and symbols. The symmetry properties of isotoxal tilings make these appropriate tools to create geometrically influenced artwork such as Escher-like patterns or aesthetically pleasing designs in the three classical geometries. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86711426","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 : 2021-10-02DOI: 10.1080/17513472.2021.1996677
J. Wilson
Mathematical aspects of the work of the German conceptual artist, Hanne Darboven, are discussed, including the role of permutation, number representation and symmetry in her early works, and the use of a checksum calculation to record calendar dates in her later works. We analyse the multiple ways she represents the checksum calculations and explore the similarities and differences of her work with mathematics. We also suggest several mathematical questions arising from her work that would be interesting to explore in a discrete mathematics, number theory or liberal arts math classroom. GRAPHICAL ABSTRACT
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Pub Date : 2021-10-02DOI: 10.1080/17513472.2021.2001962
Alexander Guerten
Before I studied mathematics, I had already finished my studies in design with a focus on Illustration and 3D-Animation. My teacher in art philosophy used to say ‘Art is always contradictory, if you are confronted with a piece of art and you can decipher it completely, you can be pretty sure that you are looking at kitsch’ (Engelmann, 2003). One should not take this statement as a general rating, since it does not distinguish between good and bad art. It also includes that kitsch could be work of high artistic quality (although one should pause for a moment to think about what this implies for math-art in general, since mathematics is not very suitable to capture contradictions). But it shows a huge difference between art and illustration: while art is about asking questions, illustration is about giving answers. When you are looking at assembly instructions for an IKEA shelf or a mathematical proof, you want the illustrations to be as clear as possible. Children’s book illustrations normally give us answers about the characters and the surrounding world while an illustration of a poem is supposed to capture the mood and rhythm of the poem. Of course these boundaries are very blurry, so in the following I want to present some illustrations that concentrate on the ‘poetic’ side of mathematical proofs. Inspired by musical compositions, Wassily Kandinsky developed a (very flexible) axiomatic system that enabled him to construct his abstract paintings. In his bookPoint and Line to Plane (Kandinsky, 1926/1955) from 1926 Kandinsky claims that points are the primal element of every painting. A line is the trace of a moving point, and the characteristics of a line or the resulting shapes are defined by the movement of the points. The combination of points, lines, and shapes on the canvas creates tension that we perceive intuitively when we study an artwork, but which in principle could be measured mathematically, if one understands the underlying grammar of the art-language. His approach to not take nature as a model for his paintings, but to instead construct his compositions out of simple geometrical forms was a radical break with the predominant traditions. He claimed to be the first, whoever painted an abstract painting. But there are other contenders who created abstract paintings around the same time, like Robert Delaunay, Piet Mondrian and Hilma af Klint, who could also be regarded as the first abstract painter, depending on your definition of abstract art. To some degree his approach resembles the work of Euclid, who a few thousand years before also developed a (very rigid) axiomatic system based on simple geometrical forms.
在学习数学之前，我已经完成了设计的学习，主要是插画和3d动画。我的艺术哲学老师曾经说过“艺术总是矛盾的，如果你面对一件艺术作品，你可以完全解读它，你可以很确定你在看媚俗”(Engelmann, 2003)。人们不应该把这句话当作一般的评价，因为它没有区分好与坏的艺术。它还包括媚俗化可以是高艺术质量的作品(尽管人们应该停下来思考一下这对一般的数学艺术意味着什么，因为数学不太适合捕捉矛盾)。但它显示了艺术和插图之间的巨大差异:艺术是关于提出问题，而插图是关于给出答案。当你在看宜家货架的组装说明或数学证明时，你会希望插图尽可能清晰。儿童读物插图通常给我们关于人物和周围世界的答案，而诗歌插图应该捕捉诗歌的情绪和节奏。当然，这些界限是非常模糊的，所以在下面我想展示一些集中在数学证明的“诗意”方面的插图。受到音乐作品的启发，瓦西里·康定斯基发展了一个(非常灵活的)公理系统，使他能够构建他的抽象画。康定斯基在1926年出版的《点与线到平面》(Kandinsky, 1926/1955)一书中声称，点是每幅画的基本元素。一条线是一个移动点的轨迹，而一条线的特征或产生的形状是由点的运动来定义的。画布上的点、线和形状的组合创造了一种张力，当我们研究一件艺术品时，我们会直观地感受到这种张力，但如果我们理解艺术语言的潜在语法，原则上可以用数学方法来衡量。他不把自然作为他绘画的模型，而是用简单的几何形式来构建他的作品，这是对主流传统的彻底突破。他自称是第一个画抽象画的人。但也有其他竞争者在同一时期创作了抽象绘画，比如罗伯特·德劳内(Robert Delaunay)、皮特·蒙德里安(Piet Mondrian)和希尔玛·克林特(Hilma af Klint)，他们也可以被视为第一个抽象画家，这取决于你对抽象艺术的定义。在某种程度上，他的方法类似于欧几里得的工作，欧几里得在几千年前也建立了一个基于简单几何形式的(非常严格的)公理系统。
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Pub Date : 2021-10-02DOI: 10.1080/17513472.2021.1979910
Maria Mannone
What do you get reversing all arrows? The drawing ‘Duality’ is an homage to mirrors, classical art themes, and abstract mathematics.I’m looking for beauty in the arts and beauty in science. It’s a ...
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Pub Date : 2021-10-02DOI: 10.1080/17513472.2021.1993657
P. Gailiunas
Helices can be found in the art and architecture of many periods, but almost always as single elements. They can be combined to make infinite structures that provide a range of possibilities for sculpture that have been little explored. The most symmetrical arrangements of helices in three dimensions can be derived from the known ways of packing rods. Some of these possibilities suggest new forms that have helices that pass through the vertices of polyhedra, and, because of the symmetry, there can be a possibility other than the standard construction of a helix through four points. One of the infinite structures is the basis for a newly described enantiomorphic saddle polyhedron that can fill space with its mirror image. GRAPHICAL ABSTRACT
{"title":"Rods, helices and polyhedra","authors":"P. Gailiunas","doi":"10.1080/17513472.2021.1993657","DOIUrl":"https://doi.org/10.1080/17513472.2021.1993657","url":null,"abstract":"Helices can be found in the art and architecture of many periods, but almost always as single elements. They can be combined to make infinite structures that provide a range of possibilities for sculpture that have been little explored. The most symmetrical arrangements of helices in three dimensions can be derived from the known ways of packing rods. Some of these possibilities suggest new forms that have helices that pass through the vertices of polyhedra, and, because of the symmetry, there can be a possibility other than the standard construction of a helix through four points. One of the infinite structures is the basis for a newly described enantiomorphic saddle polyhedron that can fill space with its mirror image. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2021-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84844069","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 : 2021-09-12DOI: 10.1080/17513472.2021.1971018
C. Yackel
Spurred by a study of producing wallpaper pattern types in itajime shibori, this paper explains how the mathematical concept of orbifold places limitations on realizing patterns in this medium. Readers are introduced to the relevant mathematics and artistic processes and their relationships. Each of the seventeen wallpaper patterns is depicted together with its fundamental domain and its orbifold. A theorem shows that at most seven wallpaper pattern types are possible if orbifolds must be folded in three-dimensional space with no cutting. Photographs of itajime shibori dyed versions of all seven are shown in the paper. GRAPHICAL ABSTRACT
{"title":"Wallpaper patterns admissible in itajime shibori","authors":"C. Yackel","doi":"10.1080/17513472.2021.1971018","DOIUrl":"https://doi.org/10.1080/17513472.2021.1971018","url":null,"abstract":"Spurred by a study of producing wallpaper pattern types in itajime shibori, this paper explains how the mathematical concept of orbifold places limitations on realizing patterns in this medium. Readers are introduced to the relevant mathematics and artistic processes and their relationships. Each of the seventeen wallpaper patterns is depicted together with its fundamental domain and its orbifold. A theorem shows that at most seven wallpaper pattern types are possible if orbifolds must be folded in three-dimensional space with no cutting. Photographs of itajime shibori dyed versions of all seven are shown in the paper. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2021-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73736631","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 : 2021-07-13DOI: 10.1080/17513472.2022.2045048
Gabriel Dorfsman-Hopkins, Shuchang Xu
We create plots of algebraic integers in the complex plane, exploring the effect of sizing the points according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize algebraic integers whose Galois group is not the full symmetric group−these integers we call rigid. We then give some analysis of the resulting images, suggesting avenues for future research about the geometry of so-called rigid algebraic integers. GRAPHICAL ABSTRACT
{"title":"Searching for rigidity in algebraic starscapes","authors":"Gabriel Dorfsman-Hopkins, Shuchang Xu","doi":"10.1080/17513472.2022.2045048","DOIUrl":"https://doi.org/10.1080/17513472.2022.2045048","url":null,"abstract":"We create plots of algebraic integers in the complex plane, exploring the effect of sizing the points according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize algebraic integers whose Galois group is not the full symmetric group−these integers we call rigid. We then give some analysis of the resulting images, suggesting avenues for future research about the geometry of so-called rigid algebraic integers. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86776477","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 : 2021-07-04DOI: 10.1080/17513472.2022.2045049
A. Holroyd
The plate trick or belt trick is a striking physical demonstration of properties of the double cover of the three-dimensional rotation group by the sphere of unit quaternions or spinors. The two ends of a flexible object are continuously rotated with respect to each other. Surprisingly, the object can be manipulated so as to avoid accumulating twists. We present a new mechanical linkage that implements this task. It consists of a sequence of rigid bodies connected by hinge joints, together with a purely mechanical control mechanism. It has one degree of freedom, and the motion is generated by simply turning a handle. A video is available at https://www.youtube.com/watch?v=oRPCoEq05Zk.
{"title":"The spinor linkage – a mechanical implementation of the plate trick","authors":"A. Holroyd","doi":"10.1080/17513472.2022.2045049","DOIUrl":"https://doi.org/10.1080/17513472.2022.2045049","url":null,"abstract":"The plate trick or belt trick is a striking physical demonstration of properties of the double cover of the three-dimensional rotation group by the sphere of unit quaternions or spinors. The two ends of a flexible object are continuously rotated with respect to each other. Surprisingly, the object can be manipulated so as to avoid accumulating twists. We present a new mechanical linkage that implements this task. It consists of a sequence of rigid bodies connected by hinge joints, together with a purely mechanical control mechanism. It has one degree of freedom, and the motion is generated by simply turning a handle. A video is available at https://www.youtube.com/watch?v=oRPCoEq05Zk.","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2021-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78991690","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 : 2021-04-03DOI: 10.1080/17513472.2021.1922239
D. Griffith
Existing interfaces between mathematics and art, and geography and art, began overlapping in recent years. This newer overarching intersection partly is attributable to the scientific visualization of the concept of an eigenvector from the subdiscipline of matrix algebra. Spectral geometry and signal processing expanded this overlap. Today, novel applications of the statistical Moran eigenvector spatial filtering (MESF) methodology to paintings accentuates and exploits spatial autocorrelation as a fundamental element of art, further expanding this overlap. This paper studies MESF visualizations by compositing identified relevant spatial autocorrelation components, examining a particular Van Gogh painting for the first time, and more intensely re-examining several paintings already evaluated with MESF techniques. Findings include: painting replications solely based upon their spatial autocorrelation components as captured and visualized by certain eigenvectors are visibly indistinguishable from their original counterparts; and, spatial autocorrelation supplies measurements allowing a differentiation of paintings, a potentially valuable discovery for art history. GRAPHICAL ABSTRACT
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