Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1734765
Gianluca Stasi
Architect by training, I have explored and expanded privately the fields of mathematic and geometry moved by personal fascination. Today they constitute a cornerstone of my architectural practice, ...
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1751564
N. Hocking
Topology and art were firmly connected as I was growing up. My artist uncle’s painting of an Alexander horn sphere hung on the dining room wall. My topologist father would delight in showing my siblings and me the strange qualities of the Mobius strip and of, what was to us, the very bizarre Klein bottle and, of course, the famous doughnut-coffee mug. His enthusiasm for these surfaces found room inmy imagination and they have been nestled in there, embedded, ever since. So too the tantalizing and challenging mystery of 4-space, the mathematician’s fourth dimension. If ever I had any doubt that topology and art were natural bedfellows, over here in London, Britain’s flagship modern art gallery, Tate Modern, held a symposium called simply; ‘Topology’. (November 2011 – June 2011). For themost part I work in 2-D and use traditional materials. I draw on fine papers with pencil, charcoal and pastel. I have several rules I impose on myself as I work. I set out a composition using the renaissance practice of rebatement. This is the geometric division of the canvas used to steer the viewer’s eye to all areas of the image and to direct the main focus on the critical parts of the narrative (Bouleu, 1963). I remain faithful to the topological rules of no tearing, cutting or intersecting of the surfaces and no puncturing either, however representing surfaces that intersect in 3-space but do not intersect in 4-space in a 2-dimensional image is a challenge to say the least. Some topological surfaces can engender so many ideas that I have to be firm and temper the wanderings of my imagination. The constraints of the topological surface in the question, the medium in use and staying true to my original inspiration present exactly the kind of challenges that I delight in. For many people even the mention of mathematics is off-putting and mathematical art can seem an oxymoron but there are ways to circumvent this reluctance. Beauty and grace are alluring and can be powerfully persuasive and with topologically derived art there is no need to apply these qualities superficially. They are built-in. As the coffee mug can morph into a doughnut, the Hopf link, two simple interlinked rings, can morph into multiple forms. In the drawing ‘Conversations in a Foreign Language; Three Solid Arguments’ (Figure 1) three solid forms bounded byHopf links are each presented from five different viewpoints. (The models for these forms are three small clay
{"title":"Conversations in a foreign language","authors":"N. Hocking","doi":"10.1080/17513472.2020.1751564","DOIUrl":"https://doi.org/10.1080/17513472.2020.1751564","url":null,"abstract":"Topology and art were firmly connected as I was growing up. My artist uncle’s painting of an Alexander horn sphere hung on the dining room wall. My topologist father would delight in showing my siblings and me the strange qualities of the Mobius strip and of, what was to us, the very bizarre Klein bottle and, of course, the famous doughnut-coffee mug. His enthusiasm for these surfaces found room inmy imagination and they have been nestled in there, embedded, ever since. So too the tantalizing and challenging mystery of 4-space, the mathematician’s fourth dimension. If ever I had any doubt that topology and art were natural bedfellows, over here in London, Britain’s flagship modern art gallery, Tate Modern, held a symposium called simply; ‘Topology’. (November 2011 – June 2011). For themost part I work in 2-D and use traditional materials. I draw on fine papers with pencil, charcoal and pastel. I have several rules I impose on myself as I work. I set out a composition using the renaissance practice of rebatement. This is the geometric division of the canvas used to steer the viewer’s eye to all areas of the image and to direct the main focus on the critical parts of the narrative (Bouleu, 1963). I remain faithful to the topological rules of no tearing, cutting or intersecting of the surfaces and no puncturing either, however representing surfaces that intersect in 3-space but do not intersect in 4-space in a 2-dimensional image is a challenge to say the least. Some topological surfaces can engender so many ideas that I have to be firm and temper the wanderings of my imagination. The constraints of the topological surface in the question, the medium in use and staying true to my original inspiration present exactly the kind of challenges that I delight in. For many people even the mention of mathematics is off-putting and mathematical art can seem an oxymoron but there are ways to circumvent this reluctance. Beauty and grace are alluring and can be powerfully persuasive and with topologically derived art there is no need to apply these qualities superficially. They are built-in. As the coffee mug can morph into a doughnut, the Hopf link, two simple interlinked rings, can morph into multiple forms. In the drawing ‘Conversations in a Foreign Language; Three Solid Arguments’ (Figure 1) three solid forms bounded byHopf links are each presented from five different viewpoints. (The models for these forms are three small clay","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"69 1","pages":"81 - 83"},"PeriodicalIF":0.2,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86629906","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 : 2020-04-02DOI: 10.1080/17513472.2020.1732803
E. Ellison
I grew up with sewing machines and power tools. The basement of my family home in Michigan was full of all kinds of drills, saws, and a sewing machine. My mother and father taught me how to use these machines. I remember always doing things with my hands and wanting to be a visual artist. When it came time to attend a university, my practical-minded mother insisted that I study ‘something that would enable me to support myself’. I laugh, as I recall my youth. I took my mother’s advice and eventually earned a B.A. in Mathematics, an M.A. in Mathematics Education, and an Ed.S. in Educational Administration. Teaching at West Lafayette High School in West Lafayette, Indiana, along with a mathematical methods class to future teachers at Purdue University, I found a strong desire to include mathematically inspired art in my classrooms. At this point, I had been investigating various media including drawing, photography, bronze, painting, and stained glass. In 1980 I discovered a book that changed my life: Geometry and the Visual Arts, by British mathematician Daniel Pedoe. Each page of Daniel’s book spoke to me. I knew I was on to something as I completedmy reading. Themedium of fabric was interesting as it could combinemathematical ideas, colour, texture, shape, perspective, and is totally hands-on. Fabric would allowme to include mathematical ideas for teaching plus give me the ability to hang the mathematical quilts in my classroom. I began generatingmathematical quilts specifically for the classroom. I co-authoredwith Dr. Diana Venters, two books on using quilts as the springboard for explaining mathematical theorems and formulas in the classroom. As students investigated the mathematics in each quilt, lesson plans evolved that could be included in a book on mathematical quilts. Mathematical Quilts andMore Mathematical Quilts resulted. I continue to learn more mathematics and generate more mathematical quilts even though I am retired. To date, I have generated 67 mathematical quilts. The quilts encompass roughly 4,000 years of recordedmathematics. Beginning in 2,000 B.C.E. to the present, 67 quilts represent most of the significant time periods for mathematics. All of my quilts are needle-turned versus using a fused raw edge technique. Ninety percent of my quilts are hand-quilted and are made of 100% cotton. The London Science Museum owns 6 of my quilts in their permanent collection.
我在缝纫机和电动工具的陪伴下长大。我家在密歇根州的地下室里堆满了各种各样的钻头、锯子和一台缝纫机。我的父母教我如何使用这些机器。我记得我总是用手做事情，想成为一名视觉艺术家。到了上大学的时候，我那务实的母亲坚持要我学“能养活自己的东西”。我笑，当我回忆我的青春。我听从了母亲的建议，最终获得了数学学士学位、数学教育硕士学位和教育学硕士学位。在教育管理。我在印第安纳州西拉斐特的西拉斐特高中(West Lafayette High School)教书，同时在普渡大学(Purdue University)为未来的教师开设数学方法课程，我发现自己有一种强烈的愿望，希望在课堂上加入受数学启发的艺术。在这一点上，我一直在研究各种媒体，包括绘画，摄影，青铜，绘画和彩色玻璃。1980年，我发现了一本改变了我一生的书:英国数学家丹尼尔·佩多的《几何与视觉艺术》。丹尼尔书中的每一页都在跟我说话。当我完成阅读时，我知道我发现了一些东西。织物的媒介很有趣，因为它可以结合数学思想、颜色、纹理、形状、视角，而且完全是动手的。织物可以让我在教学中融入数学思想，还可以让我在教室里挂数学被子。我开始专门为教室制作数学被子。我和Diana Venters博士合著了两本书，是关于在课堂上用被子作为跳板来解释数学定理和公式的。当学生们研究每一床被子里的数学知识时，课程计划也随之发展，这些计划可以被纳入一本关于数学被子的书中。数学被子和更多的数学被子。虽然我已经退休了，但我仍然继续学习更多的数学，制作更多的数学被子。到目前为止，我已经制作了67个数学被子。这些被子包含了大约4000年的有记录的数学。从公元前2000年到现在，67条被子代表了数学最重要的时期。我所有的被子都是用针翻的，而不是用融合的毛边技术。我的被子90%都是手工绗缝的，100%是棉的。伦敦科学博物馆永久收藏了我的6条被子。
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1734439
Loren Eiferman
I want to inspire in the viewer of my work a sense of wonder and awe of the natural world, as well as an appreciation of the mathematical structure of shapes and designs that are found in the world that surrounds us all. A common human experience is a simple act of picking up a stick from the ground – peeling the bark off with our fingernails and touching the smooth softwood underneath. My work taps into that same primal desire of touching nature and being close to it as well as appreciating the simple mathematical elegance of patterns and relationships that exist within nature. Trees connect us back to nature, back to this Earth. When walking in the forests surrounding my home, I am constantly picking up sticks of different sizes and lengths. My material surrounds me daily and how extraordinary is it to find something so ubiquitous and be able to create art from that. To craft my work, I usually begin with a drawing. This sketch acts like a road map for what I want my work to look like. That sketch always takes into account not only the structural form and line of the proposed sculpture but also numbers and fractions of each transition and segment that are built into the wooden sculpture. My work is not steam bent. Over many decades I have created a unique technique of working with wood – my primary material. I start out each day collecting tree limbs and sticks that have fallen to the ground. Next, I debark the branch and look for shapes found within each piece of wood. I then cut and permanently join these small shapes together using dowels and wood glue. Then, all the open joints get filled with a homemade putty and sanded. This process of putty and sanding usually needs to be repeated at least three times. It is a very time-consuming process and each sculpture takes me a minimum of a month to build. The sculpture that is being constructed appears like my line drawing but in space. I am interested in having my work appears as if it organically grew in nature, when in fact each sculpture is frequently composed of over 100 small pieces of wood that are seamlessly joined together. My influences are many – from looking at the patterns in nature and plant life on this Earth to researching the heavenly bodies in the images beamed back from the Hubble Telescope – from studying ancient Buddhist mandalas and sacred geometry throughout the ages to delving into quantum physics and string theory. All these influences inspire me daily.
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1734427
P. Ashwell
As an abstract artist inspired by a great love for the work of renaissance artist and mathematician Piero Della Francesca, I am now exploring the visual possibilities of working with the prime number series. Della Francesca’s work was ground-breaking in his acutely observational use of naturalistic colour and understanding of three-dimensional space. He used geometry and perspective to construct his pictorial spaces and their contained objects accurately. However, the viewer does not need to understand themathematics to appreciate the beauty of his art. I am using primes to construct my picture spaces; making visually stimulating art that explore pattern, spirals, sequences and areas. As well as employingmathematical concepts, it is important to me that my work reflects my own developing visual sensibilities. The mathematical element is a springboard in my creative process: discovering and manipulating the inherent patterns helps me find visual solutions and outcomes that are satisfying with or without a viewer’s awareness of the underlying maths. In this way, I trust my work has an impact on people. My first venture into primes, entitled Eratosthenes, is a six across and nine deep grid of coloured rectangles and circles. Derived from primes numbers and painted in heavy impasto, it shows how prime values beyond 2 and 3 only occur as multiples of six plus or minus 1. I named it after the ancient Greek mathematician when I later found out that this was essentially a modified Eratosthenes sieve. My Prime Marks artwork (Figure 1), created in 2010, has 72 (15× 15 cm) individual paintings, each representing one number from 1 to 72. Each prime number is represented by a unique icon. The non-prime numbers are represented by a combination of these prime icons to display their factors. For example, the primes 2 and 3 are represented by a yellow chevron and a red triangle respectively. Non-prime 6 is represented by a yellow chevron and a red triangle to show that it is made up of the prime factors 2 and 3. The 72 individual canvases can be arranged in many different grid permutations, and each time they will show a new pattern of icons. There are two arrangements shown here. The first example has eight rows of nine numbers. The top row shows numbers 1–9; the second row 10–18; and so on up to 72. The
作为一名抽象艺术家，我对文艺复兴时期艺术家和数学家皮耶罗·德拉·弗朗西斯卡(Piero Della Francesca)的作品有着极大的热爱，我现在正在探索与素数级数合作的视觉可能性。德拉·弗朗西斯卡的作品在他对自然主义色彩的敏锐观察和对三维空间的理解方面具有开创性。他运用几何学和透视法来精确地构建他的绘画空间及其所包含的对象。然而，观众不需要理解数学来欣赏他的艺术之美。我用质数来构建我的图像空间;创造视觉刺激的艺术，探索模式，螺旋，序列和区域。除了运用数学概念外，对我来说，重要的是我的作品反映了我自己正在发展的视觉感受。数学元素是我创作过程中的跳板:发现和操纵固有模式帮助我找到视觉解决方案和结果，无论观众是否意识到潜在的数学。这样，我相信我的工作对人们有影响。我对质数的第一次探索，名为埃拉托色尼，是一个6宽9深的彩色矩形和圆圈网格。它由质数推导而来，用浓重的画法绘制，展示了超过2和3的质数如何只能作为6的倍数加减1。我以古希腊数学家埃拉托色尼的名字命名，因为我后来发现这实际上是一个改良过的埃拉托色尼筛子。我的Prime Marks作品(图1)创作于2010年，有72幅(15× 15厘米)的独立画作，每幅画代表1到72之间的一个数字。每个素数都由一个独特的图标表示。非素数由这些素数图标的组合来表示，以显示它们的因数。例如，质数2和质数3分别用黄色的v形和红色的三角形表示。非质数6用黄色的v形和红色的三角形表示，表示它是由质数因子2和3组成的。这72张单独的画布可以排列成许多不同的网格排列，每次它们都会显示出新的图标模式。这里显示了两种安排。第一个例子有8行，每行9个数字。最上面一行是数字1-9;第二行10-18;以此类推，直到72。的
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1732805
Jean Marie Constant
Scientific inquiry and art are not mutually exclusive. Science is built on facts and based on knowledge, observation, and experiment. Art originates in imagination, experience, and feeling. Both tap distinct sources of information and creativity. Nonetheless, art combined with science greatly enrich the public discourse and Society itself. At the time I entered art school, mathematics was not supported in the curriculum, which made it challenging for us to develop the cognitive skills necessary to understand complex mathematical problems or the nascent computer technology. My interest at the time was already leaning toward the communication component of visual art. Semiotics and visual communication principles base their findings on proven, repeatable facts. Looking for a productive alternative to developmy skills in that direction, I started exploring on my own various principles of Euclidian and non-Euclidian geometry from a perspective highlighted by the Bauhaus in the 1920’s (Bauhaus Verbund Office, 2019) promoting a closer relationship between art, science and technology. Later in my career, I fortuitously met two mathematicians who altered deeply my perception and comprehending of this field of scientific investigation. The first one, Alex Bogomolny (2018), introduced me to the dynamic of mathematical reasoning, describing and solving in simple and clear terms a series of Sangaku Japanese problems from the Edo period I was studying for a design class. He encouraged me to explore the tablets’ unique geometry with my own vocabulary and colour cues to solve the problem. An approach that was so rewarding that I put it into practice in my design classes to enrich students’ appreciation of the connections between science and visual communication. Similarly, in the early aughts, Dr Richard Palais (2004) developed amathematical visualization program that introduced me to the notion of space curves, polyhedra, and surfaces in simple but striking visualizations. He encouraged me to share my results in privileged forums such as ISAMA and Bridges. Inspired by the work of Sequin, Fathauer, Kaplan among many, I started to convert abstract mathematical concepts into meaningful art statements, and doing so, expanded substantially the scope and depth of my research. The example below demonstrates how an inspirational series of lectures by Dr Sarhangi, Jablan, and Sazdanovic (2005) on colour-contrast modularity presented at several Bridges
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1734766
de-Wit Lee
I came to appreciate mathematics decades after my compulsory high-school math lessons were over. With a longer life experience, I came to see that math can be a fascinating and beautiful intellectual project and process for understanding the invisible foundations of everything in the universe. Aside from usingmath for practical daily applications, I understand the significance of mathematics on a purely intuitive level. Math is never the first thing that I think about when I make art, but it undergirds almost everything that I create. My work—in the form of paintings, drawings, site-specific installations, and public artworks—stems from patterns and traces of growth and transformation in the natural world and the built environment. As a child of a biologist, I grew up seeing electron micrographs and lab specimens, and much of my work refers obliquely to scientific images and ideas. It also reflectsmy long-term interest in the substance and subject of water and related themes, like fluid dynamics and features of watery environments. Through my art-making process, I interpret existing surfaces that record the effects of natural phenomena, employing photographs or drawn documents. From these sources, I develop works that aim to reveal and interpret the evidence of forces of nature on humanmade and natural structures. In my works, masses of lines evoke various influences: organic forms like plants, hair, muscles, and fungi; natural systems such as waves and wind currents; geological strata; topographical maps; and sound. These linear networks are often based on hand-drawn records of physical effects of nature in my immediate surroundings—like a bowed window frame, a sinking floor, or the decaying walls in my former studio. My process includes making tracings and rubbings of surfaces like wood grain, cracking plaster, corroding metal, and eroded stone. I think of these marks as the calligraphic signatures of quotidian natural effects and as interpretations of the material evidence of time. I also see structures and patterns of nature as very complex manifestations of mathematical formulae and processes, at scales both minute and vast. Throughmy work, I create intuitive interpretations of scientific data and evidence—and, by extension, of mathematical truths. By making works that respond to seemingly non-measurable phenomena like
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1737897
M. Reynolds
In January 2019, I turnedmy interests in art and geometry frommy work with incommensurable ratios and the grids they make to a theorem by Thales of Miletus. Thales observed that in a semi-circle, as AC in Figure 1a, any point, B, on the circumference of that semicircle, AC, when drawn to the ends of the diameter, AC, will always make a ninety-degree angle at that point, B. Because there are an infinite number of points on the circumference of the semi-circle, an infinite number of right triangles can be generated, as in Figure 1b. In Figure 1c, it follows then that in a complete circle, any right triangle, AKM, by rotating it a half-turn about the centre, O, of this circle, will produce a rectangle, AKMZ. This diagram shows one easy way to achieve this rotation a halfturn: draw a line from point K through the centre of the circle, O, to R. Because of the infinite number of points on the circle, an infinite number of rectangles – all the rectangles of the world in fact – can be generated. The result of my studies is a new series of drawings and watercolours entitled, ‘Thales Series: All the Rectangles of the World’ (ATROTW for convenience). When I began my series, I realized that any and all rectangles I drew using thismethod have three common features: (a) They share a common diagonal; (b) This diagonal is equal to the diameter of the generating circle; and, (c) This diagonal is also the hypotenuse of a right triangle. While other construction methods can produce axially-aligned rectangles as well as radial, rotational, and reflection symmetries in the circle, my interests so far have centred on generating specific ratios and families of rectangles into the circle using these three features of Thales’ theorem. I continue to work with the diagonal/diameter/hypotenuse relationship because I like the dynamic and unique appearance of the artworks. I also like the challenges and aesthetic considerations presented in the Thales construction.
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1737899
C. Séquin
I am a very visually oriented person. Geometry has been in my blood since high-school, when I was given a copy of Hermann Weyl’s book “Symmetry” [12]. Formal mathematical proofs do not appeal to me until I can get some visualization model that supports that proof on an intuitive level. This is why I have been captivated by topics such as regular maps, graph embeddings, and mathematical knots. Corresponding visualization models have led to geometrical sculptures that convey an aesthetic message even to people who do not know the underlying mathematics. Conversely, abstract geometrical artwork by artists such as Brent Collins and Charles O. Perry have prompted me to discover the underlying mathematical principles and capture them in computer programs, which then produce more sculptures of the same kind.
我是一个非常注重视觉的人。高中时，我得到一本赫尔曼·魏尔(Hermann Weyl)的书《对称》(Symmetry)[12]，从那时起，我就对几何产生了兴趣。正式的数学证明对我没有吸引力，除非我能得到一些直观的模型来支持这些证明。这就是为什么我对规则地图、图形嵌入和数学结等主题着迷的原因。相应的可视化模型产生了几何雕塑，即使对不了解底层数学的人也能传达美学信息。相反，布伦特·柯林斯(Brent Collins)和查尔斯·o·佩里(Charles O. Perry)等艺术家的抽象几何艺术作品促使我发现了其中潜在的数学原理，并将它们捕捉到电脑程序中，然后再用电脑程序制作出更多同类的雕塑。
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1730547
G. Greenfield
My interest in algorithmic, generative and evolutionary art stems from my exposure to artists featured in the SIGGRAPH art exhibitions of the early 1980s such as Roman Verostko, Hans Dehlinger, Yoichiro Kawaguchi, Mark Wilson, Jean-Pierre Hébert, Karl Sims, andWilliam Latham to name just a few. My own computer generated artworks arise from visualizations of mathematical, physical or biological processes. My objective is to draw the viewer’s attention to the complexity and intricacy underlying such processes. Previously, in this journal, I have written about minimalist art derived from maximal planar graphs (Greenfield, 2008). Elsewhere, I have written about various generative art projects using cellular automata (Greenfield, 2016, 2018, 2019). Here, I will provide details about an artwork from a recent project onminimalist art derived from the so-called ‘rotor router’ model used for simulating deterministic random walks in the plane (Doerr & Friedrich, 2009; Holroyd & Propp, 2010). I first became aware of this model thanks to an archiv preprint of Neumann, Neumann, and Friedrich (2019). Consider a 200 × 300 toroidal grid such that each cell has four rotors that advance independently. Assume the rotors have 8, 5, 4 and 4 segments numbered 1–8, 1–5, 1–4 and 1–4, respectively. For each cell, randomly initialize its rotor settings and colour the cell grey. Next, select four cells to receive ‘painting objects’. The painting objects have finite tapes over the alphabet (R)ight, (D)own, (L)eft, (U)p. There are purple, blue, green and orange objects with tapes of length 8, 5, 4 and 4 respectively. At each time step, those cells with objects assume the colour of the object, use the value of the appropriate rotor as an index for decidingwhere to send the object, and then advance the appropriate rotor. For example, using the randomly chosen cell positions (52,68), (32,222), (65, 71) and (32,246) plus the randomly generated tapes DLULL, DUURLDDR, RUDL and DULU for the purple, blue, green and orange objects, respectively, after 15,000 time steps the random walk painting on the left of Figure 1 is obtained. At first glance, it may not be clear that I have specified a two-dimensional cellular automaton. Space prohibits providing the formal details, but if one thinks about what is happening from the point of view of the cells this claim should seem plausible. The random painting on the left in Figure 1 was selected from an initial randomly generated population
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