Pub Date : 2020-05-19DOI: 10.1080/17513472.2020.1766348
J. M. Campbell
The development of effective ways of depicting the structure of finite graphs of large order forms a very active and dynamic area of research. In this article, we explore the use of colour parameterizations of the entries of graph distance matrices to depict the structure of graphs with vertex sets of large cardinality, producing many new works of mathematical art given by the application of colour processing functions on matrices of this form. Computer-generated works of art of this form often reveal interesting patterns concerning the corresponding graphs. GRAPHICAL ABSTRACT
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Pub Date : 2020-04-20DOI: 10.1080/17513472.2020.1751575
M. Khorami
ABSTRACT Space Harmony is a theory and practice that explores universal patterns of movement in nature and of man. It is studied by artists who are interested in understanding patterns of harmony and balance. Rudolf Laban created this theory and is credited with Laban Scales; these are series of movements in space that increase spatial awareness and a sense of balance in the body. Knot Theory is a branch of Topology that studies mathematical knots. In this paper, we explore the relationship between these two seemingly unrelated fields and demonstrate some of the contributions that they make to one another. More specifically, we introduce the notion of Harmonic Embeddings as a generalization of Laban scales. This gives us an interesting mathematical context to study scales and Space Harmony in general. GRAPHICAL ABSTRACT
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1729302
J. Growney
Electronics play the central role in implementing new features in mobile machines, raising their performance, flexibility and reliability. A current trend is the connection of on-board electronics to the outer world. The advantages range from preventive maintenance and optimising the operation modes to implementing machine learning and autonomous operation
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1734518
Martin P. Levin
In 1970 Dr. Georg Unger (1909–1999) of Dornach, Switzerland, suggested to me the Platonic solids as a topic worthy of contemplation. I was both startled and intrigued to hear a trained mathematician speak thus about such an elementary topic in mathematics. Soon thereafter I read George Adams’ Physical and Ethereal Space (Adams, 1965), with its imaginations of geometric forms created by lines and planes coming in from the infinitely distant plane, and I also came across L. Gordon Plummer’sMathematics of the Cosmic Mind (GordonPlummer, 1970); the theosophical symbolism seemed tome a bit contrived, but I found the drawings of nested Platonic solids to show some wonderful and surprising geometry. Beautiful geometric forms were swimming in my mind. I wanted to make some models to show to my students, but how could I make them physically, so they would actually hold together. Moreover, I wanted little material and clean lines that emphasize the geometric forms, so the viewer is stimulated to inwardly imagine the pure geometric forms; it is that inner activity that engages one and makes the subject meaningful. So, how to make them physically? After some trial and error, I eventually settled on metal tubes connected with bent wires and glue, with more tubes suspended inside on taut wires. When teaching projective geometry and group theory, the models captivate students’ attention and make the concepts very accessible, and in art galleries the viewers gaze with interest and wonder. Looking from different perspectives shows the viewer striking patterns. For that to work, however, precision is needed to make all of the tubes and wires line up exactly. The compound of five regular tetrahedra is very well known, models typically made in cardboard with solid faces. Figure 1 is a photo of this form cast in bronze. Figure 2 shows the same five tetrahedra, with brass tubes for edges. Figure 1 has solid faces. It adorns our home garden where the aging natural patina and the shifting daylight create varied and beautiful effects. However, due to the opaqueness of the solid faces, one cannot see a whole tetrahedron from any one perspective, and one cannot see at all the shape of the inner core. Figure 2 is actually a tensegrity figure. The 5 tetrahedra do not touch one another, but are suspended from one another with wires that form the edges of a dodecahedron on the outside. Moreover, aluminium tubes, suspended on diagonal strings, form an icosahedron
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1734767
C. Bowen
I decided to studymeteorology after a frightening encounter with a severe thunderstorm in the spring of 2011. I declared a physics major the next fall, despite never having taken precalculus. Along the way I fell in love with math and physics in their own right. As a visual learner, much of my understanding came through deriving concepts by drawing, but my frustrations with the loss of information inherent to presenting 3D concepts through 2D means led me to first investigate the use of sculpture as a way of solidifying my understanding. After encountering Oliver Byrne’s 1849 illustrated version of Euclid’s Elements, I realized it was possible to create visual aids that simultaneously have enough pedagogical value for use in a classroom and enough artistic merit that they would not look out of place in the living room of someone with no mathematical inclinations. This led to my current workwhich focuses on the use of cheap and easily availablematerials to create beautiful but practical visualizations that serve as concrete, tangible illustrations of otherwise abstract, cerebral concepts in analysis and mathematical physics. Robert Sabuda’s elaborateWizard of Oz pop-up book was hugely inspirational as a poor student: the idea that such dynamic 3D illustrations could be created with a material as cheap, widely available, and humble as paper was powerful to me. While still in school, I began dabbling in paper engineering, excited I had found a sculptural media that I could easily afford. But even after I was no longer constrained by financial necessity, my fixation on cheapmaterials remained because, in imposing such restrictions onmyself, I was giving myself creative challenges that forced me to find novel solutions, some of which required the invention of entirely new sculptural techniques. Since graduating with a double major inmath and academic physics and a studio artminor inDecember 2016,mymaterial repertoire has grown to include plastic beads, embroidery floss, 3D printed PLA, clear plastic cocktail straws, motherboard washers, acrylic rod, Copic alcohol ink markers, their refill inks straight from the bottle, and Mylar plastic film. The combination of alcohol ink and Mylar has especially captured my imagination, and I have made it something of a mission to see howmany different topics in math and physics I can illustrate using the two. Among my favourite pieces borne out of this endeavour is this hanging mobile featuring six of the atomic orbitals of hydrogen (Figure 1), created using a sculptural technique of my own invention.
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1729059
Melissa Fleming
Nature has always inspired my work. Long intrigued by its processes, most of my art is an inquiry into the transient and often unseen aspects of the natural world. An interdisciplinary education, including history, art, and science has influenced my way of seeing. It has taught me to look for interconnections between and across various fields of study. As a result, the influences on my work are diverse, incorporating ideas from the philosophical concept of the Sublime, art movements such as Romanticism and Abstraction, as well as modern environmental science and mathematics. Art and mathematics are seemingly unrelated areas of study, but share a common goal, which is to better understand and describe the world around us. While I locate my work mainly at the intersection of art and science, I have gained a deeper understanding of the natural world by learning more about various mathematical concepts and theories. Math, after all, is considered the ‘mother of all sciences’. Incorporating math and science into my artwork, I aim to inform people about the wonders and workings of nature and inspire new perspectives and understanding of the subject. Under Glass, my series of sculptural assemblages, highlights the many layers of complexity and almost continuous state of change present in the natural world. Attracted to these transient processes, our observations of them, and the ideas of nineteenth century citizen science, I collected natural objects and placed them under Victorian-style glass domes. Under glass, the objects are singled out for close examination and highlight the act of intense seeing (Tufte 2006) which is common to the practice of both art and science. Each seemingly simple object coupled with an engraved label on its dome seeks to explore the duality of perception and reality. One of the pieces in this series is titled Fibonacci Sequence (Figure 1). It consists of the cross-section of a nautilus shell with the first few numbers of the Fibonacci sequence – one of the world’s most famous mathematical formulas – engraved on its glass dome. Examples of the Fibonacci sequence and its associated ratio phi ( ), also known as the Golden Ratio, are found frequently in nature. It is seen, for example, in the spiral growth pattern of the scales of pinecones and the seeds of sunflowers. However, it is most famously associated with nautilus shells. Composed of chambered sections that provide buoyancy
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Pub Date : 2020-04-02DOI: 10.1080/17513472.2020.1734280
P. Stampfli
I have always been interested in geometry. When I was about 12 years old, my grandfather Oskar Stampfli, an excellent teacher and mathematician, showed me the tilings of the plane with regular polygons and that there are only five Platonic solids. The self-similar shapes of crystals and ferns were fascinating to me. Later, I discovered the geometric art of M. C. Escher as well as the concrete art of Max Bill and Verena Loewensberg. I admired their work. But then I was also disappointed, as the underlying geometrical ideas were too simple. On the other hand, I realized that these paintings and prints need a lot of work and talent. Yet, I was dreaming of creating images based on more sophisticated geometry and using less time-consuming manual labour. Meanwhile, Penrose and others discovered quasiperiodic tilings and Benoit Mandelbrotmade self-similar fractal structures popular. At the university, I studied non-Euclidean geometry. All these ideas are inspirations for mathematical art that goes beyond periodic ornaments. However, doing the images by hand takes a lot of time and is not accurate enough. Then came powerful personal computers. They can rapidly generate complicated images and allow extensive explorations. It is now possible to zoomnearly without end into details of an image. Thus, I can now realize my dreams. Usually, I beginwith some vague questions:Howcan I decorate a tilingwith fragments of a photo, such that they fit the symmetry of the tiling and that the resulting image appears to be continuous? Is there an iterative procedure tomake an image that resembles a snowflake? What happens to an image after multiple reflection in distorting mirrors? I prefer to map photos onto geometrical structures rather than doing abstract visualization. This makes more natural looking images and recognizable real world details make a surreal effect. For periodic and quasiperiodic tilings of the Euclidean plane, I use reflection at straight lines and thus I can directly make a collage of small pieces of a photo. Others are doing similar work. Frank Farris uses wave functions instead of mirrors for mapping photos, as you can see in his book ‘Creating Symmetry’ (Farris, 2015). Thus, he creates images for all wallpaper groups of the plane. With mirrors, I can only make a small subset. However, with distorting mirrors, such as inversion in a circle, it is much easier to create kaleidoscopes that make hyperbolic and fractal images. These kaleidoscope distort the pieces of the photo depending on their place in the resulting image and its geometry. This is determined by a computer program. I can choose the size, orientation and position of
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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
{"title":"Prime concerns: painting number patterns","authors":"P. Ashwell","doi":"10.1080/17513472.2020.1734427","DOIUrl":"https://doi.org/10.1080/17513472.2020.1734427","url":null,"abstract":"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","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79331147","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: 10.1080/17513472.2020.1734517
Anduriel Widmark
Diving deep into the patterns that make up the world is a fun way to reveal new perspectives. Paintings and sculptures create an opportunity where I can freely experiment and explore the universe. Mathematics provides an infinite realm of inspiration and helps give structure to my research. Abstraction expands on reality and presents a chance to look outside of a regular pattern of seeing. Playing with art and math often leads to unexpected questions and discoveries. For the past several years, I have been developing sculptures with symmetric arrangements of congruent cylinder packings restricted to only three and four directions. Hexastix and Tetrastix are periodic non-intersecting arrangements of cylinder packings that are of particular interest to me. Packing problems are an important class of optimization problems that have a visually rich history in mathematics. These homogeneous rod packings have been described by Conway in The symmetries of things (Conway, Burguel, & Goodman-Strauss, 2008) and by O’Keeffe in The invariant cubic rod packings (O’Keeffe, Plevert, Teshima,Watanabe, & Ogama, 2001). The structures described can be built easily with a little patience and present fairly stable configurations that naturally have some rigidity when compressed. Finite groupings of these cylinder packings can be joined in various ways to produce some interesting nets, helices, and polyhedrons. The large variety of options for the shape, configuration, and colouration of these structures provides ample space for artistic creativity. Finding ways to classify and develop new cylinder arrangements starts with sketching patterns of intersecting hexagonal prisms on paper. After some basic symmetry is worked out, I build a small series of models using an inexpensive material, mainly toothpicks or pencils. I develop the most appealing of these models further with diagrams that symmetrically connect the ends of the rods to create knots. Themodels and diagrams are then used to guide the creation of larger sculptures made out of glass. Straight, clear rods of borosilicate glass are cut to shorter segments before being organized using clamps and string to replicate themodel’s geometry. I use a propane andoxygen torch to melt the ends together in an orderly way. Using a flame that is over 2000 degrees
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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.
{"title":"Mathematics as a window into the art of design and form","authors":"Loren Eiferman","doi":"10.1080/17513472.2020.1734439","DOIUrl":"https://doi.org/10.1080/17513472.2020.1734439","url":null,"abstract":"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.","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85514088","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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