Pub Date : 2020-06-26DOI: 10.1080/17513472.2020.1734270
Alejandro Robles
Months ago I was willing to start a new artistic project with a new thematic. But, affected by the digital era, I foundmyself suspicious about everything. The media we consume, our relationships with others or ourselves . . . It seems like fiction has surpassed reality. Thus, I had to get down to the basics: our planet is real, and so the physics and the time within it. And we humans have created a language in order to understand all of it. That is how looking for something authentic I immersed myself in the world of mathematics. One day, after a conversation about prime numbers with my parents (both mathematicians), I felt an impulse, the need of making a drawing. What would be the result of studying these numbers from an artistic perspective? I drew the numbers doing a spiral form, remarking the prime numbers in a different color. The structure was incredible, the patterns seemed to appear and disappear. It looked like order was fighting against chaos. I started to search for new studies about prime numbers. Within that investigation, I found connections and different points of view. And also that the amazing structure I drew was no original. A famous mathematician Stanislaw Ulam had already drawn the same spiral in 1963 (Weisstein). The discovery upset me for a while, but I kept working and experimenting. From the artistic field I happily broke again into Esther Ferrer’s work. Years ago I had been amazed by the similarity of Perfiles (Ferrer, 1982) and my previous project (Un)conscious Moments (Ontiveros Robles). It was an amazing feeling to now meeting her Prime Number Poems. It was like walking a path that she had already walked.
{"title":"A new visual order for prime numbers","authors":"Alejandro Robles","doi":"10.1080/17513472.2020.1734270","DOIUrl":"https://doi.org/10.1080/17513472.2020.1734270","url":null,"abstract":"Months ago I was willing to start a new artistic project with a new thematic. But, affected by the digital era, I foundmyself suspicious about everything. The media we consume, our relationships with others or ourselves . . . It seems like fiction has surpassed reality. Thus, I had to get down to the basics: our planet is real, and so the physics and the time within it. And we humans have created a language in order to understand all of it. That is how looking for something authentic I immersed myself in the world of mathematics. One day, after a conversation about prime numbers with my parents (both mathematicians), I felt an impulse, the need of making a drawing. What would be the result of studying these numbers from an artistic perspective? I drew the numbers doing a spiral form, remarking the prime numbers in a different color. The structure was incredible, the patterns seemed to appear and disappear. It looked like order was fighting against chaos. I started to search for new studies about prime numbers. Within that investigation, I found connections and different points of view. And also that the amazing structure I drew was no original. A famous mathematician Stanislaw Ulam had already drawn the same spiral in 1963 (Weisstein). The discovery upset me for a while, but I kept working and experimenting. From the artistic field I happily broke again into Esther Ferrer’s work. Years ago I had been amazed by the similarity of Perfiles (Ferrer, 1982) and my previous project (Un)conscious Moments (Ontiveros Robles). It was an amazing feeling to now meeting her Prime Number Poems. It was like walking a path that she had already walked.","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"24 1","pages":"113-115"},"PeriodicalIF":0.2,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81848110","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 : 2020-05-22DOI: 10.1080/17513472.2020.1766340
M. L. Spreafico, E. Tramuns
We illustrate a project in art, maths, and origami, in the spirit of STEAM, carried out in an Italian high school. We chose the famous Van Gogh's painting: ‘The Starry Night’ and we invited 16-year-old students to cover some elements on the artwork with origami models. These models were related to engineering, architecture, design and art and, for each of them, we designed a lesson on a precise mathematical subject. In this paper, we give the details of the project and we sketch the mathematical lessons we did, giving also the instructions to fold the models. We have also analysed the answers of a questionnaire filled in by students, to check the adequacy and effectiveness of the experience. The results showed that the students welcomed this project, and improved their maths knowledge. GRAPHICAL ABSTRACT
{"title":"The Starry Night among art, maths, and origami","authors":"M. L. Spreafico, E. Tramuns","doi":"10.1080/17513472.2020.1766340","DOIUrl":"https://doi.org/10.1080/17513472.2020.1766340","url":null,"abstract":"We illustrate a project in art, maths, and origami, in the spirit of STEAM, carried out in an Italian high school. We chose the famous Van Gogh's painting: ‘The Starry Night’ and we invited 16-year-old students to cover some elements on the artwork with origami models. These models were related to engineering, architecture, design and art and, for each of them, we designed a lesson on a precise mathematical subject. In this paper, we give the details of the project and we sketch the mathematical lessons we did, giving also the instructions to fold the models. We have also analysed the answers of a questionnaire filled in by students, to check the adequacy and effectiveness of the experience. The results showed that the students welcomed this project, and improved their maths knowledge. GRAPHICAL ABSTRACT","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"1 1","pages":"1 - 18"},"PeriodicalIF":0.2,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82566867","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 : 2020-05-19DOI: 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
{"title":"On the visualization of large-order graph distance matrices","authors":"J. M. Campbell","doi":"10.1080/17513472.2020.1766348","DOIUrl":"https://doi.org/10.1080/17513472.2020.1766348","url":null,"abstract":"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","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"19 1","pages":"297 - 330"},"PeriodicalIF":0.2,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83657068","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 : 2020-04-20DOI: 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
{"title":"Space harmony: a knot theory perspective on the work of Rudolf Laban","authors":"M. Khorami","doi":"10.1080/17513472.2020.1751575","DOIUrl":"https://doi.org/10.1080/17513472.2020.1751575","url":null,"abstract":"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","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"72 1","pages":"239 - 257"},"PeriodicalIF":0.2,"publicationDate":"2020-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78096270","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: 10.1080/17513472.2020.1734280
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: 10.1080/17513472.2020.1734767
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: 10.1080/17513472.2020.1729302
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
{"title":"Everything connects","authors":"J. Growney","doi":"10.1080/17513472.2020.1729302","DOIUrl":"https://doi.org/10.1080/17513472.2020.1729302","url":null,"abstract":"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","PeriodicalId":42612,"journal":{"name":"Journal of Mathematics and the Arts","volume":"13 1","pages":"66 - 68"},"PeriodicalIF":0.2,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81690307","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: 10.1080/17513472.2020.1729059
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: 10.1080/17513472.2020.1734518
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: 10.1080/17513472.2020.1734517
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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