# Mathematical Imagery

Mathematical artists create strong, stunning works in all media and explore the visualization of mathematics

# 2010 Mathematical Art Exhibition

The Mathematical Art Exhibition held at the 2010 Joint Mathematics Meetings in San Francisco, included works in various media by 64 artists. Robert Fathauer was the curator of the exhibition, and the exhibition website was prepared by Anne Burns. The exhibition was juried by Fathauer and Burns, along with Nat Friedman and Reza Sarhangi. The 2010 Mathematical Art Exhibition Prizes were awarded. Four judges, selected by the American Mathematical Society and the Mathematical Association of America, made the following awards: First Prize to Robert Bosch, for his work, "Embrace"; Second Prize to Harry Benke for "The Vase"; and Third Prize to Richard Werner for "Meditations". The Prize "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form.

Acrylic on Wood, 5" x 14.5", 2009. Two steganographic codes, one ultilising a sculptural and one a painterly ciphertext, create a three way harmony with the encrypted data. Expressing code not solely as something visual, but also something tactile. My avenue of investigation is Steganography and the place of Cryptography in our society. Both Computers and Art are incredibly common in our culture and yet both are incredibly overwhelming to many of the people who see them daily. Thus, stenographic painting seems the aesthetic equivalent to 'https'. To that extent, the focus has been on devising encoding systems that utilize color and orientation, and then finessing them to make them sing together. --- Edward Alonzo

Knitted bamboo yarn (Southwest Trading Company Twize, in colors twurple and twocean (seriously)), 6.5" x 14.5" x 3", 2009. A $(p,q)$ torus knot traverses the meridian cycle of a torus $p$ times and the longitudinal cycle $q$ times. Exhibited here are the two chiral versions of the $(3,2)$ torus knot, knitted into their embedding tori. One can represent a $(p,q)$ torus knot on the standard flat torus by drawing a line of slope $q/p$. The designer of a knit torus must contend with thickening the line to make it visible (and appear continuous), compensating for the curvature of the spatially embedded torus, and discretizing the result onto the non-square grid formed by knit stitches. --- sarah-marie belcastro

**2010 Mathematical Art Exhibition - Second Prize**

Glicee Print. 18" x 14.8", 2009. "The Vase" is composed of a digitally modeled vase with "Lilies" which are Dini's Surfaces. A surface of constant negative curvature obtained by twisting a pseudosphere is known as Dini's Surface. Imagine cutting the pseudosphere along one of the meridians and physically twisting it. Its parametric equations are: $$x=a\cos(u)\sin(v);$$ $$y=a\sin(u)\sin (v);$$ and $$z=a{\cos(v)+\ln[\tan(v/2)]}+bu,$$ where $0 \leq u \leq 2\pi$ and $0 < v < \pi$. Take $a=1$ and $b=0.2$. "I'm primarily an artist. My shadow is mathematics. I'm helpless at preventing mathematics from intruding in my work and it's delightful to have the body of mathematics to work with. My art attempts to produce a nexus between mathematical beauty and the beauty of the natural world to produce a satisfying aesthetic experience. --- Harry Benke *

Digital print of 3D model on photographic paper, 23" x 19", 2009. Serial coordinate transformations interleaving symmetry preserving and symmetry breaking operations yield a stunning variety of forms. A sequence of fourteen such operations in 3D create this shape from the unit square. Small spheres are initially an array of 40000 normals to the surface at a distance of 0.05. Larger spheres represent hierarchical clustering centroids of the normals in their final positions. Sequences are selected with a genetic recombination function using esthetic appeal as the fitness function. This transformation sequence begins with a cylindrical transform around the z-axis, and finishes with a spherical coordinate transform and rotation around the y-axis. These images are from several large series exploring the creation of complex forms through sequences of simple operations or representations of simple relationships. --- Leo S. Bleicher

**2010 Mathematical Art Exhibition - First Prize**

Stainless steel and brass, Diameter = 6 inches, thickness = 0.25 inches, 2009. I began by converting a drawing of a two-component link into a symmetric collection of points. By treating the points as the cities of a Traveling Salesman Problem and adding constraints that forced the salesman's tour to be symmetric, I constructed a symmetric simple-closed curve that divides the plane into two pieces: inside and outside. With a water jet cutter, I cut along this Jordan curve through quarter-inch thick, six-inch diameter disks of steel and brass. By swapping inside pieces I obtained two copies of the sculpture. Here, steel is inside and brass is outside. Here's how I work: After I get an idea for a piece, I translate the idea into a mathematical optimization problem. I then solve the problem, render the solution, and see if I'm pleased with the result. If I am, I stop. If not, I revise the mathematical optimization problem, solve it, render its solution, and examine it. I do this out of a love of mathematical optimization--the theory, the algorithms, the numerous applications. --- Robert Bosch

Digital print, 20" x 24", 2009. Deterministic 3D strange attractor built with the dynamical system: $$dx/dt = 0.02 y + 0.4 x ( 0.2 - y^2 )\; (1)$$ $$dy/dt = - x + 35 z \;(2)$$ $$dz/dt = 10 x - 0.1 y \;(3)$$ Initial condition $(x_0, y_0, z_0 ) = ( 0, 0.01, 0 )$, fifth-order Runge Kutta method of integration, and accuracy = 10^{-5}. Euclidean coordinates representation : $( y, - x, z)$. I have always been fascinated by the Lorenz Attractor. I like to create and simulate systems of ordinary differential equations on my computer. A simple raylight formed by a 3D model follows intricate dynamics. Visualizing an infinite trajectory drawing elegant attractors within a limited phase of space unravels the aesthetics appeal of the Deterministic Theory of Chaos. I think discovering unexpected strange attractors by the exploration of 3D dynamical models constitutes a full artistic principle. By unconventional ways, I search beauty. --- Safieddine Bouali

Stainless steel and bronze. Direct metal print, 4" x 4"x 4", 2008. The sculpture is inspired by 3 dimensional origami construction. 30 identical ribbons bent around the surface of a cylinder are joined together to form the shape with the rotational symmetry of an icosahedron. My artistic passions are purely mathematical images and sculptures, which express a certain vision of forms and shapes, my interpretations of distance, transformations and space. In my opinion, mathematics is not simply a profession, but rather a way of thinking, a way of life. My images and sculptures are like photographs of interesting mathematical ideas. I have always been intrigued by the possibility of showing the intrinsic richness of the mathematical world, whose charm and harmony can really be appreciated by everyone. --- Vladimir Bulatov

Glass, 7.5" x 7.5", 2009. This tessellation is the traditional coloring and layout of the Churn Dash pattern. The Churn Dash quilt block is an interesting pattern that is composed of right triangles, rectangles, and a single square. This symmetrical design is based on a square and can be recolored in this manner to produce a tessellating motif. Geometric design and especially geometric patterns always intrigued me. I was excited by seeing how patterns changed as various symmetrical arrangements were applied to them. My pieces are created by layering and melting together various shapes and colours of glass in a fusing kiln, which heats the glass to 1500 degrees Fahrenheit. --- Galina Bulatova

Digital print, 19" X 13", 2009. The subgroup of Möbius Transformations that maps the unit circle onto itself is composed with Möbius Transformations that map the unit circle into itself to produce overlapping circles. The program was written in Actionscript. I am fascinated by art, mathematics and nature and try to combine all three in my work. --- Anne Burns

Digital print by PowerPoint, 17" x 22", 2008. Complete r-partite graph is the graph with vertices set consisting of r disjoint sets such that any two vertices in different sets are connected by an edge and not for vertices in the same set. The work is a complete bipartite graph, following by a rotation on each line segments. One part of vertices is positioned on two adjacent line segments with equal distance; another part of vertices is positioned on an oval. There are 27 vertices on one part and 24 vertices on the oval. Based on Structural Cloning Method (SCM) we can explore symmetry patterns and fractal patterns in different ways, and so math art becomes interesting. --- Mingjang Chen

Painted High Carbon Steel, Impala Black Granite, Height 18" Width 16" , Depth 16", 2009. Fusing math, art and aesthetics, Elements incorporates mathematically significant dimensions that add an intriguing subtlety to its construction. Euclid studied the Golden Ratio 1 to 1.618 (Greek letter Phi) for its many interesting properties as described in his manuscript Elements. The three intersecting planes that comprise Element's core are Golden Rectangles. Their intersection creates 20 equilateral triangles, drawn from their points--an Icosahedron. Further, these rectangles use dimensions from the Fibonacci Sequence providing for a variety of mathematical implications. --- Jeff Chyatte

Elephant hide paper, 9" x 9" x 9", 2009. The sculpture is a modular combination of three interacting pieces. Each piece is folded by hand from a circle of paper, using a compass to score the creases and cut out a central hole.This transformation of flat paper into swirling surfaces creates sculpture that feels alive. Paper folds itself into a natural equilibrium form depending on its creases. These equilibria are poorly understood, especially for curved creases. We are exploring what shapes are possible in this genre of self-folding origami, with applications to deployable structures, manufacturing, and self-assembly. We explore many mediums, from sculpture to performance art, video, and magic. In our artwork we look for epiphanies, challenges, and often connections and understanding to help solve problems in mathematics. --- Erik Demaine

Digital print, 5" x 7", 2009. This arrangement was inspired by the following result from the theory of binary space partitions in computational geometry: There exists a set S of n disjoint axis-parallel line segments with the property that any axis-parallel binary space partition of S has size at least $2n-O(n^{2/3})$. The construction is based on a shifted double grid made of disjoint segments. Art could come from anywhere. One just wants to be careful and not overlook it. --- Adrian Dumitrescu

Color print,11" x 11", 2007. This pattern contains lizards, fish, and bats representing the three classical elements, earth, water, and air. The pattern is inspired by M.C. Escher's Notebook Drawing Number 85. In this hyperbolic pattern, four blue lizards meet head-to-head, five red fish meet head-to-head, and three yellow bats meet head-to-head, unlike Escher's pattern in which three of each animal meet head-to-head. The symmetry group of this pattern is generated by reflections across the lines of bilateral symmetry of each of the animals; its symmetry group is the hyperbolic kaleidoscope group *543, in orbifold notation. Most of the patterns I create exhibit characteristics of Escher's patterns: they tile the plane without gaps or overlaps, and if colored, they are colored symmetrically and adhere to the map-coloring principle that adjacent copies of the motif are different colors. --- Doug Dunham

Photographic Paper, 20" X 20" , 2009. Julia sets are usually depicted two-dimensionally, either flat or as textures on other surfaces which themselves may have little to do with the Julia set. Here, we iterate the complex variable relation,

new $s = s^2 - 1.25$ thirteen times to produce a polynomial in the original variable, $s$, of degree 8192. I am interested in applying computer graphical techniques to illuminate mathematical processes. I wrote the code to first principles in the 'C' programming language. This particular image was constructed as a particle system made from 266 billion points and took 67 hours to compute. --- Jeffrey Stewart Ely

Digital print, 13" x 16", 2009. A starting knot was created that possessed sufficient geometric regularity to allow iterative replacement of a portion of the knot with a scaled down copy of the knot. Three such iterations were carried out to obtain the knot shown here. In addition, the path of the strands was smoothed out so that strand in the final knot curves gracefully, as opposed to being a series of straight line segments that change angle abruptly. The knot was constructed using the program KnotPlot and then exported to PhotoShop for touching up. --- Robert Fathauer

3/8" copper tubing, wood dowel, gorilla glue, 13" x 10" x 9", 2008. Copper tubing comes in a coil so is naturally 'rounded'. A knot diagram is coiled if the diagram can be traced in a constant clockwise direction. A coiled diagram is also referred to as being in a braid configuration. The typical diagram of a figure eight knot in a knot table is not coiled. The diagram of the figure eight knot shown above is coiled. A coiled diagram is suitable for forming a corresponding copper tubing knot since the natural rounded curvature of the tubing can be maintained as the knot is formed. I form copper tubing knots by hand. One must be careful not to "crink" the tubing. The ends are joined using a wooden dowel and gorilla glue. Knots are ideal mathematical forms for sculptures since a knot is completely three-dimensional with no preferred top, bottom, front, or back and can look completely different from different viewpoints. --- Nat Friedman

Digital art print, 21" x 24", 2009. This complex 3D object consists of four identical but differently colored separated tape-like shapes that have been interwoven with each other. The result of this arrangement is a six-pointed shape with some kind of unusual symmetry. Each of four separated constitutive parts and also the shapes at all six arms of the final complete form are exactly the same, but their different arrangement is the reason for the final unusual symmetry. Software considerably ease the imagination of complex mathematical shapes, and playing with them can be done without any limitation other than the mind. I create my basic shapes in software like Topmod and do arrangements, adjustments, texturings, and final renders as final realistic sculptures in render engines like Modo. --- Mehrdad Garousi

Original digitized machine stitched patterns on cotton reinforced by Timtex, Five moveable pieces, collapsible each 3" x 3" x 3", 2009. Conway enumerates the 7 spherical symmetries compatible with the uniform polyhedra in 'The Symmetries of Things.' Using the symmetry types these are 332, *332, 432, 3*2, *432, 532 and *532. The simple cuboctahedron exhibits the first 5 of the symmetry patterns: *432 has 48 symmetries (the full group of symmetries), *332, 432 and 3*2 have 24 (the three subgroups of index 2 = 48/24) while 332 has only 12 (the ones of index 4 = 48/12). Coloring the faces of the models for the Archimedean solids is a natural extension of my recent work with pop-up polyhedra. My mathematical art grows out of my experiences with my students and my explorations of mathematics, textiles, paper, and technology. --- S. Louise Gould

Digital Print, 10" x 10", unframed, 2009. This op art sequence shows visualizations from a simulation based on cellular morphogenesis. Concentrations of three of four cell transcription factors are interpreted as RGB values. These cell substances diffuse, interact, and express and inhibit the genes within cells that are responsible for producing such substances. Here there are two types of cells and the image was captured after the gene activation equation had been evaluated 400 times. Many of my computer generated algorithmic art works are based on simulations that are inspired by mathematical models of physical and biological processes. --- Gary R. Greenfield

Eight 9" folded paper plates, 6" x 6" x 6", 2009. Eight 9" circles are folded to a 32-frequency diameter grid (102 creases in an equilateral triangular grid matrix.) Each circle is reconfigured to a bi-tetrahedron pattern with some variations in iterations to the folding algorithm, and joined in multiples forming eight corners of the cube. For the last twenty years I have been exploring folding the circle for information to understand what is revealed through a simple and principled process of systematic folding, reconfiguring and multiple joining circles. --- Bradford Hansen-Smith

Wood, 8" x 8" x 8", 2008. The form of Dragonflies consists of twelve congruent parts arranged with octahedral symmetry in a form based on the third stellation of the rhombic dodecahedron. The laser-cut wood components are each a subset of the complete face of the stellation, designed so that it does not intersect with the other eleven identical copies of itself. Interweaving and assembling the rigid physical parts was an interesting challenge. As a sculptor of constructive geometric forms, my work deals with patterns and relationships derived from classical ideals of balance and symmetry. Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic. --- George W. Hart

Single sheet of Canford paper, wet-folded, 9" x 9" x 9", 2006. A Hamilton cycle on the cube has eight edges. Therefore, a regular octagon could be folded to mimic the path such a cycle traces on the cube. This piece represents a solution using folded concentric octagons, producing the illusion (?) of negative curvature. The piece was folded from a large regular octagon, approximately two feet in diameter. Concentric octagons were pre-creased, alternating mountain and valley folds. Then the model was collapsed and wet-folded to hold the cube Hamilton cycle shape. Part of the charm of paper folding is its capacity for simple, elegant beauty as well as stunning complexity, all within the same set of constraints. This mirrors the appeal of mathematics quite well. The constraints that origami provides challenges the artist in a way similar to being challenged by a mathematical problem. --- Thomas C. Hull

Digital Print, 10" x 16", 2009. The boundary between land and sea is complex, like a fractal. At the bottom of this drawing we have land, represented by rigid lines and shapes. At the top, there is the sea, swirling around. In between is the boundary, where the right angles gradually give way to curves. There is self-similarity, as one would expect in a fractal. While I try to render mathematical ideas in my art, I also realize that the artistic process is itself a lot like the mathematical process. Sometimes the original 'problem' needs to be modified after careful 'research'. To me, the final product is a lot like a theorem. --- Karl Kattchee

Oil on linen, 23.6 x 23.6 inches, 2009. This painting is a partial projection of a wireframe deltoidal icositetrahedron. A deltoidal icositetrahedron dual polyhedron which looks a bit like an overinflated cube which has all of its edges bisected. The 24 faces are deltoids. The short and long edges of each kite are in the ratio 1.00:1.29. The deltoidal icositetrahedron is a crystal structure formed by the minerals analcime and garnet. After painting the projection of the Deltoidal Icositetrahedron wireframe on the canvas, I took the artistic freedom of creating new subfaces, occupying complementary locations by colouring up the spaces obtained by the intersections of the wireframe. I like to say that as an artist, I discover mathematics through art. --- Nuria Juncosa

Digital print, 20" x 24" , 2009. Two fractals are combined to mimic the shape of the jellyfish used to create this image. The "head" fractal uses the famous dragon curve iteration. Here, the first iteration maps the negative-sloped diagonal of the starting photograph to the lower edge, and also maps the same diagonal to the left edge. The "tail" fractal uses a double spiral iteration. The original photograph was taken at Monterey Bay Aquarium by the artist. I began iterating photographs into fractals after watching a video on fractals, in which the point was made that whether you started with a single segment or a 2-dimensional photograph, the resulting fractal was the same. I find that using photographs in the iterations can make the fractals much more captivating than if they were created with abstract geometry. I work intensely between Photoshop and The Geometer's Sketchpad to create these images. --- Kendra Lockman

Digital Print, 12 x 12 inches, 2009. "Gong Gan" employs a dodecahedron as its base form. I saw polyhedra as a form of tessellation on a sphere (spherical tessellation) and just by replacing each pentagon face on the dodecahedron with a module that tessellates within the pentagon and with the adjacent ones as well I was able to create this pleasant looking sculpture. I create most of my artwork using Autocad and touch up in Photoshop afterward. --- Ghee Beom Kim

One uncut square of paper, 8" by 8" by 8", 2009. This piece uses intersecting pleats to create tension within the folded sheet and encourage three-dimensionality. In addition to sharp points created by stretching pleats close to the four corners of the original sheet, it also features a joint where the centers of two opposite sides of the square are held together by a folded lock mechanism, creating the appearance of a non-simply connected surface. I fold (mostly flat and mostly paper) surfaces into interesting shapes. To do this, I use sequences of pleats to arrange layers so that they create tension that forces the material towards a curved surface. The mathematics show up in many ways, but the two of my favorite are the combinatorics in the arrangement of pleats and the mathematical physics in understanding the forms preferred by the paper when folded. --- Goran Konjevod

Digital print, 1995/2008. "Perspicuous" is a geometrical composition where a basic decagonal shape exhibits fivefold symmetry. The straight lines inside reveal the richness and interconnectedness of the geometrical shapes (pentagons, triangles, Penrose rhombs and kites), we can observe golden mean relations. [My] interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon...concentrate on melding art, science, mathematics and technology. --- Matjuska Teja Krasek

Digital print, 13" x 9", 2009. This plain-weaving model is automatically generated from the algorithm in the paper Cyclic plain-weaving on polygonal mesh surfaces with graph rotation systems. Cycles of the plain-woven object are created from any manifold-mesh surface by twisting every edge of the manifold mesh and convert the plain-weaving cycles to 3D thread structures. Users interactively control the shape of the threads and the size of gaps with a set of parameters creating a wide variety of unique plain-weaving patterns. This generated weaving model has 16 identical closed cycles with user control over the width, displacement, and curvature of the weaving yarns. This piece is rendered and produced through Maya and Photoshop. --- Christine Liu, Ergun Akleman, Qing Xing

Digital print, 15" x 7.5", 2007. All permutations (minus symmetrical redundancies) of alternating upward & downward semi-circular paths around & between 3 points. "I employ mathematically ordered shape relationships and carefully balanced color relationships in my compositions. These call for both objective logic and subjective perception in the decoding of each composition's organizing principles. My studio work is accomplished in both traditional painting media and digital prints. --- James Mai

Solid model of layered polymer resin created via stereolithography, 7" x 7" x 6", 2005. Costa's minimal surface is the first example of a complete, embedded minimal surface of finite total curvature to be discovered. This surface admits an explicit parameterization in terms of elliptic functions via the Weierstrass representation for minimal surfaces. The topology of the surface is that of a torus with three punctures, but its embedding is rather difficult to grasp visually from a typical graphical image. Hence we provide a rendering in the form of a solid model, the data for which was created with Mathematica. In addition to helping the viewer better grasp the underlying mathematics, the process of creating the work of art brings pleasure to the mathematical artist, who must be creative in his use of computational tools in order to achieve the desired outcome. --- O. Michael Melko

Photographic print, 17" wide x 17" high, 2007. In this image, I brought the notion of a Sudoku puzzle to a 4 x 4 grid, where I used shapes instead of the digits 1 - 4. I retained the requirement that each element of the four-character alphabet appear once and only once in each row, column, and in each of the four 2 x 2 sub-grids. In addition, I added an element of layering: Each finished image is a composition of four layers, with each layer being its own solved Sudoku grid. My work is composed primarily of computer generated, mathematically-inspired, abstract images. I draw from the areas of geometry, fractals, numerical analysis, and physics, and combine these ideas with image-processing technology. [M]athematics is, in many cases, a metaphor for the beauty and complexity in life. This is what I try to capture. --- Kerry Mitchell

Photograph, 11" x 17", 2009. The image is 12 pictures stitched into a 360 degree panorama, then a modified Rectangular to Polar transformation to produce the box. "Trained as a Computer Scientist (BCS), Work as an Engineer (MSc), and as a Professional Artist for 5 years. I create art because there is beauty in all things, and I enjoy exploring new techniques in camera and post processing. Getting people to view the world as a surprise is the fun part. --- Ralph Nevins

Archival Print, 15" x 15", 2009. The algorithm I used to make this image is based on the Hilbert space filling curve, discovered by mathematician David Hilbert. The image is drawn in a custom software program I wrote myself. This version of my program recursively subdivides spaces within the total space to be filled and runs the algorithm to fill the smaller spaces separately. Each smaller space is centered on a point on the larger curve causing the smaller renderings to intersect the larger one in interesting ways. I write software to make art. --- Don Relyea

Layered drawing: Ink drawings on mylar and paper,16" X 16", framed 20" x 20", 2009. My aim in this layered drawing is to explore geometric patterns and shapes with interweaving rhythms which change spatial locations as they weave in and out from concave to convex. My sculptures, paintings and drawings are metaphors for the complexity and underlying order reflected in many patterns in nature and the mathematical coherence found in natural forms. They represent invisible forces made visible. --- Irene Rousseau

Print of digital art, 24" x 20" (framed), 2008. This is the computed Fourier transform of a constant linear measure placed on a piecewise-linear approximation to the space-filling Sierpinski Curve. The curve itself is shown in the lower-left corner. The reduced art appears gray, but in the original each pixel has a hue determined by its complex phase. The transformation was computed by the Geometric Nonuniform Fast Fourier Transform. Over the course of earning my Ph.D., I've become fascinated by the fact that in generating images for the most utilitarian of purposes (debugging, testing hypotheses, and the like) the most useful images are usually also the most aesthetically pleasing. --- Ian Sammis

Digital print, 16" X 20", 2008. "Hope" is an artwork based on the "Modularity" concept using triangles and rhombuses as its motifs in three colors. The "Modularity" concept has been presented in "Modules and Modularity in Mosaic Patterns," Journal of the Symmetrion (Symmetry: Culture and Science), Volume 19, 2008, and "Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical, and Theoretical Considerations," by Sarhangi, R., S. Jablan, and R. Sazdanovic, Bridges Proceedings, 2004. In this figure, except for the corners with constant color, the two compound triangles (modules) are in a positive-negative color relationship with respect to each other. Using these two modules in a rotational fashion, results in the pattern in the artwork. I am interested in Persian geometric art and its historical methods of construction, which I explore using Geometer's Sketchpad and PaintShopPro. --- Reza Sarhangi (1952-2016)

Digital print, 20" x 20", 2009. The pattern consisting of triangles and circles introduced into the fundamental domain emphasizes four and six fold rotational symmetry of the (4,4,4,6) tessellation. The interplay of the white weave and the pattern reinforces the underlying structure. My inspiration stems from the rich geometric structures found in tessellations of the hyperbolic plane and my area of research- knot theory. Mathematical objects can be manipulated in many ways (superimposing, dualizing, breaking symmetry) to create aesthetically pleasing computer graphics brought to life by the unusual combination of colors. --- Radmila Sazdanovic

Digital Print, 24" x 15.6", 2009. This image consists of a large number of circles. To describe the set of circles let $[a,b,c,d]$ represent the circle whose points are the zeros of the bivariate polynomial $p(x,y)=a(x^2+y^2)+bx+cy+d$. If $a, b$, and $c$ are relatively prime integers then I call the circle a "reduced rational" circle. The drawing then consists of reduced rational circles such that $a^2+b^2+c^2 \leq 9^2$, as viewed through a rectangle whose lower left vertex is $(0.01,0.21667)$ and whose upper right vertex is $(0.395,0.46667)$. The darkness of each circle depends inversely on its radius and on the term $a^2+b^2+c^2$. Most of my mathematical art has its origins in images I make to help me understand the solution to some problem I am facing in my work as a computer scientist. I find myself interested in images that are a manifestation or rediscovery of the complexity that is inherent in even simple mathematical areas. --- Stephen Schiller

Plaster model, hand painted, 5" tall, 2004. The complete graph K12 connects 12 vertices with 66 lines. Of course, in the plane this graph cannot be drawn without many crossings. A surface needs to be at least of genus 6 to allow a crossing-free embedding. With this model I have attempted to find the embedding of highest possible symmetry. The model has the 12-fold symmetry of the oriented tetrahedron. The 44 countries bounded by the 66 lines, and colored differently, are all 3-sided. My professional work in computer graphics and geometric design has also provided a bridge to the world of art. My profession and my hobby interests merge seamlessly when I explore ever new realms of 'Artistic Geometry'. --- Carlo Séquin

Design drawing (not computer generated), 21" x 22" (framed), 2009.

Design drawing (not computer generated), 21" x 22" (framed), 2009.

Design drawing (not computer generated), 21" x 22" (framed), 2009.

Design drawing (not computer generated), 21" x 22" (framed), 2009.

Acrylic on Plexiglas, 36" x 48", 2008. "As a model, this painting represents a space time continuum, the singularity of beginning and end points are of an anomaly. In the painting I suggest the dynamics of a space that continually expands the structural space and geometrical fragmentation in space-time from a Big Bang. Line segments in the universe are interconnected, even when they appear to be separate from one another. In space-time curvature the geometrical strain illustrates a deformation of space. My art combines both ancient and modern mathematical foundations ranging from Pythagoras to Einstein. --- Clifford Singer (All Copyrights Reserved to Clifford Singer.)

The above ""Spirals in Chaos--The Dance in Four Movements: Design Drawings" are by artist Mickey Shaw. "Spirals are curves emanating from central points, progressively growing further away as they revolve around the point. These drawings are one of a kind renditions of spirals, but created in reverse direction from outer edges into a central point. Some variations resembling Sinusoidal, Archimedean and Hyperbolic spirals and even an occasional pseudospheres are created. Drawings are created on a weighted drawing board suspended from a pole with an attached arm holding a pen. The board is set in motion by hand. My inspirations are drawn from nature, mathematics and science. I believe that art is an intrinsic aspect of all visual experiences and mathematics can provide a basis for understanding and recreating those same experiences. The spiral design drawings convey a two-dimensional visualization and exploration of this interconnection. --- Mickey Shaw

[Above:] Bead-crochet (glass beads, thread) , 11.25" x 11.25", 2008-2009. A series of "map-coloring" bead-crochet bracelets. "The first three are examples of maps on the torus where each of seven ?countries? shares a border with all six others. Such patterns prove that at least seven colors are necessary for map coloring on the torus [Heawood]. The fourth bracelet design is an embedding of the complete graph on seven vertices [K7] on the torus. The artists wish to acknowledge the extraordinary seven-color torus designs by Norton Starr (painted hydrostone), Carolyn Yackel (crocheted yarn) and sarah-marie belcastro (knitted yarn), which inspired our development of these patterns in bead-crochet. "Bead-crochet bracelets are made by crocheting a strand of beads into a cylinder and sewing the ends together to form a torus. Visualizing finished designs from the linear strand or from 2-D patterns can be quite challenging. Our design explorations started with a desire to create novel patterns that went beyond those we found in books. Noting that bracelets are topological tori, Sophie and Ellie went hunting for mathematics to inspire new patterns and found Susan's seven-color tori website. Susan joined the quest to design the ideal 7-color torus bracelet, adding mathematical insight that gave rise to more perfect symmetry and better understanding of the relationships between designs. The four bracelets represent our collective steps in this process. As a set, they enhance our enjoyment of the beauty of the patterns, the pleasure of the craft, and the insights that come from the puzzle-solving design process. --- Susan Goldstine

Giclee digital print, 22" x 14", 2009. Ribbons of Rhythm (foreground image with detail behind) is an exploration of the aesthetic qualities of Penrose tiling. David Austin in "Penrose Tilings Tied up in Ribbons," describes the ribbons: "Opposite sides of a rhomb are parallel to one another. Therefore, if we begin with a rhomb and a pair of opposite sides, we may form a "ribbon" by adding the rhombs attached to that pair of opposite sides and then continuing outward." The print reveals only a single family of parallel ribbons in one orientation, however there are another four orientations associated with a five-fold Penrose tiling. --- Paul Stacy

(Pressurized) ball point pen on paper, 14.25" wide by 15.25" high, 1973. This was drawn by computer-controlled pen on a CalComp Drum plotter at the University of Waterloo. It consists of several hundred concentric star images, with their 'radius' varied sinusoidally so as to create the shadow effects of darker and lighter regions. The end result is like an unrealistically precise charcoal drawing. The computer afforded a degree of control that made possible my use math functions to provide desired transitions between dark and light regions. 'Experiment in Shading' is one consequence of that initiative. --- Norton Starr

Movie Clip, 2009. "Tying and untying" is a short movie addressing one of the principal questions in knot theory-unknotting and distinguishing knots. More precisely, we illustrate John H. Conway's classification of knots into knot and link families. Mathematical ideas permeate vivid animations and music creating visual-acoustic symphony. See video on YouTube. --- Victor Stipsic, Marko Vujic, Radmila Sazdanovic

Acrylics on Polyethylene sphere, 10" diameter sphere, 2008. The "Perspective Sphere" is the story of perspective. It shows a 360 degrees in all directions cityscape which is organized with a six point perspective system. This means, every line drawn on the sphere makes a greater circle and every cubical building projects to all six vanishing points which are equally spread around the sphere. This piece also shows examples within the spherical painting of a one point perspective, a two, three, four, five and six point perspective. All are sectioned off with the use of color. I paint what are called Termespheres. These are inside out total worlds that are painted on spheres that hang and rotate from ceiling motors. These spherical paintings show you up, down and all around environments. I feel this exploration has opened up a new way to see the world and its geometry tells me it is more than just art. --- Dick A. Termes

Laser-cut acrylic, 6.5" x 5.5" x 6", 2007. The successful application of a pattern to repeat across the faces of a polyhedron is determined by the pattern's underlying lattice structure and its inherent symmetry operations. Only pattern classes containing six-fold rotation are applicable to patterning icosahedron. Icosahedron #1 exhibits a p6 pattern cut from the faces of the solid. Centres of six-fold rotation in the pattern become axes of five-fold rotation at each vertex and all other rotational symmetries are preserved. I am fascinated by the fundamental concept of symmetry and its application in the creation of patterns. This recent work explores the possibilities of patterns repeating in three-dimensions, around the faces of mathematical solids. --- Briony Thomas

Watercolor on paper, 8.5" x 8.5" x 8.5", 2009. This polyhedron is comprised of ten tetrahedra. Two mirror-image compounds of five tetrahedra are merged to form the solid. When the polyhedron is rendered in a single color it is difficult to distinguish the individual tetrahedra, in part because some pairs of faces are coplanar. To help the viewer resolve this visual puzzle, the ten tetrahedra have been painted with distinct patterns and colors, which are suggestive of the Sun and the nine planets. The overall star-like quality of the polyhedron, and the tight entwining of the tetrahedral "planets", is evocative of our solar system. I love the symmetric beauty of polyhedra and enjoy using paper to create models to study. This model pays homage to Renaissance depictions of the solar system that used various polyhedra to model the celestial bodies. --- Eve Torrence

Acrylic on canvas, 20" x 24", 2008. The Fermat Point of a triangle is the point of least total distance from the vertices of a given triangle. The painting depicts that the Fermat Point of a triangle can be obtained by constructing equilateral triangles on each side and then joining the vertices of the original triangle and the equilateral triangles. It also shows that circles with the sides of the triangle as chords also intersect at the Fermat Point. I seek to depict interesting mathematical truths, curiosities and puzzles in simple, visually descriptive ways. Mathematical amusements inspire the color and form in my paintings, and I try to strike a balance between the simplicity of the concepts and their depiction in art. --- Suman Vaze

Computer image printout glued on cartonboard panel (vZome program by Scott Vorthmann), 17" x 12" (A3, 420x297 mm), 2009 (2005 for the concept of the Golden Pyramid, shown at Bridges London). My 'Golden Pyramid' is a truss that can project (when viewed from underneath, at a precise, quite near point, orthogonally to the back golden triangular face) into the K5 graph (pentagram inscribed in a pentagon) with remarkable proportions (two equilateral and two golden triangles on a golden rectangular base featuring its two diagonals). The spacecraft looking model shown has its struts built here, kind of fractally, from 463 overlapping tiny golden pyramids (that can be 3D-copied). --- Samuel Verbiese

Stirol cube with plastic foil, 2, 5" x 2, 5"x 2, 5", 2009. Example of extending pattern of "48 different squares" over the surface of RUBIK's 4x4x4. Each square of the set appears twice on the 96 tiles of the cube. There are various symmetries on the sides of the cube and between the sides also. So there is more than one coherent and continuous arrangement. I deal with diagonally striped, coloured squares. [These squares assign a restricted de Bruijn sequence $\{S(k,n)\}$. There are $[k(k-1)^{n-1}]/2$ distinct squares, where $k$ is the number of colours, $n$ is the number of stripes.] Last year I studied the geometric shape of arrangements of the squares [in case of $k=3, n=6, S(3,6)=48$] with coherent pattern on the plane. Presently my aim is filling surfaces of solid figures with these squares. Here is one of them. --- Anna Virágvölgyi

Inkjet on transparencies, mounted on acrylic layers in an acrylic frame hung by a laced beadchain, 11" x 9", hung by a beadchain, plus 11" x 7" explanatory placard 2009 (updated from 2002). Symbols from logic map a classical aphorism about watching one's tongue into a visually recursive statement. In addition, the colors of red and green play on the binary nature of electronic gates. An accompanying placard gives the aphorism and lists the symbols used. The art offered is a melding of symbolism from science and mysticism. The flexibility of computer-aided design and execution supports this blend of ancient and modern expression. Math is fun, and art can help make that clear. When it can also take a supportive, spiritual, inspirational, cooperative color, then it is a harbinger of a new humanity. I hope to make art that way. --- Benjamin Wells

Papercraft, 12 inches in diameter, 2009. "Geodesic domes are well known as architectural structures, but generally they exhibit only triangular grids. My main interest, however, has been in having geometric patterns projected onto a spherical surface. The icosahedron is most frequently used for this purpose, but other polyhedrons can serve just as well for the same purpose. 'Geodesic Cuboctahedron 7 frequency' is the cuboctahedron in a 7 frequency basket weave pattern with 6 squares of one color and 12 rectangles of 6 other colors projected onto the surface of the cuboctahedron's circumsphere. --- Magnus Wenninger

**2010 Mathematical Art Exhibition - Third Prize**

Plastic and wood, two pieces, each 6" x 7" x 7", 1998. The two pieces give alternate views of the same three-dimensional surface. The sculpture has been used for classroom illustrations of the concept of partial derivatives as well as integration of functions of two variables. Since the construction is with clear plastic, a myriad of delightful views of intersecting curves can be found allowing the viewer to hypersee the surface. The work that I do now is becoming a blend of my interest in math and my love of nature, with a little bit of steam-punk influence creeping in as well. --- Richard Werner

Digital print, 16" x 13", 2009. This fractal image is based on a Julia set, visible in dark blue along the intersections of the loops. The loops, which all begin and end on the Julia set, also exhibit self-similarity, and hint at the existence of more small loops hidden behind the larger loops. I have always been fascinated with repetition, abstraction, and the search for pattern, and this is what drew me to mathematics. As a mathematical artist, this love of repetition and detail has lead me to fractal art. As a mathematician teaching at an art college, some of my work is pedagogical in the sense of illustrating mathematical concepts in aesthetically pleasing ways; some of it is purely visual play. --- Jennifer Ziebarth

* Harry Benke (1949-2014) For information on original works by Harry Benke please contact julianne at visualimpactanalysis dot com