The connection between mathematics and
art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius ands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

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"Sanctuary," by Nicholas Rougeux2139 views"Sanctuary" consists of 19 layers, each one of which contains variations of the orbit trap algorithm. The traps are geometric shapes placed in the complex plane that end iteration of a point when its orbit falls within the shape, hence the name. The shape, size, and location of the traps permit Nicholas Rougeux to control the appearance of each of the layers, which are then combined together as if they were transparencies held up to light. Nicholas Rougeux, a North American web developer, reinforces in this picture the idea of a sanctuary by including smooth curves on the sides that simultaneously create sensations of protection and welcome. The mild colors also help to obtain the objective of evoking a comfortable place where spirits are free.

"Rhombic Triacontahedron III," by Vladimir Bulatov (2007)2130 viewsMetal sculpture, 4.0" diameter. "Stellation of rhombic triacontahedron with 30 identical rhombic faces makes base for this sculpture. All internal intersections of rhombic faces were carefully eliminated by cutting away parts of rhombuses. The resulting 3D body was given organic shape by replacing straight faces with smooth subdivided surface. 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." --- Vladimir Bulatov, Independent Artist, Corvallis, OR

"Potemkine," by Etienne Saint-Amant2124 viewsEtienne Saint-Amant is a Canadian scientist passionate for art and mathematics. He has had various exhibitions both individual and collective, he has presented conferences on art and mathematics and he has appeared in numerous programs on radio and television. His work can be seen on CD and book covers, calendars and web pages. "Potemkine" is a pseudo-abstract composition that portrays the intense emotion lived during the rebellion of the battleship Potemkin in the port of Odessa, Ukraine, in 1905. It brings to mind the ship enveloped in smoke, the flying projectiles, the din of the battle: a scene of the terrible emotional conflict of the Russian troops brought about by the orders to quash the rebellion and those sentiments confronted by compassion towards their compatriots. The image was created to commemorate, a century later, these events.

"Saw," by Mike Field (University of Houston)2115 views"Saw" is a Symmetric Fractal with 11-fold rotational symmetry constructed using methods based on iterated function systems. The image was created many years ago when I was at the University of Sydney, Australia, and appears in Symmetry in Chaos (Mike Field and Marty Golubitsky, OUP, 1992).

--- Mike Field

"The Regular Hendecachoron," computer model by Carlo Sequin, University of California, Berkeley.2097 viewsThis hendecachoron (a literal translation of "11-cell" into Greek) is a regular, self-dual, 4-dimensional polytope composed from eleven non-orientable, self-intersecting hemi-icosahedra. This object also has 11 vertices (shown as spheres), 55 edges (shown as thin cylindrical beams), and 55 triangular faces (shown as cut-out frames). Different colors indicate triangles belonging to different cells. This intriguing object of high combinatorial symmetry was discovered in 1976 by Branko Grünbaum and later rediscovered and analyzed from a group theoretic point of view by geometer H.S.M. Coxeter. Freeman Dyson, the renowned physicist, was also much intrigued by this shape and remarked in an essay: "Plato would have been delighted to know about it." The hendecachoron has 660 combinatorial automorphisms, but these can only show themselves as observable geometric symmetries in 10-dimensional space or higher. In this image, the model of the hendecachoron is shown with a background of a deep space photo of our universe, to raise the capricious question, whether this 10-dimensional object might serve as a building block for the 10-dimensional universe that some string-theorists have been postulating.

A more detailed description and visualization of the 11-Cell, describing its construction in bottom-up as well as in top down ways, can be found in a paper by Sequin and Lanier: “Hyperseeing the Regular Hendecachoron”. There are additional images and VRML models for interactive inspection here. --- Carlo Sequin

"A Piece of Hyperspace," by Sarah Mylchreest and Mark Newbold2088 viewsThe quilt depicts a polyhedron known as the Great Triambic Icosidodecahedron. It was paper-pieced by Sarah Mylchreest from a design generated by Mark Newbold using his "Hyperspace Star Polytope Slicer" Java applet. It won a ribbon in the 2002 Vermont Quilt Festival. The Dogfeathers.com site has a description of the quilt pattern.

--- Photograph and image copyright 2005 by Mark Newbold, dogfeathers.com.

"Enmpperaltta," by Inigo Quilez2077 viewsInigo Quilez is an engineer born in the Basque Country, Spain, who actually works in Belgium designing virtual reality tools. The word that titles the picture, Enmpperaltta, signifies nothing; it is simple a permutation of the French word "L'Appartement." The reason is the obsession shown by the author while trying to buy the perfect apartment in Brussels; that goal was finally achieved and he celebrated with this image. Enmpperaltta is in fact a still frame from an animation calculated by means of proprietary software written in the C language from a variant of the well-known Pickover algorithm, a formula that generates shapes resembling those produced by mixing fluids, for example liquids of different colors. To generate the image, the formula was repeated three times with slightly altered parameters, each in a separate process, and applied to the three basic components of color in the image: red, green and blue, that are combined together to produce the final result.

"The Net," by Mehrdad Garousi (2008)2074 viewsDigital art print, 24" x 18.5". "This image exhibits a very complex, yet ordered series of lonely fibers that are woven in each other. This generated lacy net is not flat and goes to infinity at the center and also many times in each of its main arms. Another wonderful mathematical and artistic representation is where hexaploid weaving is modified into a triple one without cutting or deleting any fibers. Self similarity is the main property of this work, as any small hole in the main arms is nearly similar to the whole image. Having experimented with other media, I chose mathematical fractal image making as one of the newest and most wonderful common areas between mathematics and art." --- Mehrdad Garousi, Freelance fractal artist, painter and photographer, Hamadan, Iran

Circle 52014 viewsComputers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

"Kissing in Motion"2003 views"Kissing in Motion" shows the motion of the "shadows" of kissing spheres in a deformation pointed out by J.H. Conway and N.J.A. Sloane, following an observation of H.S.M. Coxeter. The sequence is left-right, right-left, left-right (sometimes called boustrophedon). The image accompanies "Kissing Numbers, Sphere Packings, and Some Unexpected Proofs," by Florian Pfender and Günter M. Ziegler (Notices of the American Mathematical Society, September 2004, p. 873).

--- Bill Casselman

"Poincare," by Mary Candace Williams. Quilt copyright 2005 Mary Candace Williams; photograph by Robert Fathauer.1993 viewsThis is a hyperbolic design so it is as if a sphere was mapped onto a plane. The printed fabric has distorted spheres. This quilt is unusual in that it is pieced from the outside to the center.

--- Mary Candace Williams

"Totem," Harry Benke, Visual Impact Analysis LLC (2008)1993 viewsArchival digital print, 19" x 13.6". "'Totem' represents the frontier, the uncharted, the often surprising and almost mystic nature of mathematical discovery. The totem is composed of ellipsoids ((x2/a2)+(y2/b2)+(z2/c2)) = 1, ray-trace rendered over an algorithmically generated fractal skyscape. Atmospheric effects were calculated as well such as scattering, moisture etc. The totem signifying the last guidepost to the unknown." --- Harry Benke, freelance artist/mathematician, Novato, CA (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.

"A Mathematician's Nightmare," by JoAnne Growney (2008)1985 viewsLaser print on paper, 15 1/2" x 17 1/2" . "The poem, 'A Mathematician's Nightmare,' introduces a version of the unsolved Collatz Conjecture which asserts that when prescribed operations are iterated on any positive integer, the sequence produced will eventually reach 1. The prescribed operations are these, for any starting positive integer n: if n is even, replace n by n/2 (i.e., decrease n by half); if n is odd, replace n by (3n+1)/2 (i.e., increase n by half and round up to the next integer); my exhibit-entry displays both the poem and a graph of the sequence of iterations applied to the integer 27." --- JoAnne Growney, Poet, Professor Emerita, Department of Mathematical Sciences, Bloomsburg University. Residence: Silver Spring, MD

"Embrace," by Robert Bosch (Oberlin College, Oberlin, OH)1965 views2010 Mathematical Art Exhibition, First Prize.

Stainless steel and brass, Diameter = 6 inches, thickness = 0.25 inches, 2009. 2010 Mathematical Art Exhibition, First Prize. 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. All artists are optimizers. All artists try to perform a task--creating a piece of artwork--at the highest level possible. The main difference between me and other artists is that I use optimization explicitly. 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. Often, I need to go through many iterations to end up with a piece that pleases me. I do this out of a love of mathematical optimization--the theory, the algorithms, the numerous applications. --- Robert Bosch (Oberlin College, Oberlin, OH) www.dominoartwork.com

"Snowflake Model 8," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)1948 viewsIn nature roughly a quintillion molecules make up every crystal that falls to earth, with the shape dictated by temperature, humidity and other local conditions. How such a seemingly random process produces snowflakes that are at once geometrically simple and incredibly intricate has captivated scientists since the early 1600s. Now we have simulated their 3D growth using a computational model that faithfully emulates both the basic shapes and the fine details and markings of the full range of observed forms. Our model is driven by diffusion-limited attachment of micron-scale blocks of ice; read about the underlying mathematics at http://psoup.math.wisc.edu/Snowfakes.htm. --- David Griffeath