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 bands, 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.

"Escher's 'Relativity'," copyright Andrew Lipson. Made of Lego ®2361 viewsDaniel Shiu and I worked on this as a joint project. There are no camera tricks, but the picture has to be taken from exactly the right place, and that was a challenge in itself. Unlike many of Escher's other "impossible" pictures (like "Ascending and Descending"), there is actually no optical illusion involved here. Gravity seems to be working in three different directions simultaneously, but the picture shows a perfectly self-consistent physical scene. So modelling it should certainly be feasible. But while Escher's picture has three different "up"s, Lego® isn't quite so flexible. See photos of the construction in progress. Lego® is a trademark of The Lego Group. On my website I post images of M.C. Escher's original works (C) Cordon Art, Baarn, the Netherlands on his website, used with permission, so that you may compare with the Lego® creations. All rights reserved. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)

"Eifiona," by Tina Oloyede2353 viewsTina Oloyede left her profession as a medical doctor to become a self-taught fractal artist, a passion and obsession since 1999. Residing in England, she balances her artistic activity with the care of her young family. She is actually one of the most versatile and publicly-appreciated fractal artists. For this picture she used 13 different formulas: 7 for building the basic structure of the image, 3 for adding different textures, and another 3 for controlling the coloring of the image. The name of the picture, "Eifiona," is the Welsh name of a friend of the artist, who ordered the image with one condition, that it be of "The Autumn" and in return granted absolute freedom to make the design and finish of the image. Tina Oloyede's capacity for artistic expression is unquestionable; it is impossible to see this picture without an autumnal image appearing in our mind.

"Tribute to Zemela," by Joseph Presley2351 viewsJoseph Presley has worked with traditional art forms since he was a child, but discovered his favorite form of expression in fractal art, which produces the sensation of painting with the same tools that make nature beautiful. This image was generated by means of a variant of the Barnsley formula, "IFS-Barnsley-JockIII," written by Jock Cooper, and colored basically with the algorithm "fBm Popcorn Traps," written by Mark Townsend. The name of the picture, Tribute to Zemela, refers to an artist friend of the author, Lisa Thallauer, for whom Joseph Presley designed the image, being inspired by an imaginary wooden object.

Hopf Fibered Linked Tori2348 views"Hopf Fibered Linked Tori," by The
3DXM Consortium

The Hopf map maps the unit sphere in four-dimensional space to the unit sphere in three-dimensional space. The four tori linked in this image are made up of fibers, or pre-images, of the Hopf map. In this visualization, each fiber has a constant color and the color varies with the distance of the fibers. Any two of the four tori are linked, as are any pair of fibers on a given torus. See more surface images on the 3D-XplorMath Gallery.

--- adapted from "Hopf Fibration and Clifford Translation of the 3-Sphere," by Hermann Karcher

"Toroweave," by David Bachman (Pitzer College, Claremont, CA)2344 views10 x 25 x 25 cm, 3D printed "sandstone" (gypsum powder + binder), 2016

These pieces were each created from two copies of a diamond tiling of a torus. The interior of each face of the tilings was removed, and the resulting webs were alternately offset in opposite directions to form a weave. Complementary colors are used to distinguish the two interlinked forms, which are completely disjoint. -- David Bachman

"Rhombic Triacontahedron III," by Vladimir Bulatov (2007)2326 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

"Sanctuary," by Nicholas Rougeux2322 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.

"Saw," by Mike Field (University of Houston)2311 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

"Potemkine," by Etienne Saint-Amant2299 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.

"The Regular Hendecachoron," computer model by Carlo Sequin, University of California, Berkeley.2290 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

"The Net," by Mehrdad Garousi (2008)2290 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 52276 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"2275 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

"A Piece of Hyperspace," by Sarah Mylchreest and Mark Newbold2260 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 Quilez2248 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.