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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Parametric Breather2300 views"Parametric Breather," by The 3DXM Consortium.

This striking object is an example of a surface in 3-space whose intrinsic geometry is the hyperbolic geometry of Bolyai and Lobachevsky. Such surfaces are in one-to-one correspondence with the solutions of a certain non-linear wave-equation (the so-called Sine-Gordon Equation, or SGE) that also arises in high-energy physics. SGE is an equation of soliton type and the Breather surface corresponds to a time-periodic 2-soliton solution. See more pseudospherical surfaces on the 3D-XplorMath Gallery.

--- Richard Palais (Univ. of California at Irvine, Irvine, CA)

"Eights," by George W. Hart (www.georgehart.com)2289 viewsThis six-inch diameter paper sculpture is made of sixty identically shaped parts. Parts of any one color form a type of tetrahedron, and there are five such, deeply interlocked. No glue is used; they parts just hook into each other. I call this type of design "modular kirigami". It took me about four hours to assemble after several hours of false starts and figuring out how to do it. I generated a computer-rendered view down a five-fold axis. The "8"-shaped parts each link with many others. So they could not be made as single pieces of paper unless they were glued or taped together after being linked. But I wanted to be a purist and use no glue or tape, so I designed the parts as two overlapping "3"-shaped pieces.

--- George W. Hart (www.georgehart.com)

"Rosetta," by Edward Alonzo (Artist, University of Vermont)2281 viewsAcrylic 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 current avenue of investigation is Steganography and the place of Cryptography in our society. Encryption has become incredibly powerful and equally incredibly common place. The hidden nature of steganography is because either the cryptographer decides to do it, or in the more common case of "https" because the user is ignorant of its existence. The ignorance in the second case is due to the overwhelming complexity of computers and computations done by them. Which is akin to the overwhelming complexity of art and decisions made by artists. 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 (Artist, University of Vermont) http://www.sirhair.com/

"Asundriana," by Janet Parke2280 viewsJanet Parke, born in Memphis (USA), has passed the major part of her life as a ballet dancer, choreographer, and dance professor. In 1999 she began to exhibit and sell her fractal art, characterized by an extraordinary sensitivity and coloring style unknown until then. Janet Parke replaces the characteristic loud and bright colors of the first generations of fractal art with smooth, rich tones and delicate shades. Her style will be imitated by a new generation of fractal artists. "Asundriana" is based on a variant of the Julia set ( z -> z-squared + c ) such that the parameters c and z are manipulated to produce distortions in the typical spiral structures of this set. The name of the image comes from the word asunder, since the structure of the image seems to fold into and separate from itself.

"Rhombic Dodecahedron I," by Vladimir Bulatov (2008)2237 viewsMetal sculpture, 4.5" diameter. "The base of this sculpture is rhombic dodecahedron (polyhedron with 12 rhombic faces with cubical symmetry). Each of the 12 faces was transformed into a curved shape with 4 twisted arms, which connects to other shapes at vertices of valence 3 and 4. The boundary of the resulting body forms quite a complex knot. 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

"Three Link Chain," by Jarke J. van Wijk (Technische Universiteit Eindhoven). Image courtesy of Jarke J. van Wijk.2233 viewsThis knot consists of three similar links, and is threefold-symmetric. The surface shown is a Seifert surface, an orientable surface bounded by the links. Considering only the links, it is hard to imagine that such a surface does exist. However, in the 1930's, the German mathematician Herbert Seifert presented an algorithm to find such surfaces for any knot or link. This image was made with a tool called SeifertView.

--- Jarke J. van Wijk

"Escher's 'Ascending and Descending'," copyright Andrew Lipson. Made of Lego ®2222 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 the final photograph was slightly distorted to emphasize the perspective effect. I'm especially pleased with the way the roof in the top left of the picture came out. 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)

"The Susurrus of the Sea," by George W. Hart (www.georgehart.com)2199 viewsSoft waves, suggestive of both sky and water, travel around the globe along six different criss-crossing equators. The susurrus (murmur) of the sea is suggested as a sense of harmony in this sphere. Technically difficult, the 60 transparent blue acrylic plastic components had to be made very precisely to fit together. Heat-formed, the components were formed and assembled on special jigs which imparted the proper dimensions and angles. Mathematically, the blue spirals are helixes that follow the edges of an icosidodecahedron. This is a polyhedron that was known to the ancient Greeks, but the oldest known drawing of it is by Leonardo da Vinci. Formally constructed of triangles and pentagons (which show up here as the openings) it can also be seen as an arrangement of six equatorial regular decagons. Each equator makes ten twists in a complete path, crossing the other five equators at two opposite points. If one "walks along" a dark blue edge, making right-angle turns where edges meet, one traces a large five-pointed star before returning to one’s starting point.

--- George W. Hart (www.georgehart.com)

"Untitled," by Mark Townsend2185 viewsMark Townsend is a complete fractal artist who combines a refined technique with a marvelous creativity. This versatile Australian programmer has designed dozens of formulas for the program Ultra Fractal, but also gets part of his fame as the author of the popular software Apophysis. Mark Townsend is one of the authors who has contributed much to the recognition of fractal art, providing a work that is both innovative and at the same time personal. For this image, he tried to create shapes that did not appear to be made with a computer. The lines were included to emphasize the two-dimensional nature of the image.

Hopf Fibered Linked Tori2158 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

"Escher's 'Relativity'," copyright Andrew Lipson. Made of Lego ®2153 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)

"Crocheted Lorenz manifold, white background," by Hinke Osinga, in collaboration with Bernd Krauskopf, Department of Engineering Mathematics, University of Bristol (www.enm.bris.ac.uk/staff/hinke/crochet/)2147 viewsDr. Hinke Osinga and Professor Bernd Krauskopf (Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics.

The white background in the photograph brings out the rotational symmetry of the Lorenz manifold and gives an idea of the structure of the mesh.

For more information, the crochet pattern and mounting instructions, see: http://www.enm.bris.ac.uk/staff/hinke/crochet/.

"Snowflake Model 1," by David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis)2137 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

"Tribute to Zemela," by Joseph Presley2120 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.

"Rhombic Triacontahedron III," by Vladimir Bulatov (2007)2120 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