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.

"Cob," by Stephen Luecking, DePaul University, Chicago, IL (2008)Giclee print, 13" x 13". "Images begin as super ellipses constructed from bezier curves in which the weight and position of the control points are randomized, using a random number generator to induce eccentricity. The eccentric curves are then layered subjected to various improvised Boolean and path edits. The results are not intended to be read as mathematical objects, thus the randomizing and improvising procedures. Rather the goal is to seek out visual tensions implicit in the relationship between the curves and the tondo format, between the wholeness of the circle and the fragmentation in its interior." --- Stephen Luecking, Professor of Computer Graphics, School of Computing and Digital Media, DePaul University, Chicago, IllinoisApr 14, 2009

"Overlapping Circles I," by Anne Burns, Long Island University, Brookville, NY (2008)Digital print, 13" x 12". "This is an iterated function system made up of Mobius Transformations, programmed in ActionScript. I began my studies as an art major; later I switched to mathematics. In the 1980's I bought my first computer and found that I loved programming and could combine my all of my interests: art, mathematics, computer programming and nature." --- Anne Burns, Professor of Mathematics, Long Island University, Brookville, NYApr 14, 2009

"Rhombic Triacontahedron III," by Vladimir Bulatov (2007)Metal 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, ORApr 14, 2009

"Rhombic Dodecahedron I," by Vladimir Bulatov (2008)Metal 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, ORApr 14, 2009

"Recursive Construction for Sliding Disks," Adrian Dumitrescu, University of Wisconsin, Milwaukee (2008)Digital print, 11" x 5". "Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk slides in the plane without intersecting any other disk, so that its center moves along an arbitrary (open) continuous curve. One can easily show that 2n moves always suffice, while the above construction shows pairs of configurations that require 2n-o(n) moves for this task, for every sufficiently large n. Disks in the start configuration are white, and disks in the target configuration are shaded. " --- Adrian Dumitrescu, University of Wisconsin, Milwaukee Apr 14, 2009

"The Lake," by Harry Benke, Visual Impact Analysis LLC (2007)Digital C-print (laser exposed photographic paper, i.e. Lightjet print), 15" x 12". "'The Lake' is an object rising from ripples in a lake. The object is formed by placing 5 pointed stars on the transparent faces of a dodecahedron. The sine wave and harmonic ripples in the lake as well as the dodecahedron elements are rendered 3D models. The models are digitally composed with a scanned background. The mountains could also be fractal and algorithmically generated, but in this work the mountains are part of the base background scan which gives a better sense of depth to the artwork." --- Harry Benke, freelance artist/mathematician, Novato, CA (1949-2014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.Apr 14, 2009

"Totem," Harry Benke, Visual Impact Analysis LLC (2008)Archival 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.Apr 14, 2009

"Recursive Figure-8 Knot" by Carlo Sequin, University of California, BerkeleyIn some depictions of a mathematical knot, some of the meshes formed between the criss-crossing strands resemble the overall outline shape of the whole knot. It is then possible to fit a reduced copy of the knot into every one of these meshes and reconnect the strands so as to obtain again a mathematical knot consisting of a single closed strand. Then this process can be continued recursively resulting in a self-similar pattern. This general process was applied to the 4-crossing Figure-8 knot. But rather than performing this process in a drawing plane as outlined above, subsequent generations of reduced knot instances were placed in planes that are roughly perpendicular to one another, resulting in a truly 3-dimensional sculpture. --- Carlo Sequin Jul 02, 2008

"Arabic Icosahedron" by Carlo Sequin, University of California, BerkeleyMoorish patterns found in the Alhambra often depict lattices of interlocking knots. Here such a pattern composed of interlocking trefoil knots has been wrapped around an icosahedron. Each of the 20 faces is replaced with a trefoil knot, which interlocks along the triangle edges with three adjacent trefoils. The exact nature of the linking between adjacent trefoils leaves some freedom to the designer: In the simplest case two adjacent trefoils interlock with just one lobe each. In the "Arabic Icosahedron" they are linked with two lobes each, resulting in a much tighter meshing. --- Carlo Sequin Jul 02, 2008

"Birds in the Sky" by Carlo Sequin, University of California, BerkeleyThe surface of a sphere is divided into 24 identical regions with the same symmetries as an oriented octahedron. These tiles have bird-like shapes inspired by the work of M.C. Escher. Half the tiles are yellow and have a relief pattern that clearly identifies them as birds. The other 12 tiles are blue without a special relief pattern; they can thus be seen as either the shadows or profiles of birds, or alternatively as blue sky background. --- Carlo Sequin Jul 02, 2008

"Doubly Impossible Staircase," by Jean-Francois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)Traversing along the outside, the stairs always rise; but traversing along the inside, they always descend. Finally, alternating between the exterior and interior, it behaves like a normal staircase. -- Jean-Francois ColonnaJun 20, 2008

"Symmetry Mobius," by Mary Candace Williams; photograph by Annette Emerson.In order to keep the mobius as a band, I used only the eleven symmetries that are not based on a hexagon. The fabric was chosen for its mathematical content. -- Mary Candace WilliamsJun 20, 2008

"Tumbling Escher," by Mary Candace Williams. Quilt copyright 2006 By Mary Candace Williams; photograph by Annette Emerson.If you look at the quilt at a perpendicular angle you have a traditional diamond tessellation known as Tumbling Block. From the side, however, it rises up and back into the quilt; thus a nod to Escher's "Reptiles" in which the drawn lizard rises up and out and back into the drawing board. --- Mary Candace Williams

Jun 19, 2008

"Artistic View of the Klein Bottle," by Jean-Francois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)In mathematics, the Klein Bottle is a non-orientable surface, i.e. a surface with no distinct "inner" or "outer" sides. Other related non-orientable objects include the Mobius strip and the real projective plane. Whereas a Mobius strip is a two-dimensional object with one side and one edge, a Klein bottle is a three-dimensional object with one side and no edges.Jun 16, 2008

"Bidimensional Visualization of the Verhulst Dynamics," by Jean-Francois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)In this image, grey, orange, and red represent negative Lyapunov exponents; yellow, green, and blue represent positive Lyapunov exponents. The two groups of colors distinguish stable systems from chaotic ones.Jun 16, 2008