
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 awardwinning 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 mathematicsorigami, computergenerated
landscapes, tesselations, fractals, anamorphic art, and
more.
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"The Lake," by Harry Benke, Visual Impact Analysis LLC (2007)Digital Cprint (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 (19492014) 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, raytrace 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 (19492014) For information on original works by Harry Benke please contact julianne@visualimpactanalysis.com.Apr 14, 2009


"Recursive Figure8 Knot" by Carlo Sequin, University of California, BerkeleyIn some depictions of a mathematical knot, some of the meshes formed between the crisscrossing 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 selfsimilar pattern. This general process was applied to the 4crossing Figure8 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 3dimensional 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 birdlike 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 JeanFrancois 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.  JeanFrancois 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 JeanFrancois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)In mathematics, the Klein Bottle is a nonorientable surface, i.e. a surface with no distinct "inner" or "outer" sides. Other related nonorientable objects include the Mobius strip and the real projective plane. Whereas a Mobius strip is a twodimensional object with one side and one edge, a Klein bottle is a threedimensional object with one side and no edges.Jun 16, 2008


"Bidimensional Visualization of the Verhulst Dynamics," by JeanFrancois 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


"Artistic View of a Bidimensional Texture," by JeanFrancois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)This image was obtained by means of a selftransformation of a fractal process.Jun 16, 2008


"Clover51," by JeanFrancois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)This image shows the lack of associativity for addition and multiplication inside a computer. In order to be able to obtain the exact same results over the years for a certain computation, I did include the definition of some "devices" in my own programming language, which allow the definition of the precise order of the arithmetic operations: +, , *, and / (by the way, parentheses won't do that, for example, X=A+(B+C) does not mean T=B+C then X=A+T).
This opens the door to something very powerful: The possibility to dynamically redefine the arithmetic used when launching a program. This picture and "Clover52" are the results of the combination of eight elementary pictures: 3clover, 4clover, ... ,10clover with substitutions like (A+B) > MAX (A,B), (A*B) > (A+B).Jun 16, 2008


"Clover52," by JeanFrancois Colonna (Centre de Mathematiques Appliquees, Ecole Polytechnique)This image shows the lack of associativity for addition and multiplication inside a computer. In order to be able to obtain the exact same results over the years for a certain computation, I did include the definition of some "devices" in my own programming language, which allow the definition of the precise order of the arithmetic operations: +, , *, and / (by the way, parentheses won't do that, for example, X=A+(B+C) does not mean T=B+C then X=A+T).
This opens the door to something very powerful: The possibility to dynamically redefine the arithmetic used when launching a program. This picture and "Clover51" are the results of the combination of eight elementary pictures: 3clover, 4clover, ... ,10clover with substitutions like (A+B) > MAX (A,B), (A*B) > (A+B).Jun 16, 2008


"Coordinate Axis, Highly Unlikely Square and Highly Unlikely triangle" in seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd InfinitumThese three pieces are woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. The Coordinate Axis shows the basic structure of box stitch, and is also suitable for a game of children's Jacks. The Highly Unlikely Square and Triangle are beaded versions of the Impossible Triangle of Roger Penrose that was made famous by M.C. Escher. Compared with a regular square frame or triangular frame like you might hang on your wall, these frames have one quarter turn on each side. To see the effect of these twists, imagine painting a regular square frame with four colors to identify four paths: inside, outside, front and back. A similar coloring on the Highly Unlikely Square identifies four paths or faces, one of which is outlined with gold seed beads. Starting at the corner closest to the camera traveling clockwise, the golden face is outside, back, inside, front. In fact, all four faces are congruent. The effect of the quarter turns on the Highly Unlikely Triangle is different; there is only one face that travels around the triangle four times.  Gwen L. Fisher (www.beadinfinitum.com)Apr 07, 2008


"Mobius Frame with 2 Holes (View I)" in seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd InfinitumThis Mobius Frame is woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. This Mobius Frame represents two distinct mathematical objects. First, one can view this object as assembled from cube (or cubelike) shapes. We might be tempted to try to build such an object from wood using three long beams and two short beams. However, like the Impossible Triangle, this Mobius Frame cannot be built in 3D using all straight lines and right angles. In connecting the beams at their ends, the sides of the beams need to twist. The flexibility of the thread connecting the beads allows the beaded frame to twist to accommodate the necessary turns to build this object in 3D. The second way to view this object is to see it as a patch of an infinite surface with no thickness and two holes. Since the surface has no thickness, ignore the layer of purple beads in the middle. The blue and green coloring of the largest faces shows that this surface has two distinct faces.  Gwen L. Fisher (www.beadinfinitum.com)Apr 07, 2008


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