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.

"Hilbert Cube 512"4952 views"Hilbert Cube" is a space-filling recursive curve in 3 dimensions in analogy to the famous Hilbert curve in the plane. Special care has been taken never to place more than 3 coplanar line segments in sequence. At the largest recursion step the geometry has been slightly altered so as to obtain a closed loop. In the proper parallel projection one can see that the 2 halves of this sculpture are connected by only 2 tube segments. This piece of art gives me the association of an abstract, constructivist model of the human brain. See more views of the
"Hilbert Cube 512". --- Carlo Sequin

"Poincare FishDish," by Carlo Sequin, University of California, Berkeley3679 viewsA tiling with regular heptagons does not fit into the Euclidean plane, since 3 times the dihedral angle of the heptagon exceeds 360 degrees. But if we are willing to introduce a progressive scale factor, then the whole hyperbolic plane can be fit into the Poincaré disc. Here is a visualization of a {7,3} tessellation where 3 heptagons join at every vertex, using a tiling motif inspired by the famous Dutch artist M.C. Escher. Each heptagon is cut into 7 identical pizza slices with irregular boundaries in the shape of fish that properly interlock with one another. See more tiling patterns on the Poincare disc. --- Carlo Sequin

"Lizard Tetrus," by Carlo Sequin, University of California, Berkeley3566 views24 Lizard tiles, inspired by one of the many planar tilings by M.C. Escher, are mapped around a rounded tetrahedral frame of genus 3. This tiling is a contorted version of the pattern of 24 heptagons displayed on the surface of the marble sculpture "Eight-fold Way" by Helaman Ferguson. That sculpture celebrates Felix Kelin's famous "Quartic Curve" which achieves the maximal symmetry of 168 automorphisms possible on a genus-3 surface. Read more about patterns on the Tetrus surface.. Thanks to Pushkar Joshi and Allen Lee for their help with mapping Escher tiles onto the tetrus. --- Carlo Sequin

"Knot divided" (snow sculpture), by Carlo Sequin (University of California, Bekeley), Stan Wagon (Team Captain), John Sullivan, Dan Schwalbe, and Rich Seeley3058 viewsCan a DIVIDED KNOT be NOT DIVIDED? When carving this sculpture out of a 10x10x12 foot block of hard compacted snow, we started with the simplest possible knot: the overhand knot, also known as the trefoil knot. We then split lengthwise the whole ribbon forming the three big loops. But there is a twist that may lead to surprises: The original knotted strand was actually a triply twisted Moebius band! Thus the question: Does our cut separate the structure into two pieces, or does it form a single, highly knotted twisted strand? Read more about this snow sculpture. --- Carlo Sequin

"The Regular Hendecachoron," computer model by Carlo Sequin, University of California, Berkeley.2313 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

"Recursive Figure-8 Knot" by Carlo Sequin, University of California, Berkeley1848 viewsIn 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

"Arabic Icosahedron" by Carlo Sequin, University of California, Berkeley1827 viewsMoorish 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

"Birds in the Sky" by Carlo Sequin, University of California, Berkeley1609 viewsThe 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

"125 tetrahedra in 25 projected 5-cells," (I) by Carlo Séquin, University of California, Berkeley986 viewsThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This view (I) is featured on a printed AMS 125th anniversary poster, free upon request.

"125 tetrahedra in 25 projected 5-cells," (II) by Carlo Séquin, University of California, Berkeley683 viewsThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.

"125 tetrahedra in 25 projected 5-cells," (III) by Carlo Séquin, University of California, Berkeley642 viewsThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.

"125 tetrahedra in 25 projected 5-cells," (V) by Carlo Séquin, University of California, Berkeley627 viewsThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.

"125 tetrahedra in 25 projected 5-cells," (IV) by Carlo Séquin, University of California, Berkeley616 viewsThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.

"125 tetrahedra in 25 projected 5-cells," (VII) by Carlo Séquin, University of California, Berkeley591 viewsThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.

"125 tetrahedra in 25 projected 5-cells," (VI) by Carlo Séquin, University of California, Berkeley584 viewsThe 5-cell (the four-dimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a three-sided bi-pyramid. Twenty-five of those cells again outline the same projection of a 5-cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine. --- Carlo Séquin

This work is featured on a printed AMS 125th anniversary poster, free upon request.