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