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, Berkeley
In 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