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.

"Seifert surfaces for torus knots and links," by Saul Schleimer (University of Warwick, UK) and Henry Segerman (University of Melbourne, Australia)

Four pieces: 111mm x 111mm x 105mm, 125mm x 130mm x 99mm, 125mm x 125mm x 118mm, 103mm x 103mm x103mm, PA 2200 Plastic, Selective-Laser-Sintered, 2012

As elegantly discussed in Ghys' 2006 ICM plenary talk, the natural parameterization of the Seifert surface for the trefoil knot uses Eisenstein series of lattices in the plane. This was generalized by Milnor to all (p,q) torus knots; he replaces Eisenstein series by certain fractional automorphic forms. Tsanov reduces the construction of these forms to finding an analytic description of the universal covering of the orbifold S^2(p,q,infinity) by the hyperbolic plane. Mainly following Lehner, we find a Fourier series for the covering map. Combining these ideas, we obtain a map from a hyperbolic triangle T_H, with angles pi/p, pi/q, and 0, to a domain T_S in S^3; rigid symmetries of T_S in S^3 generate the Seifert surface. Using Schwarz-Christoffel theory we uniformize T_H by a Euclidean triangle T_E having angles pi/p, pi/q and pi(1-1/p-1/q). In this way we transfer decorations on T_E to the Seifert surface; for these sculptures we use a subdivision of T_E into 15 congruent triangles. -- Saul Schleimer and Henry Segerman