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.
"Mobius Frame with 2 Holes (View II)," by Gwen Fisher (www.beadinfinitum.com)
Materials: seed beads, Nymo nylon thread
This Mobius Frame is woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space 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 cube-like) 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 Fisher (www.beadinfinitum.com)