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.
"Gaussian Wave Packet Sculpture," by Chet Alexander (University of Alabama, Tuscaloosa)
Wood (birch, walnut, maple, ebony), 9" x 11" x 10", 2006
Mathematics of the Wave -Packet Sculpture: In this sculpture, mathematics was used to calculate the Gaussian wave-packet model of a particle in quantum mechanics. This is accomplished by forming a linear combination of plane waves of different wave-numbers, k. A particle with mass and momentum p can have wave properties as described by the de Broglie wavelength relation λ=h/p. The Gaussian wave packet model is a way to combine the wave and particle properties of a particle of momentum p=hk localized at position x_0. The probability of finding the particle at position x_0 is given by the probability density of the particle as ІΨ(x,0) І^2~exp[-(x-x_0)^2/2(∆x)^2] , and by a Fourier transform the probability density of the particle's momentum can be written ІΨ(k) І^2~exp[-(k-k_0)^2/2(∆k)^2]. The wave packet sculpture presents a Gaussian wave packet envelope and an electromagnetic wave enclosed in the envelope. --- Chet Alexander