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.

 Explore the world of mathematics and art, share an e-postcard, and bookmark this page to see new featured works..
 FILE 383/667

# "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

 Art & Music, MathArchives Geometry in Art & Architecture, by Paul Calter (Dartmouth College) Harmony and Proportion, by John Boyd-Brent International Society of the Arts, Mathematics and Architecture Journal of Mathematics and the Arts Mathematics and Art, the April 2003 Feature Column by Joe Malkevitch Maths and Art: the whistlestop tour, by Lewis Dartnell Mathematics and Art, (The theme for Mathematics Awareness Month in 2003) MoSAIC - Mathematics of Science, Art, Industry, Culture Viewpoints: Mathematics and Art, by Annalisa Crannell (Franklin & Marshall College) and Marc Frantz (Indiana University) Visual Insight, blog by John Baez