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.

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 Superimposition of Polar Surfaces-1, by Dejenie A. LakewThe superimposition of two polar surfaces: rho = 2sin4[theta] rho = 5/3 cos4[theta] (wire-framed) with some compositions of tilts and turns. The two polar surfaces are generated in such a way that one is a derivative surface of the other but with different polar radius.
 Superimposition of Polar Surfaces-2, by Dejenie A. LakewSuperimpositions of polar surfaces. Equations: rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])), rho = exp(sin 3(exp sin 3[theta]), rho = -3cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])) with compositions of tilts and turns. Here the first polar surface is the derivative of the second surface and the third surface is a spatial reflection of the first through the origin with wire frames, but with a larger spatial radius.
 Superimposition of Polar Surfaces-3, by Dejenie A. LakewSuperimposition of polar surfaces. Equations: rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])), rho = 3 + 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])) with several compositions of tilts and turns.
 Superimposition of Polar Surfaces-4, by Dejenie A. LakewSuperimpositions of polar surfaces. Equations: rho = 10sin8[theta] rho = 10cos8[theta] with a number of compositions of tilts and turns.
 Superimposition of Polar Surfaces-5, by Dejenie A. LakewSuperimposition of three polar surfaces. Equations: rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])), rho = 3 + 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])) rho = -3cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta]) followed by many compositions of tilts and turns.
 Superimposition of Polar Surfaces-6, by Dejenie A. LakewSuperimposition of three polar surfaces. Equations: rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])), rho = 3 + 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])) rho = 6*(the outer sphere wire-framed) with many compositions of tilts and turns as rigid transformations.
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 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