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.

Mountains in SpringComputers make it possible for me to "see" the beauty of mathematics. The artworks in the gallery of "Mathscapes" were created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector.

--- Anne M. Burns

"Persian Rug (Recursian I)," by Anne M. Burns (Long Island University, Brookville, NY)An applet uses a recursive (repeatedly applied) procedure to make designs that resemble Persian rugs. You may choose 3 parameters a, b and c, and one of 6 color palettes each consisting of 16 colors numbered 0 through 15. The parameter c ( 0 through 15) represents an initial color. A 257 by 257 square is drawn in the color numbered c. Label the 4 corner colors c1, c2, c3 and c4 (at the initial stage they will all be c). then a new color is determined by the formula a + (c1+c2+c3+c4)/b mod 16 and a horizontal and vertical line that divide the original square into 4 new squares are drawn in the new color. The procedure is repeated recursively until all the pixels are filled in. Read more about "Persian" Recursians, enter the parameters and click on Draw rugs, and download a Windows Program that makes "Persian" rugs, at http://myweb.cwpost.liu.edu/aburns/persian/persian.htm. --- Anne M. Burns (Long Island University, Brookville, NY)

Circle Picture 10Computers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

--- Anne M. Burns

Circle Picture 5Computers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

--- Anne M. Burns

Circle 5Computers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

Circle 4Computers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

Circle 3Computers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

Circle 1Computers make it possible for me to "see" the beauty of mathematics. This image and all of the Circle Pictures are made by iterating systems of Mobius Transformations.

"Fractal Scene II," by Anne M. Burns (Long Island University, Brookville, NY)"Mathscapes" are created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector. See the Gallery of "Mathscapes and find citations for my articles on modeling trees, plants and mountains, and on "blending and dithering," at http://myweb.cwpost.liu.edu/aburns/gallery/gallery.htm. --- Anne M. Burns (Long Island University, Brookville, NY)

"Lilacs--an Imaginary Inflorescence," by Anne M. Burns (Long Island University, Brookville, NY)"Inflorescence" is the arrangement of flowers, or the mode of flowering, on a plant--sometimes simple and easily distinguishable, sometimes very complex. "Lilacs" is an example of an imaginary inflorescence that I have created using computer graphics techniques. Two Java applets allow users to see and draw purely imaginary inflorescences at various stages using the recursive (repeatedly applied) functions. Download the code from either applet, and see photographs of real inflorescences several imaginary inflorescences, at http://myweb.cwpost.liu.edu/aburns/inflores/inflores.htm. --- Anne M. Burns (Long Island University, Brookville, NY)

"Imaginary Garden," by Anne M. Burns (Long Island University, NY)"Mathscapes" are created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector. See the Gallery of Mathscapes and find citations for my articles on modeling trees, plants and mountains, and on "blending and dithering" at http://myweb.cwpost.liu.edu/aburns/gallery/gallery.htm. --- Anne M. Burns (Long Island University, Brookville, NY)

"Fractal Scene I," by Anne M. Burns"Mathscapes" are created using a variety of mathematical formulas. The clouds and plant life are generated using fractal methods. The mountains are created using trigonometric sums with randomly generated coefficients; then, using 3-D transformation, they are projected onto the computer screen. Value and color are functions of the dot product of the normal to the surface with a specified light vector. See the Gallery of "Mathscapes and find citations for my articles on modeling trees, plants and mountains, and on "blending and dithering," at http://myweb.cwpost.liu.edu/aburns/gallery/gallery.htm. --- Anne M. Burns (Long Island University, Brookville, NY)