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 ands, 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.

Anne M. Burns :: Gallery of "Mathscapes", Complex Flows and More

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"Tile 7," by Anne M. Burns, Long Island University, Brookville, NYHere is a fractal tiles created with Geometer's Sketchpad. I start with a single "tile" designed using Geometer's Sketchpad. Then, using Flash Actionscript I place that "tile" in the center of the screen and surround it with 12 copies of the tile that are half the size of the original, then surround those with 36 "tiles" half the size of the second set of "tiles"; the process is continued until the tiles are too small to see. Thus we obtain a "fractal" tiling. See more fractal tiles at http://myweb.cwpost.liu.edu/aburns/. --- Anne M. Burns

"Kaleidoscope," by Anne M. Burns, Long Island University, Brookville, NYThis circle image is made by iterating systems of Mobius Transformations. Read about how this and other circle images are created and view more examples at http://myweb.cwpost.liu.edu/aburns/. --- Anne M. Burns

"June wreath," by Anne M. Burns, Long Island University, Brookville, NYThis circle image is made by iterating systems of Mobius Transformations. Read about how this and other circle images are created and view more examples at http://myweb.cwpost.liu.edu/aburns/. --- Anne M. Burns

"Circle D," by Anne M. Burns, Long Island University, Brookville, NYThe Unit Circle Group is a subgroup of the group of Mobius Transformations. Read about how this and other circle images are created and view more examples at http://myweb.cwpost.liu.edu/aburns/ --- Anne M. Burns

"Flow of complex valued functions (1)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. Burns

"Flow of complex valued functions (2)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. Burns

"Flow of complex valued functions (3)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. Burns

"Flow of complex valued functions (4)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. Burns

"Summertime," 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/. --- Anne M. Burns

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.