From fractals to cellular automata
According to the American Mathematical Society, “the connection between mathematics and art goes back thousands of years.” Below are some particularly striking pieces based on various mathematical concepts.
Hamid Naderi Yeganeh believes that finding mathematical objects that resemble real recognizable things may encourage young people to pursue mathematics. He created this image using 2,000 line segments. Yeganeh is a math student at the University of Qom in Iran, and writes equation-based computer programs that, by altering parameters, can produce thousands of shapes. He then looks over the shapes and finds the ones that speak to him.
In this case, he took k=1, 2, 3,…2000 and made the endpoints of each segment equal (cos(6πk/2000)-i cos(12πk/2000))e^(3πi/4) and (sin((4πk/2000)+(π/8))+i sin((2πk/2000)+(π/3)))e^(3πi/4).
“I change the formulas to find better shapes. But I don’t know anything about the results of the programs before running them,” he says according to Science Friday.
Image courtesy of Hamid Naderi Yeganeh
This probably isn’t what you normally picture when someone mentions a fractal — geometric shapes containing endless smaller copies of themselves. It is called an Lsystem fractal and looks oddly like a plant. It is one of the many in Dr. Daniel Ashlock and Liz Blakenship’s taxonomy of fractals.
Image courtesy of Dr. Daniel Ashlock and Liz Blakenship
A biomorph is a mathematical object that looks like a living thing. The image below was created by taking a chunk from a Newton’s method fractal and editing the colors with image-editing software.
Image courtesy of Dr. Daniel Ashlock
This image was created by Kerry Mitchell in celebration of Curiosity landing on Mars in 2012. According to Mitchell, “’Curiosity’ celebrates the unsung measure of central tendency, the harmonic mean, and the Curiosity rover’s landing on Mars in August 2012.
Image Courtesy of Kerry Mitchell
Although cellular automata sometimes generate fractals, they are not fractals per se. They are generated based on a collection of starting points (cells) and rules that define what state the cells in their neighborhood take on.
Sierpinski Triangle in the Snow
Not all math art is computer-generated. Simon Beck, who works as an orienteering mapmaker, created this breathtaking piece of art by walking in the snow wearing snowshoes. The fractal represented is a famous one called the Sierpinski Triangle. It begins with a single equilateral triangle and continues outwards with new triangles being added to the midpoint of each segment. Interestingly, the Sierpinski Triangle appears in certain cellular automata like a variation of Conway’s Game of Life.
Image courtesy of Simon Beck
If you’re inspired by all this art, why not print out some fractal coloring sheets and see where your mind takes you as you relax.