Math and Art

When I used to hear the pink-tied "mathies" at the University of Waterloo speak about the "symmetry of numbers" and the "beauty of an equation" I will admit that I really thought they'd been spending too much time crunching numbers and had completely lost touch with reality. Then a friend studying math introduced me to the art of M. C. Escher, and I was (almost) convinced that there might be something to this after all. Until this point, the closest I'd ever come to connecting math and art was with an annoying grade 8 project. Perhaps you remember using string to connect nails on a board in order to turn straight lines into curves? Well, we had to provide all the materials and tools, and the kids who cheated and bought the pre-patterned and pre-nailed hobby shop kits got the highest marks. This was not a good way to make a positive math/art connection!

If you aren't familiar with the work of Escher, here is a link to the "official M. C. Escher website". Escher was famous for his impossible art--featuring stairways that changed orientation depending on your point of reference, and transformations (tessellations) in which a repeating fish pattern might gradually coalesce into a flock of birds in flight, or a group of lizards might suddenly walk off a drawing page moving from two- to three-dimensional creatures. His tessellation work ranges from the simple to the quite complex in its geometry, and the staircases, which feature impossible spaces, seem to draw from non-existent dimensions. He also used reflection tricks, though he is less famous for these.
I could not find any public domain Escher works, so you will need to visit the link above to take a peek at his work.

Mathematics is an integral part of art, whether in the case of two-dimensional art, as in drawings, sketches, paintings, etc. or three-dimensional sculpture. Artists use perspective, vanishing points, horizon, the "rule of three", the "golden proportion" and many other mathematical tools. Geometry is an integral part of form in art. Sculptors also incorporate the use of 3-dimensional space and topology in their work. Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects (Wolfram Alpha site definition).Some examples of topology can be found here. An example of a sculpture based on the idea of a 3-dimensional spirograph can be found here.

Natural patterns, such as the patterns on cones, leaves, flower petal arrangements, formations of shells, etc. tend to follow the Fibonacci sequence. This is a sequence that starts with 0 and 1. Add those together to get the third number, which is 1. Add the last two to get the next number, which is 2. Add the last two to get 3, then 5, 8, 13, 21, 34 etc. These number can be found in many places! And their patterns make for some interesting art as well.

If you have ever heard of fractals or Mandelbrot Sets, you are familiar with Chaos theory. The book Chaos by James Gleik explains better than I can how this branch of mathematics takes seemingly random or chaotic data or systems and attempts to find the underlying pattern governing the data/system. The resulting mathematics has provided interesting equations that create beautiful patterns. Some of these can be found in nature, such as in the pattern of a shoreline, a feather or a fern leaf.
Here is an image of a Mandelbrot Set and a second image of it repeated:

Source: Wiki Commons (both images)
For more on Mandelbrot sets, see this incredible site: http://www.skytopia.com/project/fractal/mandelbulb.html
Here are two contrasting images of fractals:

 Source: PDphoto.org

Source: Wiki Commons

These may be mathematical constructs, but one would be mistaken not to also consider them beautiful, and in their own way, works of art.

So when you or your students get bored of ho-hum arithmetic and worksheets, have a little fun with some of these and see where they take you!

More math activities and links can be found here.