Irreducible fractions come up all the time in math problems, but who knew they looked so beautiful all together?
This picture, from Wolfram, is of irreducible fractions in the Complex plane.

These pictures, drawn by a harmonograph, may better be described as pictures of music. A harmonograph is a mechanical device that uses pendulums that move pen and paper simultaneously to create these images. The device was invented by a mathematician in 1844 and apparently was a popular form of entertainment at Victorian soirees.
The images vary depending on how the pendulums swing in relation to one another. Harmonic ratios used in music such as 3:2, 4:3 give the images below.

These beautiful origami pieces were constructed by mathematicians Erik and Martin Demaine (first image) and Thomas Hull (second image) to investigate the surfaces that result from different kinds of pleated folding.

"Map the infinite checkerboard to the plane so that black squares map to squares and white squares map to quadrilaterals. Such a map is a disguised version of a discrete analytic function. This is a discrete approximation to the exponential map f(z)=e^z."
source: http://www.math.brown.edu/~rkenyon/gallery/gallery.html