Nothing captures the imagination like the concept of infinity. It has fascinated poets, philosophers and mathematicians for ages. In the last half of the 19th century, Georg Cantor wrote a series of papers that put mathematical infinity on a logical footing. Cantor defined an infinite set as one that could be put in a one-to-one correspondence with a proper subset of itself. A proper subset is one where some, but not all, of the elements are removed from the set. For example, the natural numbers "1, 2, 3, 4" can be put into a one-to-one correspondence with the even numbers "2, 4, 6, 8." There are exactly as many numbers as there are even numbers. This could never happen with a finite set.
Fractals are self similar structures -- structures that look the same at different scales. The leaves of a tree often look like the entire tree, and the smallest branches look like the largest branches at a different scale. A coastline viewed from space, its seashore from a few feet and the same seashore under a microscope all appear similar. Some of the most beautiful fractals can only be created on a computer. If the points in the complex plane are run through a simple function, the points make a complex map. You can zoom in on this map and get a similar -- but not quite identical -- map at each level of resolution.
Quaternions are similar to complex numbers only there are three imaginary components -- i, j and k -- instead of the single imaginary component in complex numbers. When a complex number or a figure composed of complex numbers is graphed, it can be moved and rotated by multiplying the points by a single number. Quaternions allow you to do this in three dimensions. Computer-generated Images (CGIs) are described as quaternions and rotated and moved by multiplying each of the points by a quaternion.
At the beginning of the 20th century, David Hilbert -- the most famous living mathematician -- proposed 23 problems that he hoped would be solved in the new century. These 23 problems generated entirely new branches of mathematics and made the reputations of many mathematicians. This was big league mathematics in the 20th century. All were solved except one -- the Riemann hypothesis. At the beginning of the 21st century, seven problems were proposed by the Clay Institute. This time there was a $1,000,000 prize attached to each problem. The only problem that appeared on both lists is the Riemann hypothesis.