52. Discovery or invention?
Are mathematics and numbers out there in the Universe, waiting to be discovered? Or are they constructs applied by us to the Universe, imposed rather than uncovered? If mathematics is a process of discovery, how can we be sure that the mathematics we think we have discovered is the right one?
51. Goldbach meets Gödel
Goldbach's conjecture is that every even integer greater than 2 is the sum of two prime numbers. Proposed in 1742, it has resisted all attempts to date to prove it true or false. Thanks to Kurt Gödel, we do know that there is no way to tell, in advance, whether any given conjecture can or cannot be proved. Perhaps it's just unprovable?
50. Some brain-teasers
Amongst the immense literature on recreational mathematics and logic are a vast array of associated puzzles and brain-teasers. I've picked out just a few examples that have amused or entertained me over the years. I particularly like those that appear tricky at first sight, but which yield easily to the correct approach.
49. Complex numbers
The square root of -1 is an important concept in mathematics and physics, but it can give rise to some alarm when first encountered... in part because it lies outside of our own day-to-day experiences, and in part because it is labelled as `imaginary', and forms the basis of what are called `complex' numbers.
Sorting is the systematic rearrangement of a list into some chosen order. Commonly, we may want to rearrange a set of numbers, or a table of words or names into ascending or descending sequence. Sorting is central to many computer algorithm and database applications, but there is no unique best way of doing so.
47. The Riemann hypothesis
The importance of the Riemann hypothesis lies in its intimate connection to the understanding of the distribution of prime numbers. And its fascination is compounded by the fact that, while generally assumed to be true, it remains unproven, despite being long held as one of the biggest open problem in mathematics.
46. Elliptic curves
An elliptic curve is a particular form of cubic equation which turns out to be of spectacular importance. Originating in the search for an expression for the circumference of an ellipse, elliptic curves find use in public key cryptography, in the factorisation of large numbers, in the testing of primes and, famously, in the proof of Fermat's Last Theorem.
Except for straight lines, the simplest algebraic curves are the 'conics': the hyperbola, the parabola, and the ellipse (the circle being just a special case of the ellipse). After circles, ellipses are probably the most familiar curves in all of mathematics. And, like circles, their applications in mathematics and physics are many and varied.
44. Phi and the Fibonacci series
The 'golden ratio', or divine ratio, often referred to as phi, is the irrational number 1.618033... It appears in countless places in mathematics, most notably in the Fibonacci series, originally formulated in the context of the growth of a population of rabbits. It also appears, often alongside the Fibonacci series, in art and Nature.
16. Voronoi diagrams
Place a number of objects anywhere in a plane. For each object there is a region consisting of all points of the plane closer to that object than to any other. The regions associated with each object with this proximity property are called Voronoi cells. They turn out to have many, and often surprising, real world applications.
28. Triangle centres
Draw straight lines from each vertex of a triangle to the mid-point of their opposite sides. Whatever the triangle, these three lines always meet in a single point, the triangle's centroid, corresponding to its centre of gravity. There are many other 'centres' associated with a triangle's geometry, and some beautiful properties are still being found.
41. Sphere packing
A classical problem in geometry relates to sphere-packing. How densely can a number of identical spheres be packed together in 3-dimensions? How curious it is that something everyone 'knows' intuitively - how to pack such spheres - should have required 400 years, and various great minds, to prove!