Category Archives: Uncategorized

The Platonic solids can be regarded as regular tilings of the surface of a sphere in much the same way as the square, hexagonal and triangular tilings of the plane are regular. For example, the dodecahedron is a tiling of … Continue reading →

Emma McCaughan notified me that topic of the 2012 Larmor Lecture organised by the Cambridge Philosophical Society is the Antikythera mechanism. It takes place at 5:30 pm today (Monday 8th October) in the Bristol-Myers Squibb lecture theatre, located on Lensfield … Continue reading →

Dating the discovery of the integers is difficult indeed. Positive integers have been used since time immemorial for counting, and positive rationals were later used by the Ancient Greeks to express lengths. However, this was not sufficient, as the Pythagorean … Continue reading →

If I were more diligent and could actually be bothered to tag and categorise my blog posts, this one would belong in the esoteric realm of ‘probabilistic geometry’. Suppose we have three variables, X, Y and Z, which are independent and random, determined … Continue reading →

I shall be rather busy during the next couple of months, so there may be a noticeable drop in the frequency of posts on Complex Projective 4-Space. Nevertheless, I shall endeavour to maintain a steady stream of interesting articles, assuming I … Continue reading →

We’ll begin by considering Hamiltonian quaternions, a four-dimensional extension to the complex numbers. A particularly interesting bunch of quaternions is the multiplicative group of 120 ‘icosians’. These can be thought of as the vertices of a 600-cell or, by duality, the … Continue reading →

There is a reasonably popular card game entitled 24, where players attempt to derive 24 by applying basic arithmetic operations to a set of four positive integers. For example, the provided integers may be {3, 3, 8, 8}. A solution must … Continue reading →

Near the beginning of the 20th century, a shipwreck was discovered. It contained a horde of treasure en route from Greece to Rome, including a life-size bronze head of a philosopher. The most valuable treasure, however, was a 2000-year-old analogue … Continue reading →

Continuing the theme of magic squares, we begin with the unique simplest non-trivial magic square. It contains the integers {1, 2, …, 9}, one in each square, such that the rows, columns and diagonals all sum to 15. Remarkably, the converse is … Continue reading →

If we provide a bog-standard computer with an infinite data store, it suddenly becomes a Turing machine: capable of answering any question answerable by any digital computer. Even quantum computers are no more powerful; they are merely faster. For example, a quantum … Continue reading →