Mathematical Conversations: Selections from The Mathematical Intelligencer [Hardcover]

This collection has some pretty interesting articles. Everyone should read at least van Dalen's wonderful article on how the aged Hilbert felt it necessary to dismiss Brouwer as associate editor of the Annalen. Displays of loyalty and back-stabbing worthy of a Shakespeare drama ensued, and with no less literary quality, as we may illustrate by this Einstein quotation: "I consider [Brouwer], with all due respect for his mind, a psychopath ... I would say: 'Sire, give him the liberty of a jester [Narrenfreiheit]!' If you cannot bring yourself to this, because his behaviour gets too much on your nerves, for God's sake do what you have to do." Also interesting is Williams's article on the Baptistery of San Giovanni. The baptistery is octagonal. A classic construction of a regular octagon is this: start with a square; set the radius of the compass to half the diameter and draw the circles centred at the vertices; these circles cut the square at 8 points, the vertices of a regular octagon (as is easily proved using symmetry and Pythagoras's theorem). There is also an inner octagon on the baptistery pavement, which is constructed as follows. The octagon, and thus the square from which it was constructed, has a natural inner square. And the octagon walls naturally accommodate two squares: one connecting the midpoints of the slanted sides and one connecting the midpoints of the straight sides. The inner squares of these two squares make up the inner octagon. In another article Ibragimov surveys Lie's theory of differential equations. All the transformations of the x-y-plane that leave the essence of a differential equation (including boundary conditions, etc.) invariant make a group. Surely a solution will be invariant under this group, so we can solve the differential equation by finding the group and studying its invariants. Also, whatever invariants we can find of the most general groups of physical equations will be physical laws, and Kepler's laws of equal area and T^2/r^3 may be deduced in this manner.