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There is a cluster of five British universities known as the G5, originally formed as a pressure group to extract higher student fees from the government. They are now sometimes referred to as the Golden Triangle. How, you may reasonably ask, can five universities form a triangle?
The answer is that while Oxford and Cambridge keep a respectful social distance from the others in the group, the remaining three are all clustered very near to each other in London. On a map, they look roughly like this:
Although this is almost exactly an isosceles triangle (two of the three sides have equal length; shown in red on the map), it is not, mathematically speaking, even close to being a golden one, because the angles are wrong. A golden triangle, shown by the purple lines on the map, would have angles of 72 degrees at Oxford and Cambridge and 36 in London, dividing the 180 degrees in the triangle in pleasing proportion of 2:2:1. Unfortunately, at least for people in London, for the triangle to become truly golden, the London universities would have to relocate to a spot in the English Channel, a few miles off Bexhill on Sea; but this arrangement would probably suit only the marine biologists at the institutions involved. The situation could be somewhat improved by making the triangle a right-angled one (45 degrees in Oxford and Cambridge, 90 in London); then the London institutions would only have to move north a few miles to Cricklewood ([p. Or, if they were willing, they could move just south of the M25, to Crockham Hill near Sevenoaks, and form a nice equilaterial triangle (60 degrees at all three points).
Interestingly, all these triangles are special cases of a general type of isosceles triangle. Let's divide the 180 degrees inside a triangle into N equal parts; for example, if N=5, we get 5 parts of 36 degrees each. Then we set two of the angles to M of these parts, and the other to N-2M parts. So if N=5 and M=2, we get two angles of 72 degrees and one of 36. That's a golden triangle, and it has some fascinating properties, which we'll get to. But for now, we'll generate some other "(N, M) triangles" to go alongside our (5, 2), or golden, triangle.
How small can we make N? Well, both M and N-2M have to be at least one, or our triangle will collapse into a line. So the simplest member of the family is (3, 1): three equal angles, which is an equilaterial triangle. The next simplest is (4, 1): two angles of 45 degrees and one of 90 - an isosceles right-angled triangle. For N=5, we have two choices: (5, 2), which is golden, and (5, 1), which is a flattened triangle known as a gnomon, after that pointy thing on the top of a sundial: two angles of 36 degrees and one of 108. Here they all are. Rather than writing the angles in degrees, I have shown them as fractions of the total of 180 degrees:
In the real world, the so-called "golden" triangle of universities is roughly a (15, 4) triangle: divide 180 degrees by 15 and you get 15 pieces of 12 degrees each, and then you allocate 4 of those pieces to Oxford, 4 to Cambridge and the other 7 to London*. And of course, if you let N get as high as 15, you have a lot of choices for M, so the number of triangles starts to grow fast enough that individual ones are probably not very interesting.
The (real) golden triangle and its friend the gnomon do have some strange properties, though. In both a golden triangle and a gnomon, the ratio of the length of a long side to a short side is φ, otherwise known as the golden ratio. φ has the value (1+√5)/2, and it has the property that (φ+1)/φ = φ. It occurs all over the place in both maths and the real world. And if you bisect one of the base angles (at the bottom) of a golden triangle and draw a line - shown in red on the right - from there to the opposite side, you get a smaller golden triangle below the line and a gnomon above it. And there's more; see the Wikipedia page for details.
*As a Cambridge graduate myself I cannot resist pointing out that if you allocated pieces to universities not according to geography but in proportion to the number of Nobel Prizes their researchers have won, you would still get something close to a (15, 4) triangle, but Cambridge would be at the apex, with nearly twice as many prizes as the other four universities in the group combined. Cambridge has in fact won more of the things than any other university in the world, at least according to the official counting methodology.
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