In March (2023), a page of mathematical writing by Alan Turing came up for sale at auction. It was a note about a mathematical problem, written to Rolf Noskwith probably in 1941-2 while they were both working in Hut 8 at Bletchley Park. It is phrased as a question about unit equilateral triangles in Rn (the n-dimensional real number space), the answer to which Turing then goes on to derive.
On the face of it, it is an odd question to ask. It would be fascinating to know if this was just recreational mathematics, or if Turing was modelling a problem his Hut 8 team were working on. It will take a more imaginative mathematician than I to see what that problem might have been, or whether the result is known and/or important (his explanation of the problem, as a corollary, sets out the coordinates of the centre of gravity of the unit equilateral ‘triangle’ in the first 2n-tant of Rn with one vertex at the origin).
King’s was able to buy the document, with part of a very generous 2019 bequest.
Here is what Alan wrote:
To see a transcript of it click here. I have added a few footnotes, where I needed clarification when I tried to understand it. Full disclosure: I didn’t understand the key step, or indeed any of the sentences involving the words ‘centre of gravity’. Those with better intuition in multiple dimensions will no doubt see that it’s obvious.