On Wednesday 05 March 2008, Dan Asimov wrote:
Michael Kleber wrote: ... (116 - 8 Sqrt[2] - 20 Sqrt[5] + 140 ArcCsch[2] - 40 ArcSinh[1] + 80 ArcSinh[2] + Log[32] + 10 Log[-1 + Sqrt[5]] - 15 Log[123 + 55 Sqrt[5]]) / 120 ... This answer is intriguingly random-looking. It seems to be daring us to come up with an explanation for why it should be as it is.
I don't have an answer, but I have an observation: It's a bit less random than it looks, because ArcCsch[2] = Log[1+Sqrt[5]] - Log[2] ArcSinh[1] = Log[1+Sqrt[2]] ArcSinh[2] = Log[1+Sqrt[5]] -1 + Sqrt[5] = 4/(1 + Sqrt[5]) 123 + 55 Sqrt[5] = 2^-9 (1 + Sqrt[5])^10 so we get (29 - 2 Sqrt[2] - 5 Sqrt[5])/30 + ... ... (- 2 Log[1+Sqrt[2]] + 3 Log[1+Sqrt[5]] + Log[2]) / 6 unless my algebra's gone screwy, which is always possible. Further remark: 29 = 2^2 + 5^2. Further further remark: the numbers 1, 2, sqrt(2), sqrt(5) are the distinct distances between corners of those squares. and a suggestion: Try some differently placed pairs of squares, and see how the answer varies.
How about a much simpler task: Can anyone suggest a reason for the denominator of 120 ?
See above: at least some of it is an illusion :-). -- g