Surprises come in different flavors. There are scads of theorems in which complicated integrals evaluate to surprisingly simple numbers, but I’ve seen a lot of those, so in a way I’ve stopped finding such surprises surprising. On the other hand, there aren’t a lot of nontrivial theorems about digit-patterns in integers or real numbers, so when I see something that looks like a digit-pattern, I’m surprised. (But I’ve looked at other Fibonacci numbers, and I don’t see anything going on with their digits, so I’ve become convinced that the “pattern” I saw was just a fluke.) Jim On Sat, Apr 27, 2019 at 10:35 AM Brad Klee <bradklee@gmail.com> wrote:
For comparison, let's keep the same parameters and try the more simple P(x)=x^2. Then,
Equal[ Integrate[Hypergeometric2F1[1/3, 2/3, 1, x^2], {x, 0, 1}], 2 (3 ArcSinh[1/(2 Sqrt[2])] + Log[8])/(Sqrt[3] Pi)] N[% /. Equal -> List, 20]
Maybe other list readers do not find so much fun when "chasing a trail of smoke and reason" (though it does go even higher). So let's look at another geometry:
Show[ContourPlot[ -p^2 + q^2 - (4/27)*(-3*q*p^2 - q^3)^2 == 0, {q, -2, 2}, {p, -2, 2}]]
Should we be surprised to find out that the area interior to sextic lemniscate zbVV, up to harmonic scale factor pi/sqrt(3), equals a rational number 3/2? --Brad
On Fri, Apr 26, 2019 at 11:01 AM Brad Klee <bradklee@gmail.com> wrote:
Does anyone else have an opinion? --Brad
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