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[math-fun] For an anti-diversity argument read:
by Brad Klee 28 Jan '20

28 Jan '20

https://phys.org/news/2020-01-math-teacher-plea-pi-irrational.html Where a young student is implicitly denounced as stupid for using python to compute a curve that the math teacher can easily prove circular. I am guessing that the student's real "mistake" was using floating point numbers instead of rationals or arbitrary precision, but the article doesn't say. The student doesn't need derision from the teacher, but might be well-served with an explanation of machine epsilon, and also with pointers to python calculations using fractions: https://docs.python.org/2/library/fractions.html https://www.tutorialspoint.com/python-rational-numbers-fractions The student can search for rational configurations by any expedient algorithm. Eventually, arbitrarily-large sets of high-quality data can be derived, without ever mentioning congruent triangles. The teacher should complement the student's data set, and ask them to keep refining their analysis of the data. This sort of development can and will lead to a correct proof. There's nothing else I can do, except to reject this author's plea to "keep pi an irrational number". I look forward to the next generation of creative scientists, who will undoubtedly continue to find value in thinking of pi as something other than just "an irrational number". -- Brad

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[math-fun] New Heesch tilings
by ed pegg 27 Jan '20

27 Jan '20

I've put some new Heesch tilings found by Craig S Kaplan at the top of http://www.mathpuzzle.com/ --Ed Pegg Jr

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[math-fun] This will probably be incredibly stupid when we figure it out.
by Bill Gosper 25 Jan '20

25 Jan '20

But that might be so easy. Plot says that f[InverseFunction[#1 - f[#1] &][t]] - InverseFunction[#1 - f[#1] &][t] == -t for the several different functions f that I tried. Here's f@t:=Sin[4t] <http://gosper.org/crazysinx-x.png>. (The gold overprinted the blue.) How can those two subexpressions plot so crazily, yet differ merely by the identity function? It's hard not to suspect Mathematica of malfeasance. Strangely methodical malfeasance. Regardless of the vibrations, does anyone understand this identity? Does anyone understand the vibrations? —rwg PS, did everyone (but me) get my (almost trivial) solution to c[0]:=0, c[n+1] = 2 + √c[n] ?

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[math-fun] Mystery table 2
by Keith F. Lynch 25 Jan '20

25 Jan '20

Yes, this is loosely related to my previous mystery table. Yes, it's symmetrical, unlike my previous table. Yes, the numbers on the diagonal are all integers. This operation is defined for all pairs of positive real numbers, not just for integers, and it always returns a positive real number. I'd be glad to extend this table, or to give the value for any two positive real numbers, if anyone asks. 1 2 3 4 5 6 7 8 1: 1.000000 1.545362 2.293678 3.240906 4.248364 5.259193 6.267074 7.272747 2: 1.545362 2.000000 2.576905 3.343752 4.270644 5.262645 6.267506 7.272793 3: 2.293678 2.576905 3.000000 3.589622 4.367036 5.288115 6.272582 7.273628 4: 3.240906 3.343752 3.589622 4.000000 4.596536 5.380654 6.299598 7.279833 5: 4.248364 4.270644 4.367036 4.596536 5.000000 5.600888 6.389624 7.307724 6: 5.259193 5.262645 5.288115 5.380654 5.600888 6.000000 6.603882 7.395990 7: 6.267074 6.267506 6.272582 6.299598 6.389624 6.603882 7.000000 7.606068 8: 7.272747 7.272793 7.273628 7.279833 7.307724 7.395990 7.606068 8.000000

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[math-fun] Reccurence for Pi^2
by françois mendzina essomba2 25 Jan '20

25 Jan '20

Hi, a [n+1]=2^(2*n+5)*(3-sqrt(3)*sqrt(sqrt((3-2^((-2*n)-3)*a[n])^2/3+1)+1)); a [0]=8*(3-sqrt (3)) Best regards Envoyé depuis Yahoo Mail pour Android

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[math-fun] (Log[2]-5/8) = 0.06814718 . . .
by Brad Klee 24 Jan '20

24 Jan '20

For a horrible approximation of Pi/2, and a minor headache along the way: Ann2 ={1, -4 x, 4 (1 - x) x}; ERev1 = Hypergeometric2F1[1/2, 3/2, 2, x] x; ERev2 = ERev1*Integrate[ Series[Exp[-Integrate[Ann2[[2]]/Ann2[[3]], x] ]/ERev1^2, {x, 0, 100}], x]; Limit[D[EllipticE[1 - x] - ERev2, x], x -> 0] N[Normal[ERev2 + %*ERev1 - Pi/2] /. x -> 1] Out[] = (Log[2] - 5/8) Out[] = -0.00155653 Where did this factor (Log[2]-5/8) come from? Is asymptotic analysis the only option we have left? After a few hours today, I still have yet to come up with a convincing answer, and feel thoroughly defeated (by a computer program no less). Am I missing something? Is this (Log[2]-5/8) somehow easy to derive? --Brad See also: http://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/04/01/

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[math-fun] 1/Pi^2
by françois mendzina essomba2 24 Jan '20

24 Jan '20

Hi, Dear Bill, I didn't quite understand your reccurence.It may be obvious, but not for me. here is another one I propose for 1 / Pi ^ 2. It is simpler to obtain and the method is disconcertingly simple. a[n+1]=2^(-2*n-5)/(1-sqrt(1-2^(-2*n-4)/a[n])); a[0]=1/8 n----- infinity a[n] ------- 1/Pi^2 Best regards

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Re: [math-fun] Mystery table
by Keith F. Lynch 23 Jan '20

23 Jan '20

Andres Valloud <avalloud(a)smalltalk.comcastbiz.net> wrote: > Do you know what the table represents? Yes. Let me show a little more of the table. The "all" means that all real numbers will fit there (if you write them very small). I suggest looking for patterns in where there are integers in the table. Or where a number and its negative appear in the same row. 1 2 3 4 5 6 7 8 1: all 2: -inf +inf 3: -inf 1.000000 +inf 4: -inf 0.000000 2.261406 +inf 5: -inf -0.349081 1.000000 3.074601 +inf 6: -inf -0.523305 0.523305 1.564328 3.795684 +inf 7: -inf -0.626499 0.268479 1.000000 2.000000 4.495526 +inf 8: -inf -0.694242 0.109189 0.694242 1.350067 2.385122 5.190739 +inf 9: -inf -0.741892 0.000000 0.500000 1.000000 1.643796 2.748616 5.884929 10: -inf -0.777116 -0.079600 0.364842 0.777116 1.249756 1.908997 3.102748 11: -inf -0.804148 -0.140255 0.265003 0.621297 1.000000 1.469217 2.158924 12: -inf -0.825509 -0.188046 0.188046 0.505587 0.825509 1.192579 1.671024 13: -inf -0.842792 -0.226700 0.126799 0.415933 0.695796 1.000000 1.366735 14: -inf -0.857046 -0.258628 0.076822 0.344232 0.595108 0.857046 1.155985 15: -inf -0.868994 -0.285461 0.035215 0.285461 0.514414 0.746112 1.000000 16: -inf -0.879146 -0.308343 0.000000 0.236333 0.448129 0.657173 0.879146 1 2 3 4 5 6 7 8

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[math-fun] Draft of essay about two proofs of a rectangle tiling theorem
by James Propp 23 Jan '20

23 Jan '20

I'm hoping to post the essay "What Proof is Best?" on my blog in a few days; comments are very welcome! The current draft is at http://jamespropp.org/mathenchant/057-draft2.pdf Thanks, Jim

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[math-fun] Mystery table
by Keith F. Lynch 23 Jan '20

23 Jan '20

Who will be the first to explain what this is a table of? (As always with tables, it's best viewed in a fixed-width font.) 2 3 4 5 6 7 8 9 3: 1.000000 4: 0.000000 2.261406 5: -0.349081 1.000000 3.074601 6: -0.523305 0.523305 1.564328 3.795684 7: -0.626499 0.268479 1.000000 2.000000 4.495526 8: -0.694242 0.109189 0.694242 1.350067 2.385122 5.190739 9: -0.741892 0.000000 0.500000 1.000000 1.643796 2.748616 5.884929 10: -0.777116 -0.079600 0.364842 0.777116 1.249756 1.908997 3.102748 6.578811

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