October 2012 - mathfun@mailman.xmission.com

Re: [math-fun] parametrizing the Koch snowflake curve
by Bill Gosper 24 Apr '15

24 Apr '15

Fourier epicycles thru nth: (d16) sum(((product((1 - (( - 1)^j * tan(((%pi * (2 * k + (1/3)))/(2^j)))/(sqrt(3)))),j,1,inf)) * (%e)^(%i * (6 * k + 1) * t)/((2 * k + (1/3))^2)),k, - n,n) See http://www.tweedledum.com/rwg/cog.htm Also http://gosper.org/fst.dvi Pix: http://gosper.org/fst1.png ,http://gosper.org/fst2.png , http://gosper.org/fst3.png ,http://gosper.org/fst4.png . (For parametric, take Re and Im.) Rich's base 4 recursion loops for rational arg. Find the fixed point of the pending stack ops to simulate infinite recursion, and get an exact val on the curve. Equate to this Fourier series for a *non-obvious* identity. --rwg > If you view the starting triangle as running from 0 to 3, > and write the fraction base 4, you can use the > fraction digits to select which part of the (sub)edge > you are on. The orientation of the edge can be calculated > from the 1s and 2s of the high order portion of the fraction, > and that should give a somewhat kludgy sum for the (x,y) > coordinates of the image point. > RWG likely has some nice FFT clock sum expression. > > Rich > > > -----Original Message----- > From: math-fun-bounces(a)mailman.xmission.com [mailto: math-fun-bounces(a)mailman.xmission.com] On > Behalf Of N. J. A. Sloane > Sent: Thursday, February 25, 2010 12:43 PM > To: math-fun(a)mailman.xmission.com > Cc: njas(a)research.att.com > Subject: Re: [math-fun] parametrizing the Koch snowflake curve > > Jim Propp asked: > >> Does anyone (Bill Gosper, maybe?) know an explicit parametrization > of the Koch snowflake curve? > > The Cantor set consists of the points 0 <= x <= 1 which have a base 3 "decimal" expansion that > does not contain a 1. Is there a similar characterization of the Koch snowflake? > > Neil Sloane > >

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[math-fun] That fractal pentagram again
by Bill Gosper 09 Apr '15

09 Apr '15

For some pictorial. mildly trick questions: http://gosper.org/foupent.pdf (1.6M) --rwg

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[math-fun] why can a line of slope 10^n determine n decimal digits of π?
by Gary Antonick 19 Jun '14

19 Jun '14

Just noticed this yesterday and can't figure out why it works. Draw a circle with radius 1 around 0,0. Draw a line with slope -10 that hits the circle at 1,0. This line will also hit the circle again and make a little arc. The length of the arc will be about 1/31 the perimeter of the circle. Now try again with a line of slope -100. The line will create an arc 1/314 of the circle. Starting to look a bit like pi. Am wondering why this is. In case I made a mistake here's what I'm trying to describe.. if the circle were a clock the first line would cut an arc from 15 minutes (3:00) to almost the 13 minute mark.

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[math-fun] sqrt(pi)
by Bill Gosper 24 Mar '13

24 Mar '13

Pairwise combining the terms of A&S 6.5.31 (upper Gamma CF), then adding last week's lower gamma CF: In[185]:= Equal[Gamma[ z], ((y^z*((1/(cfk[n*(z - n), -z + y + 2*n + 1, List[n, Infinity]] - z + y + 1)) + (1/(z + cfk[n*y, z - y + n, List[n, Infinity]] - y))))/(E^y))] z 1 Gamma[z] == (y (------------------------------------------- + -y + z + cfk[n y, n - y + z, {n, Infinity}] 1 y -----------------------------------------------------------)) / E 1 + y - z + cfk[n (-n + z), 1 + 2 n + y - z, {n, Infinity}] where cfk:=ContinuedFractionK and y>0 can be chosen to optimize convergence. Specializing, In[186]:= % /. z -> 1/2 /. y -> 1 Out[186]= 1 Sqrt[Pi] == (-------------------------------------- + 1 1 -(-) + cfk[n, -(-) + n, {n, Infinity}] 2 2 1 ------------------------------------------) / E 3 1 3 - + cfk[(- - n) n, - + 2 n, {n, Infinity}] 2 2 2 In[192]:= N[List @@ %186 /. cfk[num_, den_, _] :> ContinuedFractionK[num, den, {n, 9}]] Out[192]= {1.77245, 1.77244} This is the first non-asymptotic sum or CF I've seen for sqrt(pi). But the sum behind the lowergamma cf has been there all along, so I must not have been paying attention. --rwg

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[math-fun] another (illustrated) commercial
by Bill Gosper 31 Oct '12

31 Oct '12

for matrix products <http://gosper.org/foumat.png>. It's a ridiculous png because Mac Mathematica 8+ simply evaporates when I try to save as pdf. If anyone wants the pdf, I believe Neil's Windows 7 Mma 8 will do the job. If anyone wants the 44.7M nb, I'll put it on gosper.org. --rwg

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[math-fun] World's fastest number game wows spectators and scientists
by Henry Baker 30 Oct '12

30 Oct '12

FYI -- http://www.guardian.co.uk/science/alexs-adventures-in-numberland/2012/oct/2… http://www.youtube.com/watch?v=7ktpme4xcoQ http://www.youtube.com/watch?v=p_Terdik7aA http://www.youtube.com/watch?v=_vGMsVirYKs World's fastest number game wows spectators and scientists Flash Anzan, in which contestants add up numbers with an imaginary abacus, reveals the astonishing potential of the human brain Â and makes for a breathtaking spectacle too As I mentioned in a previous post, a million Japanese children every year learn the abacus, which they call soroban. The high point of the abacus calendar is the All Japan Soroban Championship, which took place earlier this year in Kyoto. And the high point of the championship is the category called "Flash Anzan" Â which does not require an abacus at all. Or rather, it requires contestants to use the mental image of an abacus. Since when you get very good at the abacus it is possible to calculate simply by imagining one. In Flash Anzan, 15 numbers are flashed consecutively on a giant screen. Each number is between 100 and 999. The challenge is to add them up. Simple, right? Except the numbers are flashed so fast you can barely read them. I was at this year's championship to see Takeo Sasano, a school clerk in his 30s, break his own world record: he got the correct answer when the numbers were flashed in 1.70 seconds. In the clip below, taken shortly before, the 15 numbers flash in 1.85 seconds. The speed is so fast I doubt you can even read one of the numbers. Click here to see a longer clip from 2011, showing Sasano break the world record at 1.80 seconds. Note that the format of the competition is a bit like an arithmetical version of a spelling bee. The remaining contestants are sitting in chairs. The numbers are flashed. The contestants write down their answers and exchange papers for marking. The result is displayed on the screen, and those who got the correct answer stand up. Flash Anzan was invented a few years ago by an abacus teacher, Yoji Miyamoto, who wanted to design a maths game that was only solvable by calculation with an imaginary abacus, a skill known as anzan. When the contestant sees the first number he or she instantly visualizes the number on the imaginary abacus. When they see the second number they instantly add it to the number already visualized, and so on. At the end of the game the contestants cannot remember any of the numbers, or the intermediate sums. They only retain the final answer on the imaginary abacus. Performing arithmetic using an imaginary abacus is the fastest way to perform mental calculations. Earlier this year the Japanese Naofumi Ogasawara won the Mental Calculation World Cup using the technique. The previous winner, Priyashi Somani, from India, did too. When I returned to London, I met up with Brian Butterworth, professor of cognitive neuropsychology at University College London and the author of The Mathematical Brain, and showed him some video clips of Flash Anzan. He was flabbergasted. "I don't see how you can represent whatever that number was on a mental abacus faster than you can say it," he said, adding: "A lot of money should be spent doing research on how the brain can manage to do this, because I think this is a really extraordinary thing!" In 2008 I visited Yoji Miyamoto's abacus school in Tokyo and he showed me another fascinating aspect of Flash Anzan. Since abacus and anzan calculation use a different part of the brain from pencil and paper arithmetic, as discussed here, it is possible to play language games while playing Flash Anzan. In this clip the two girls are adding up 30 digits in 20 seconds while simultaneously playing "shiritori", a Japanese game in which you must say a word beginning with the last syllable of the previous word. If you want to practise Flash Anzan there are several sites on the internet and some mobile apps. How about a British contestant at the All Japan Soroban Championship in 2013? I was in Japan making a radio documentary about numeracy, Land of the Rising Sums, which will be broadcast on BBC Radio 4 at 11am on Monday October 29. Posted by Alex Bellos Monday 29 October 2012 03.09 EDT guardian.co.uk

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[math-fun] Look-and-saay (kind of) self-describing array -- with even terms
by Eric Angelini 29 Oct '12

29 Oct '12

Hello Math-Fun, I've drawn this, last Saturday: 2 2 4 2 2 4 6 2 4 4 2 6 8 2 6 4 4 6 2 8 10 2 8 4 6 6 4 8 2 10 12 2 10 4 8 6 6 8 4 10 2 12 14 2 12 4 10 6 8 8 6 10 4 12 2 14 16 2 14 4 12 6 10 8 8 10 6 12 4 14 2 16 18 2 16 4 14 6 12 8 10 10 8 12 6 14 4 16 2 18 20 2 18 4 16 6 14 8 12 10 10 12 8 14 6 16 4 18 2 20 ... Top line says: I see two integers "2" so far Next line says: I see four integers "2" and two integers "4" so far Next line says: I see six integers "2", four integers "4" and two integers "6" so far etc. The counts are thus cumulative. And the (easy-to-build) array infinite. Who will produce a similar array (self-describing and infinite), containing at least one odd integer? I couldn't. Best, É.

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[math-fun] Applicants fail Williams Formula 1 math test
by Henry Baker 28 Oct '12

28 Oct '12

FYI -- I guess it would be fruitless to ask about differential equations. http://www.telegraph.co.uk/finance/jobs/9638334/Williams-F1-struggling-to-f… E.g., 1. 44 + 12 + 35 = ? 2. 735 - 576 = ? 3. 7523 x 8 = ? 4. 7713 / 9 = ? 5. 2 + 3 = ? 6. Write 0.3 as a fraction 7. The square of 7 is = ? 8. The cube of 2 is = ? 9. What is 11% of 250 = ? 10. Increase 64 by 12.5% = ? 11. Find the value of x^2 + 2xy + y^2 where x = 6 and y = -3 12. 5/16 / 1/8 = ? 13. If log 3 = 0.4771 and log 4 = 0.6020, log 40 = ? 14. If log 3 = 0.4771 and log 4 = 0.6020, log 0.12 = ? 15. The cube root of 27 is = ?

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Re: [math-fun] another (illustrated) commercial
by Fred lunnon 27 Oct '12

27 Oct '12

Suppose I should have thought of that --- thanks! WFL On 10/27/12, Hans Havermann <gladhobo(a)teksavvy.com> wrote: >> Tantalising postage stamps, at 9% of fullsize --- which Firefox in >> its wisdom has decided is the best I'm going to see! > > Most web browsers are preferenced to shrink large pictures to window > size until one clicks on the picture. > > Hans >

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[math-fun] Math paper generator
by Eric Angelini 26 Oct '12

26 Oct '12

... this is pure math-fun, http://thatsmathematics.com/mathgen/ Best, É.

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