February 2016 - mathfun@mailman.xmission.com

[math-fun] π puzzle
by Bill Gosper 19 Feb '16

19 Feb '16

How does this work? In[787]:= Nest[#/(Sqrt[1 - #] + Sqrt[1 + #]) &, 1`33, 98] 2^99 Out[787]= 3.141592653589793 (I confess not understanding where half the significance goes. Forcing Mathematica to retain 33 digits throughout only rescues three or four.) --rwg

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[math-fun] computational complexity of "reciprocated subset sum"
by Warren D Smith 18 Feb '16

18 Feb '16

Given N numbers x1, x2, ..., xN, I want to compute the sum 1/(x1+x2+..+xN) + 1/() + 1/() ... where my sum has 2^N - 1 summands, one for each nonempty subset of {1,2,3,...,N}, and the denominators are the sums of the x's with indices in each subset. Obviously, my sum can be computed in O(2^N) operations. I want to know if there is any way to do it faster.

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[math-fun] San Jose Math Decagon
by Bill Gosper 17 Feb '16

17 Feb '16

A couple of years ago Julian and I showed some animations to the San Jose Math Circle. gosper.org/decazoom.gif suggests why the area of a circle is the radius times half the circumference, and displays the continued fraction of a very lame approximation to π. --rwg

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[math-fun] Draft of my February, 2016 blog post
by James Propp 16 Feb '16

16 Feb '16

I've posted a draft of my essay "When Not to Expect What You're Expecting" and would appreciate comments: https://mathenchant.wordpress.com?p=562&shareadraft=56c20a66c9fbd <https://mathenchant.wordpress.com?p=482&shareadraft=568d547f0934f> If you send comments via WordPress, keep in mind that they're anonymous by default, which may suit you or may not. I plan to publish, as always, on the 17th. Jim Propp

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[math-fun] If circles were square, π would = 2√2
by Bill Gosper 15 Feb '16

15 Feb '16

gosper.org/pi=2rt2.gif --rwg

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[math-fun] Linkification of A=f(bh)
by Hilarie Orman 15 Feb '16

15 Feb '16

It is easier to understand the theorem than the url encoding. Email readers, when presented with odd urls, can be stumped when trying to make them clickable. Using "reserved" characters (per rfc3986) in urls is asking for trouble. The equals sign and the parens are in the "reserved as delimiters" set, and it is standard procedure to "percent encode" them if they are supposed to be interpreted as ordinary characters. In my personal email code, I assume that parens are not part of a url. It is likely that the sender failed to terminate the url with whitespace and instead started a parenthetical comment after the url. This happens with "." and "," and ";" all the time. Hilarie

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[math-fun] 3-D Euclidean geometry: exscribed tetrahedron puzzle
by Fred Lunnon 15 Feb '16

15 Feb '16

The following problem (or rather its solution) constitutes the crux of what appears to be a new result in elementary 3-space Euclidean geometry. However its solution proves so neat that I'm posing it on the list, ahead of the solution to follow in a few days. [A handy excuse for further procrastination over the next instalment of Poncelet pontification.] Gloss (0) : Given an arbitrary tetrahedron A , denote by B the inscribed tetrahedron with vertices the centroids of faces of A . It is well known that B is similar to A , but scaled down to 1/3 ; and an analogous result holds for any dimension n , with scale-down 1/n . Given A and a chosen vertex A_0 , a tetrahedron is `exscribed' when its vertices meet the extended face planes of A , but lie outside face plane F_0 opposite A_0 . Problem (1) : Construct explicitly some exscribed C congruent to A . For example, grey A and yellow C in diagram https://www.dropbox.com/s/blcihoao8t0k3lh/exscribe2.png Problem (2) : Generalise the result to a simplex in Euclidean n-space, for n >= 3 . Question (3) : Investigate whether C is the smallest possible exscribed on F_0 and similar to A . (Probably, but I don't know either.)

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[math-fun] 8x8 = 5x13
by Bill Gosper 15 Feb '16

15 Feb '16

You've all seen this, but maybe your kids haven't: gosper.org/8x8=5x13.gif --rwg (Preserving old Macsyma animations while I still can.) (With the help of a suggestion by Howard Cannon that I'm annoyed at not having thought of myself.)

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[math-fun] gear generator
by Stuart Anderson 15 Feb '16

15 Feb '16

http://geargenerator.com/#200,200,100,6,1,4,269.0999999999988,5,1,8,2,4,27,… Matthias Wandel's Gear Generator is a tool for creating involute spur gears <http://en.wikipedia.org/wiki/Involute_gear> and downloading them in SVG format. In addition it lets you compose full gear layouts with connected gears to design multiple gears system with control of the input/output ratio and rotation speed. Gears can be animated with various speed to demonstrate the working mechanism. video of Windows version of Gear generator https://www.youtube.com/watch?v=9MycFQKnZkA -------------------------- Stuart Anderson

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Re: [math-fun] [math - fun] A = bh/2
by Bill Gosper 14 Feb '16

14 Feb '16

Gene, My post has nothing to do with parallelograms. It simply says any two triangles with the same base and height can be trivially interdissected. So one of them might as well be half a rectangle. --rwg On 2016-02-14 15:31, Eugene Salamin via math-fun wrote: > Bill, > I didn't see at all how the animated picture illustrated your post. > It would be much more clear if you could present a fixed picture of > the parallelogram with a line bisecting it into the two triangles, and > a picture of a parallelogram with a line indicating the triangle to be > chopped off. > > -- Gene > > > > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Sent: Sunday, February 14, 2016 1:33 PM > Subject: [math-fun] A=bh/2 > > Remember when they taught you that the area of a triangle is half > the area of the parallelogram you get by gluing two copies together? > And the area of the parallelogram is the area of the rectangle you get > by chopping a triangle off one end and gluing it on the other? Did > anybody complain that this doesn't work for sufficiently skewed > parallelograms? Fix: gosper.org/A=f(bh).gif --rwg

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