September 2014 - mathfun@mailman.xmission.com

[math-fun] quadratic characters
by Warren D Smith 29 Sep '14

29 Sep '14

Baker's question was exactly what Lehmer's bicycle-chain sieve and related later machines (Univ. of Manitoba machine based on shift registers) were intended to solve.

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[math-fun] Fwd: gluing a 3-3-3 triangle onto a 3-5-7
by Bill Gosper 29 Sep '14

29 Sep '14

E.g.,In[162]:= triangs[144, 135, 31] Out[162]= {ArcCos[475/486], ArcCos[-(5/27)], ArcCos[7/18]} In[144]:= Rationalize[#, 99^-9] & /@ Ratios[%] Out[144]= {3400276089/412519223, 2/3} I.e., two of the angles of a 31:135:144 triangle are in ratio 2:3 . --rwg ---------- Forwarded message ---------- From: Mike Hirschhorn <m.hirschhorn(a)unsw.edu.au> Date: Sun, Sep 28, 2014 at 7:10 PM Subject: RE: gluing a 3-3-3 triangle onto a 3-5-7 To: "billgosper(a)gmail.com" <billgosper(a)gmail.com> Dear Bill, I wrote something related to this some years ago. Please see gosper.org/paper166.pdf. It appeared in the (British) Mathematical Gazette). Best wishes, Mike H.

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[math-fun] FullSimplify embarrassment contest
by Bill Gosper 29 Sep '14

29 Sep '14

What is the "simplest" 0 FullSimplify can't see? Extra credit if FunctionExpand fails too. I'll start the bidding with In[97]:= FullSimplify[FunctionExpand[Subtract @@ %]] Out[97]= ArcCos[-(7/9)] - 2 ArcSec[3] In[98]:= TrigExpand[Cos[%]] Out[98]= 1 --rwg

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[math-fun] The shape of pebbles
by Warren D Smith 29 Sep '14

29 Sep '14

Smooth pebbles on the beach are not spherical. They tend to be shaped more like oblate ellipsoids. But not axially symmetric. Obviously the population of shapes & sizes evolves as the water grinds them against each other. Is there one attractor-shape, or one at any given pebble-mass? Or can we understand the shape distribution? Here's a partial explanation. Spheres are unstable. If they get a bit oblate, then pebble will tend to lie on the flat side to lower its center of gravity, hence will grind itself flatter. But they don't get unboundedly flatter & flatter approaching a disc. Why not? Wear rate of surfaces (thickness loss per unit time) seems to be proportional to normal pressure times sliding speed. If the oblate does happen to tilt vertical, which will happen more often for disc like shapes, then get much higher pressure and probably also higher slide speed, which will grind it to become less oblate. So if it is too oblate it will evolve to be less so.

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Re: [math-fun] gluing a 3-3-3 triangle onto a 3-5-7
by Leo Broukhis 28 Sep '14

28 Sep '14

Michael Kleber writes: > http://www.mathpuzzle.com/triangle.html <http://www.mathpuzzle.com/triangle.html> says "As it happens, a [7, 7, 13] > triangle can be divided into two non-similar triangles with integer sides: > [3, 5, 7] and [3, 7, 8]", but does not mention the relation between the > two. Boggle. It does, see http://www.mathpuzzle.com/ITKK.gif there. Leo

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[math-fun] gluing a 3-3-3 triangle onto a 3-5-7
by Bill Gosper 28 Sep '14

28 Sep '14

makes a 3-7-8, thus both the latter have a rational angle. Did everybody but me already know this? --rwg

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Re: [math-fun] Julian's off to Cambridge
by Bill Gosper 28 Sep '14

28 Sep '14

Which school? Jim Propp (currently residing in the nearby town of Belmont, but emotionally still a resident of Cambridge) PS: Or did you mean Cambridge, England? Um, the Goucher reference was intended to suggest the Trinity flavor. Jack Holloway, Stu Nelson, Mike Beeler, & I (circa '66) rented a house in Belmont (from a Dr. Quigley). Across Maple Street from the IOOF lodge. I thought the address was 24, but IOOF is listed 16, so I'm confused. New England Telephone thoughtfully recycled the phone # of a popular local business: 484 3232 . We eventually gave up and started answering the phone "Plywood Ranch". This didn't sit well with N.E. Tel when they called to ask about our illicit phone extensions (which they detected remotely via line induction.) In those days it was illegal to connect your own "instrument" to the phone system. But what really got "the Phoneys" curious was Nelson making hundreds of touchtone calls to the now infamous Area Code 809, which was unlisted and just coming on line. He was trying to figure out where it was, and eventually heard an operator say "Charlotte Amalie". But NE Tel's billing department had no idea where 809 was, nor how to charge for it. --rwg On Saturday, September 27, 2014, Bill Gosper <billgosper(a)gmail.com> wrote: Hopefully he, like APGoucher, will be spared enough time to still have some math-fun. Before leaving, he and Timothy Ngotiaoco, another local whizz kid, have been helping out Dan Goldston: http://arxiv.org/abs/1409.2151 . --rwg

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[math-fun] Julian's off to Cambridge
by Bill Gosper 27 Sep '14

27 Sep '14

Hopefully he, like APGoucher, will be spared enough time to still have some math-fun. Before leaving, he and Timothy Ngotiaoco, another local whizz kid, have been helping out Dan Goldston: http://arxiv.org/abs/1409.2151 . --rwg

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[math-fun] Fwd: [n.wildberger@unsw.edu.au: Video of Jim and Norman's debate on infinity]
by meekerdb 27 Sep '14

27 Sep '14

An interesting debate (I think Norm won too). Brent ----- Forwarded message from Norman Wildberger <n.wildberger(a)unsw.edu.au> ----- Date: Fri, 26 Sep 2014 02:20:26 +0000 From: Norman Wildberger Subject: Video of Jim and Norman's debate on infinity Hi everyone, Earlier in the week Jim Franklin and I had a robust discussion on the topic `Infinity: does it exist??' in the Pure Maths seminar. Thanks to all who came and showed an interest in our opinions, and thanks for the many questions and comments afterwards. The video is now at the School YouTube site, at https://www.youtube.com/watch?v=5CiiGdaYEPU Unfortunately our microphone could not pick up audience questions and comments very well, and in fact our battery died near the end anyway. So apologies for those who asked interesting questions and comments, and we invite you to post them directly on the comments section of the video, where we can try to answer them! Best Regards, Norman Wildberger ----- End forwarded message -----

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[math-fun] Observation on Stern-Brocot tree
by David Wilson 26 Sep '14

26 Sep '14

It appears that the sum of the nth row of the Stern-Brocot Tree is (3*2^(n-2) - 1)/2. I assume this is known, but I don't know how to prove it.

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