September 2016 - mathfun@mailman.xmission.com

[math-fun] "What Big Bang?" --Pascal
by Bill Gosper 11 Sep '16

11 Sep '16

Math-fun moderator Rich Schroeppel had an early mentor who was a Mathematical Creationist. He taught that, in the beginning, the Void was a bilaterally infinite string of cells containing zeros, and every day, God would simultaneously replace each cell with the sum of its immediate neighbors. Then, exactly once, God accidentally said "0+0=1", and Pascal's Triangle exploded into existence. A similarly quaint belief is still taught in the Book of Wolfram. The actual reason God paused on the seventh day was to gaze back over the previous eternity, In[686]:= FoldList[#2 - #1 &, 0, {0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 0, 0, 0}] Out[686]= {0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0, 0} In[687]:= FoldList[#2 - #1 &, 0, %] Out[687]= {0, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 0, 0, 0} In[688]:= FoldList[#2 - #1 &, 0, %] Out[688]= {0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0} In[689]:= FoldList[#2 - #1 &, 0, %] Out[689]= {0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0} In[690]:= FoldList[#2 - #1 &, 0, %] Out[690]= {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} In[691]:= FoldList[#2 - #1 &, 0, %] Out[691]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} In[692]:= FoldList[#2 - #1 &, 0, %] Out[692]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1} In[693]:= FoldList[#2 - #1 &, 0, %] Out[693]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12} In[694]:= FoldList[#2 - #1 &, 0, %] Out[694]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 6, -10, 15, -21, 28, -36, 45, -55, 66, -78} In[695]:= FoldList[#2 - #1 &, 0, %] Out[695]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -4, 10, -20, 35, -56, 84, -120, 165, -220, 286, -364} . . . satisfied that time, like space, was bilaterally infinite, and miracle-free. --rwg

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Re: [math-fun] A distant branch of my family is named Bellows.
by Bill Gosper 11 Sep '16

11 Sep '16

On 2016-09-10 22:00, Hans Havermann wrote: >> Including one who, honest to God, was christened Sniffin Kelly Bellows. > > Is this the one (Delaware County NY, 1842-1928) who died of "an > accumulation of gas"? No. He was indeed from upstate NY, but died in the 70s or 80s of an accumulation of carcinogens from home-rolled cigarettes. --rwg

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[math-fun] A distant branch of my family is named Bellows.
by Bill Gosper 10 Sep '16

10 Sep '16

Including one who, honest to God, was christened Sniffin Kelly Bellows. But they were not mathematicians--the Bellows Theorem is actually about bellows. It states that some nonconvex polyhedra (e.g. http://gregegan.customer.netspace.net.au/SCIENCE/Steffen/Steffen.html) can actually flex, but not breathe. Thus, PolyhedronData["Steffen"] could produce a tetradecahedron GraphicsComplex with a symbolic parameter which would wash out under the "Volume" operation. Corollary: A geometrically (and musically) ideal accordion would be silent. --rwg

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[math-fun] A problem about finite sets
by Victor Miller 09 Sep '16

09 Sep '16

Say that a finite set of integers, A, has *property P* is there are two non-empty disjoint subsets S and T of A, such that sum(S) = sum(T) (where sum(X) is the sum of the elements of X). 1) Show that every 10 element subset of {1,...,100} has property P (this is fairly easy). 2) Is it true that every 9 element subset of {1,...,100} has property P? Victor

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[math-fun] Hadamard matrix update
by Veit Elser 09 Sep '16

09 Sep '16

All of you will be relieved to know there is an alternative hypothesis on the supply of Hadamard matrices. If we are willing to admit transient behavior at low orders, then the entropy of Hadamard matrices shows scaling behavior when we allow for it to grow as N^phi for an exponent phi less than 2. Again, using available data up to N = 32, a value of phi between 1.4 and 1.8 seems to be singled out. Here’s what I get for phi=1.6: N log_2(num(N)) / N^phi 4 1.04302 8 1.15610 12 1.20338 16 1.25669 20 1.25455 24 1.27412 28 1.25239 32 1.25612 There’s a strong transient up to N=12. What’s slightly remarkable is how the near convergence of the subsequent values strongly depends on phi. This hypothesis, should it be true, raises two points: (1) The exponent phi should be added to the list of “basic things about Hadamard matrices we have no clue about”. (2) Hadamard hunters may wish to cast a wider net. Almost all searches (such as for order N=668) have been for constructions based on sequences, that is, a much lower entropy source corresponding to phi=1. -Veit

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[math-fun] Epicycles
by James Propp 09 Sep '16

09 Sep '16

Has anyone gone back to the data compiled by early (pre-Kepler) astronomers and created better epicyclic approximations to planetary orbits using Fourier methods? Jim Propp

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[math-fun] Moore curve drawn with epicycles
by James Propp 09 Sep '16

09 Sep '16

Can Bill Gosper or anyone else explain what's going on with https://twitter.com/algoritmic/status/772699702064254976 ("Moore curve drawn with epicycles")? Jim Propp

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[math-fun] imminent world shortage of Hadamard matrices?
by Veit Elser 08 Sep '16

08 Sep '16

The Hadamard matrix conjecture holds that such matrices exist for all orders that are divisible by 4. After surveying what’s been done on the classification/enumeration of Hadamard matrices (e.g. http://neilsloane.com/hadamard/) I’ve felt that what’s humbling about the conjecture is that we lack the knowledge to prove the existence of even a single Hadamard (at each possible order) when the evidence points to a very rapid growth in their number. Now I’m not so sure. Let x = log_2(N), N = order of Hadamard (a multiple of 4), and y = (1/N^2)log_2(num(N)), where num(N) is the number of Hadamard matrices of order N. Sequence A206711 gives num(N) for N = 1, … , 32. If you plot y vs. x you get a very straight line: y = 0.78785 - 0.09458 x. Taking this literally, there should be a maximum in the number of Hadamard matrices at order N = 196, and beyond that the number plummets, vanishing at around N = 322. The available constructions (beyond this number) would then represent isolated points. -Veit

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Re: [math-fun] Gaa, engine BRAKing!
by Bill Gosper 06 Sep '16

06 Sep '16

On 2016-09-05 07:26, Henry Baker wrote: > You can thank the EPA and their 100,000 mile "no pollution", > "no maintenance" requirement. > > Although aimed at reducing pollution, the EPA requirements > had the effect of tightening up quality all throughout > the engine & air intake & exhaust systems. > > For example, you used to have to replace your muffler > every 15 minutes, and that is no longer required, either. > > Of course, this mandated increase in quality (including > crash tests, etc.) also had the effect of raising the > price of a minimal "automobile" from ~$5k to ~$20k. > > These regulations also had the (perhaps unintended) > effect of *standardizing* engines, drive trains, etc., > and suppressing innovation. Who could afford the cost > of proving compliance? E.g., look what happened to poor > Mazda with their Wankel rotary engine. (I'm not a huge > fan of the Wankel, but it did represent an attempt at > innovation.) > > One can only wonder at why it took the 21st C. & Tesla > before someone finally got serious about electric > vehicles. Perhaps heavy regulation exacts an > enormous toll on innovation? > > --- > > BTW, when was the last time you ever heard of a ball > bearing failing? High quality ball bearings are > now assumed, but prior to & during WWII, high quality > ball bearings were unobtainable. The U.S. put a huge > priority on bombing German ball bearing plants, because > they were so ubiquitous and so important. Don't forget the strategically critical Japanese pachinko industry: https://www.youtube.com/watch?v=CFs4zOC2Yg0 (More seriously: https://www.youtube.com/watch?v=19duYMdiXi0 ) --rwg > > At 08:07 PM 9/4/2016, Richard Howard wrote: >>Of course. >>Had to use special "break-in" oil. >> >>Now engines are machined so tightly when they come off the line that they >>use 0W oil--sort of like the consistency of water. >> >>--R >> >>On Sun, Sep 4, 2016 at 9:51 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >>> Who remembers when we had to break engines in? --rwg > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Gaa, engine BRAKing!
by Henry Baker 05 Sep '16

05 Sep '16

You can thank the EPA and their 100,000 mile "no pollution", "no maintenance" requirement. Although aimed at reducing pollution, the EPA requirements had the effect of tightening up quality all throughout the engine & air intake & exhaust systems. For example, you used to have to replace your muffler every 15 minutes, and that is no longer required, either. Of course, this mandated increase in quality (including crash tests, etc.) also had the effect of raising the price of a minimal "automobile" from ~$5k to ~$20k. These regulations also had the (perhaps unintended) effect of *standardizing* engines, drive trains, etc., and suppressing innovation. Who could afford the cost of proving compliance? E.g., look what happened to poor Mazda with their Wankel rotary engine. (I'm not a huge fan of the Wankel, but it did represent an attempt at innovation.) One can only wonder at why it took the 21st C. & Tesla before someone finally got serious about electric vehicles. Perhaps heavy regulation exacts an enormous toll on innovation? --- BTW, when was the last time you ever heard of a ball bearing failing? High quality ball bearings are now assumed, but prior to & during WWII, high quality ball bearings were unobtainable. The U.S. put a huge priority on bombing German ball bearing plants, because they were so ubiquitous and so important. At 08:07 PM 9/4/2016, Richard Howard wrote: >Of course. >Had to use special "break-in" oil. > >Now engines are machined so tightly when they come off the line that they >use 0W oil--sort of like the consistency of water. > >--R > >On Sun, Sep 4, 2016 at 9:51 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> Who remembers when we had to break engines in? --rwg

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