- mathfun@mailman.xmission.com - mailman.xmission.com

[math-fun] gyroelongated bifastigium
by Bill Gosper 14 Sep '16

14 Sep '16

is nonconvex, but the name is irresistible. gosper.org/gyroelongatedbifastigium.png Resistince is futile. --rwg

4 4

[math-fun] Packings of Regular Pentagons in the Plane
by Hans Havermann 14 Sep '16

14 Sep '16

The pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice (formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons) has now been proved. http://arxiv.org/abs/1602.07220

1 0

[math-fun] Although there's nothing wrong with my car's 3rd gear,
by Bill Gosper 13 Sep '16

13 Sep '16

I never use it when driving home along my street. I'm always in either 2nd or 4th until I reach my driveway. The nearest traffic light is miles away. Explain. --rwg

7 8

Re: [math-fun] Although there's nothing wrong with my car's 3rd gear,
by Bill Gosper 13 Sep '16

13 Sep '16

On 2016-09-13 16:02, Tom Karzes wrote: > Ok, second attempt: Could you be at the bottom of a hill, using 2nd > for braking when there are cars ahead of you? > > Tom > Partial credit. (I should have added that there's almost never a car in front of me. That could get me into 3rd, but I don't think it's ever happened.) Hilarie's been on my street and knows the answer. And NeilB for sure--from his driving lesson there! My street branches steeply off another steep street (with a stop sign at the top). Approaching from the top, there is a rollercoaster down, slightly left onto my street, then suddenly up. I shift straight from 1st to 4th on the down, shoot onto my street, and roll over the hill, which is shallow near my house. Approaching my street from the downhill side entails a sharp, steep right--a bit exciting even in 2nd gear--leaving inadequate rpm to upshift afterward. Near the inside curb there are several feet of nearly 20% grade. I'm trying to perfect a slinky that will function there. --rwg > Bill Gosper writes: > > I never use it when driving home along my street. I'm always in either 2nd > > or 4th until I > > reach my driveway. The nearest traffic light is miles away. Explain. > > --rwg

1 0

[math-fun] Eight prime square theorem?
by David Wilson 12 Sep '16

12 Sep '16

Lagrange's four square theorem states that every nonnegative integer is a sum of four squares. Empirically, it looks as if every nonnegative integer is a sum of 8 squares of primes (counting 1 as prime). It looks as if arbitrarily large numbers require 8 terms.

3 5

[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

4 4

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

3 4

[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

2 1

[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

8 9

[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

2 1