June 2015 - mathfun@mailman.xmission.com

26 Jun '15

To whom it may concern. Here is a game, Leap Frog, which may not be original. It is played on a strip of squares. A move is to place a bean on an unoccupied square. So far we have one of the many manifestations of She-Loves-Me-She-Loves-Me-Not, but there is a second type of move. A player may jump a bean onto an unocupied square, provided there is at least one intervening square and all such intervening squares are occupied by beans. To ensure that the Ending Condition is satisfied, we will say that such jumping moves may not be reversed. If you make that ``may not be IMMEDIATELY reversed'' or ``may not be reversed by the opposite player'' then these versions may be of interest, but I suspect that ties occur for n > 5. For 0 < n < 5 the game is equivalent to S-L-M-S-L-M-N, since jumping moves need not arise. For example, for n = 4, 1 & 3 are good replies to one another, as are 2 & 4. n = 5 is an N-position. For example, placing a bean on square 3 wins. The moves 1 & 5 are good replies to each other, while 2 (or 4) can be met by 1 (or 5), since the jumping moves 2 to 4 or 1 to 4 are met by the jump 3 to 5. n = 6 is beginning to get difficult by hand. Over to you! Fame & fortune await a complete analysis! R.

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[math-fun] Possible use for a" spacefilling curve"
by Bill Gosper 25 Jun '15

25 Jun '15

(A simple zigzag would probably do.) Silting up: https://en.wikipedia.org/?title=Three_Gorges_Dam#Erosion_and_sedimentation Keep the water behind it moving at high speed by confinement to a long, tortuous course. The confining structures could be flimsy compared to the actual dam. --rwg

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[math-fun] Pegg's unit Heawood
by Warren D Smith 25 Jun '15

25 Jun '15

I suggest you color the two kinds of vertices red & blue, not merely red. Excellent embedding.

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[math-fun] Pisot & Salem numbers
by Warren D Smith 25 Jun '15

25 Jun '15

As JA pointed out, at the end of https://en.wikipedia.org/wiki/Rauzy_fractal they say "For any unimodular substitution of Pisot type (??meaning what??), which verifies a coincidence condition (apparently always verified), one can construct a similar set called "Rauzy fractal of the map". They all display self-similarity and generate, for the examples below, a periodic tiling of the plane." Then they give 4 pictures, which you can click on to see enlargements. These pictures seem rather unconvincing to me, by the way. My own alleged tiling ("example #1") based on the "plastic number" 1.3247179572... presumably is of that ilk? Google then suggested that WP Thurston found my plastic-number-based tiling, or something like it, in the 1980s. It also found this paper: http://math.tsukuba.ac.jp/~akiyama/papers/Min_pisot.pdf which in fig 1 and fig 2 page 5 gives pictures of a tile arising from the "plastic number." Their allowed bitstrings forbid 11, 101, 1001, and 10001 as substrings? This sure does not look like it tiles plane by translation in a periodic way, but it does tile nonperiodically via 3 tile-scalings. They claim the Haussdorf dimension of the boundary of this tile is 1.10026... Wikipedia also claims the Rauzy tile will "tile the plane periodically by translation" in addition to the nonperiodic tiling I had in mind, involving 3 tile scalings and both translation & rotation. You can see pictures of both kinds of Rauzy tilings here: http://www.cant.ulg.ac.be/cant2009/lec-as1.pdf The "cut and project" method of generating a quasicrystal is, 1. start with some point-lattice in a higher dimension. 2. make a plane, or anyhow affine subspace, which is "irrationally oriented" with respect to the lattice directions. 3. Consider the lattice points nearby (e.g. within distance K of) that subspace. 4. project them orthogonally into the subspace. 5. result is a point set in your desired dimension, which is aperiodic, obeys point-density upper and lower bounds, but acts periodic in many approximate ways, e.g. fourier transform. This is simplest to think about for 1-dimensional quasicrystal arising from 2-diml lattice. I have heard claims that every quasicrystal can be regarded as arising in this way. To the extent that is true, "quasicrystallography" is the same thing as "crystallography with higher dimensions permitted." This view seems to have been adopted by at least one international society for crystallographic froobozzles. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Fano Plane puzzle
by David Wilson 24 Jun '15

24 Jun '15

Let F be the Fano Plane. Let FL be the set of 7 lines in F. Let FP be the set of 7 points in F. For L in FL, let P(L) be the set of 3 points on L. Now let SP be a set of 7 points in general position in R^3. Let m : FP ó SP be a bijection. For each line of L of F, let C(L) be the circle in R^3 through the 3 points m(P(L)). Let SL = C(FL). Let S = (SL, SP). S is then a model of the Fano Plane in R^3 with circles for Fano lines and points for Fano points. If we scale S so that the largest circle has radius 1, how large can we make the radius of the smallest circle of S by judicious choice of SP?

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[math-fun] Imaginary gotchas
by Bill Gosper 24 Jun '15

24 Jun '15

An identity you can't formally conjugate: Out[9]= ProductLog[1,-1/2-I Log[π]/π]==I π/2-Log[π] In[10]:= Conjugate/@% Out[10]= Conjugate[ProductLog[1,-1/2-I Log[π]/π]]==-I π/2-Log[π] Why did it chicken out on the lhs? OK, we'll do it manually. In[11]:= %%/.I->-I Out[11]= ProductLog[1,-1/2-I Log[π]/π]==I π/2-Log[π] Nothing happened! This is because I is not a Symbol: In[12]:= FullForm[{-I,I/2}] Out[12]//FullForm= List[Complex[0,-1],Complex[0,Rational[1,2]]] OK, dammit, In[13]:= %%/.Complex[x_,y_]->x-I*y Out[13]= False Whaddya mean False?? OK, paralyze the = sign: In[14]:= MapAt[HoldForm,%%%,0] Out[14]= Equal[ProductLog[1,-1/2-I Log[π]/π],I π/2-Log[π]] In[15]:= %/.Complex[x_,y_]->x-I*y Out[15]= Equal[ProductLog[1,-1/2+I Log[π]/π],-I π/2-Log[π]] Yes! Why was this so hard? Now unfreeze the = sign: In[16]:= ReleaseHold[%] Out[16]= False To quote a certain duck, WAK! Spoiler: Conjugating this requires a change of branch: In[17]:= ReleaseHold[MapAt[-#&,%%,{1,1}]] Out[17]= ProductLog[-1,-1/2+I Log[π]/π]==-I π/2-Log[π] which is what In[10] Conjugate would have said, had it been smarter. --rwg

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[math-fun] Graphene for desaliniation membranes
by Henry Baker 24 Jun '15

24 Jun '15

FYI -- We were talking about desalinization technologies & energies a few weeks ago. http://www.technologyreview.com/questionoftheweek/538596/what-will-it-take-… What Will It Take to Commercialize Better Desalination Technologies? An update on research into whether graphene could make it easier to remove salt from seawater. By Kristin Majcher on June 18, 2015 A rendering of the structure of graphene. http://www.technologyreview.com/sites/default/files/images/graphenex299.jpg The World Health Organization estimates that 750 million people lack access to safe water sources, while many more have insufficient supplies because of droughts like the one in California. Creating fresh water through desalination of the ocean or briny water on land is still expensive, mainly because of the energy required to push water through membranes that filter out the salt (see ÂHow Can Desalination Become Cheaper?Â and ÂDesalination Out of DesperationÂ). In recent months, scientists have published research that advances our understanding of the prospects for better membranes made out of the superstrong and lightweight material graphene. http://www.technologyreview.com/questionoftheweek/532891/how-can-desalinati… http://www.technologyreview.com/featuredstory/533446/desalination-out-of-de… Single crystals In the March 23 issue of Nature Nanotechnology, researchers at Oak Ridge National Laboratory described how they made a graphene membrane for desalination from layering carbon atoms in a honeycomb structure. These atoms form a hexagon-shaped crystal that measures about 0.1 millimeters in width and length, with holes smaller than a nanometer designed to let water through and block salt. A desalination membrane made of this graphene crystal would ideally have to be measured in meters to work in a commercial plant, says Oak Ridge researcher Ivan Vlassiouk. He says the team has scaled this membrane material up to several millimeters so far. http://www.nature.com/articles/nnano.2015.37 Sealing leaks MIT researchers have shown it is possible to use sheets of graphene as a desalination membrane by attaching it to a polycarbonate support structure. However, defects tend to form in the graphene, which can weaken the membrane and possibly let salt or other contaminants through. In an April 27 Nano Letters paper, the MIT researchers and authors from Oak Ridge and Saudi ArabiaÂs King Fahd University of Petroleum and Minerals appear to have found a way to fix this. The team filled in the larger defects with nylon and deposited hafnium metal followed by a layer of oxide to smaller defect areas. http://pubs.acs.org/doi/abs/10.1021/acs.nanolett.5b00456 Computer design GrapheneÂs strength, thinness, and chemical properties could make it the ÂultimateÂ membrane material for desalination, MIT researchers David Cohen-Tanugi and Jeffrey Grossman wrote in a paper reviewing some of the progress researchers have made. They describe how computer simulations have helped researchers understand the chemical properties of the nanopores poked into the material; how water flows through these openings; and how well the membranes can retain their strength over time. The paper was featured in the June 15 issue of Desalination. http://www.sciencedirect.com/science/article/pii/S001191641500003X? This next link gets you a copy of the full paper. (The following link is correct, but the file is really a pdf file. Change the file extension to .pdf after downloading to be able to read this article. http://zeppola.mit.edu/pubs/David_Desal_2015 ) The Takeaway: Oak RidgeÂs Vlassiouk says it could take at least a decade to commercialize graphene desalination. While graphene could theoretically make for membranes that process water more quickly with less energy, the cost savings that would be associated with using them remain unclear. Do you have a big question? Send suggestions to questionoftheweek(a)technologyreview.com. --------------------------- https://newsoffice.mit.edu/2012/graphene-water-desalination-0702 Massachusetts Institute of Technology A new approach to water desalination Graphene sheets with precisely controlled pores have potential to purify water more efficiently than existing methods. David L. Chandler, MIT News Office July 2, 2012 The availability of fresh water is dwindling in many parts of the world, a problem that is expected to grow with populations. One promising source of potable water is the worldÂs virtually limitless supply of seawater, but so far desalination technology has been too expensive for widespread use. Now, MIT researchers have come up with a new approach using a different kind of filtration material: sheets of graphene, a one-atom-thick form of the element carbon, which they say can be far more efficient and possibly less expensive than existing desalination systems. When water molecules (red and white) and sodium and chlorine ions (green and purple) in saltwater, on the right, encounter a sheet of graphene (pale blue, center) perforated by holes of the right size, the water passes through (left side), but the sodium and chlorine of the salt are blocked. ÂThere are not that many people working on desalination from a materials point of view,Â says Jeffrey Grossman, the Carl Richard Soderberg Associate Professor of Power Engineering in MITÂs Department of Materials Science and Engineering, who is the senior author of a paper describing the new process in the journal Nano Letters. Grossman and graduate student David Cohen-Tanugi, who is the lead author of the paper, aimed to Âcontrol the properties of the material down to the atomic level,Â producing a graphene sheet perforated with precisely sized holes. They also added other elements to the material, causing the edges of these minuscule openings to interact chemically with water molecules Â either repelling or attracting them. ÂWe were very pleasantly surprisedÂ by how well graphene performed compared to existing systems in computer simulations, Grossman says. One common method of desalination, called reverse osmosis, uses membranes to filter the salt from the water. But these systems require extremely high pressure Â and hence, energy use Â to force water through the thick membranes, which are about a thousand times thicker than graphene. The new graphene system operates at much lower pressure, and thus could purify water at far lower cost, the researchers say. While reverse osmosis has been used for decades, Âreally basic mechanisms of separating salt from water are not well understood, and they are very complex,Â Cohen-Tanugi says, adding that itÂs very difficult to do experiments at the scale of individual molecules and ions. But the new graphene-based system, he says, works Âhundreds of times faster than current techniques, with the same pressureÂ Â or, alternatively, the system could run at similar rates to present systems, but with lower pressure. The key to the new process is very precise control over the size of the holes in the graphene sheet. ÂThereÂs a sweet spot, but itÂs very small,Â Grossman says Â between pores so large that salt could pass through and ones so small that water molecules would be blocked. The ideal size is just about one nanometer, or one billionth of a meter, he says. If the holes are just a bit smaller Â 0.7 nanometers Â the water wonÂt flow through at all. Other research groups have worked to create pores in graphene, Cohen-Tanugi says, but at very different sizes and for very different purposes Â for example, making much bigger holes to filter large molecules such as DNA, or to separate different kinds of gases. The methods used for those processes were not precise enough to make the tiny holes needed for desalination, he says, but more advanced techniques Â such as helium-ion bombardment to make precise holes in graphene, chemical etching and self-assembling systems Â might be suitable. For now, Grossman and Cohen-Tanugi have been doing computer simulations of the process to determine its optimal characteristics. ÂWe will begin working on prototypes this summer,Â Grossman says. Because graphene is the subject of research into many different applications, there has been a great deal of work on finding ways of making it inexpensively and in large quantities. And for desalination, because graphene is such a strong material Â pound for pound, itÂs the strongest material known Â the membranes should be more durable than those presently used for reverse osmosis, Grossman says. In addition, the material needed for desalination does not need to be nearly as pure as for electronic or optical uses, he says: ÂA few defects donÂt matter, as long as they donÂt open it upÂ so that salt could pass through. Joshua Schrier, an assistant professor of chemistry at Haverford College, says, ÂPrevious simulations had studied the flow of water through very small holes in graphene, and the design of pores that selectively allow ion passage, but Â despite the social and engineering relevance to desalination Â nobody had thought to examine the intersection of these two fields.Â The work by the MIT team could open a whole new approach to desalination, he says. Schrier adds, ÂManufacturing the very precise pore structures that are found in this paper will be difficult to do on a large scale with existing methods.Â However, he says, Âthe predictions are exciting enough that they should motivate chemical engineers to perform more detailed economic analyses of Â water desalination with these types of materials.Â The work was funded by the MIT Energy Initiative and a John S. Hennessy Fellowship, and used computer resources from the National Energy Research Scientific Computing Center. Topics: GrapheneMaterials scienceMIT Energy Initiative (MITEI)Nanoscience and nanotechnologyWaterDesalinationResearchJeffrey GrossmanResearch Laboratory of Electronics Related Paper: "Water Desalination across Nanoporous Graphene" Jeffrey Grossman David Cohen-Tanugi Research Laboratory of Electronics Department of Materials Science and Engineering MIT Energy Initiative

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[math-fun] To demonstrate a multipronged identity
by Bill Gosper 24 Jun '15

24 Jun '15

when FullSimplify can't canonicalize down to a single form, e.g., IIn[296]:= FullSimplify[%, H > 0 && H \[Element] Integers] Out[296]= {1/8 Sinh[H Log[3 + 2 Sqrt[2]]]^2, 1/16 (-1 + Cosh[H Log[17 + 12 Sqrt[2]]]), 1/8 Sinh[2 H ArcSinh[1]]^2, ChebyshevU[-1 + H, 3]^2} you could just subtract one from the rest and hope FullSimplify finds, in this case, three zeroes. Alternatively, you can really punish FullSimplify for its laziness: In[298]:= FullSimplify[%.{x - y, y - z, z - w, w - x}] Out[298]= 0 Proof: In[77]:= Reduce[ForAll[{x, y, z, w}, {a, b, c, d}.{x - y, y - z, z - w, w - x} == 0], {a, b, c, d}] Out[77]= b == a && c == a && d == a Or In[78]:= Resolve[ForAll[{x, y, z, w}, {a, b, c, d}.{x - y, y - z, z - w, w - x} == 0]] Out[78]= a - b == 0 && b - c == 0 && a - d == 0 && c - d == 0 --rwg

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Re: [math-fun] Pisot & Salem numbers
by Warren D Smith 24 Jun '15

24 Jun '15

Each Pisot number arises as a limit point (and from both sides) of Salem numbers. But yes to JA: I should have excluded (or included, anyhow said something special about) Pisot numbers when I was writing that characterization of Salem numbers. Siegel also showed the 2nd smallest Pisot number was 1.38027756 obeying x^4=x^3+1. Apparently it is not known whether the Pisot numbers are the full set of Salem limit points, or whether there are more. Also, amazingly enough, it is known the Pisot numbers are a closed set. It seems clear Pisot and Salem numbers have a lot to do with both recursive tilings and quasicrystals. But I certainly do not fully understand any of these things.

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Re: [math-fun] Fano Plane puzzle
by Bill Gosper 24 Jun '15

24 Jun '15

Is it possible to get that close point (https://www.dropbox.com/s/ymf2f9k4mcgmhac/fano7pt7rg.gif?dl=0) to actually lie on the circle? Then it would lie on four circles, and be the fourth point of one of them. Presumably this fails the definition of a Fano Plane, but does it have the (projective?) properties that motivated the definition? Seems to me like an interesting object, if it exists. --rwg On 6/23/15, Dan Asimov <asimov(a)msri.org> wrote: By the way, is this 4th point's closeness really a problem? WFL> It's largely an aesthetic matter: such collisions do make the diagram difficult to interpret. And, is it really close in R^3 or only close in the displayed projection? WFL> Note that all configurations so far constructed by myself are _planar_! [ DWW has raised the question of whether they might be "shaken out" (as it were) so as to properly occupy 3-space --- which appears likely, though I do not currently know for sure. ] DanA> Just clicked on this but got a 404 "The file you are looking for has either moved or been deleted: error." Synching error apparently --- try instead https://www.dropbox.com/s/jeyqj76ng4z42s7/fano7pt7rg_2.gif?dl=0 WFL

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