June 2015 - mathfun@mailman.xmission.com

[math-fun] The "God problem" -- designing the universe as a linear programming problem
by Warren D Smith 21 Jun '15

21 Jun '15

Suppose you are God. You want to design a universe. (You may not believe in God, intelligent design, etc. I'm ignoring that debate here as much as I can. I'm just trying to investigate the problems you would face if you were such a God.) Well, almost all the known physics formulas out there, are of the form SomethingImportant = PRODUCT_j (FundamentalConstant_j)^(R_j) where the R_j are known rational numbers, and the FundamentalConstants are things like "electron mass" or "pi." In the rare cases where formula is not of this form, it is usually because it is a sum of two such formulas, in which case you usually can either (a) neglect one of the two terms, or (b) optimize some design parameter which trades off one term versus the other. Either way, the formula then becomes of our kind. Now, if you are a God with certain Goals (such as, creating humans) then you need to choose the FundamentalConstants in such a way that various inequalities are obeyed that enable those goals. Such as, the lifetime of a star capable of incubating life, needs to exceed the time it takes for humans to evolve. The mass of such a star, needs to exceed the minimum possible star-mass but be below the maximum possible star-mass. Etc. The observation I want to make is: this is a LINEAR PROGRAMMING PROBLEM. I.e. the logs of the fundamental constants are the variables, and the inequalities are constraints. If you now want to assess "how likely" our life-friendly universe was, or how likely other observed properties of universe are, so you can compute how much of a "miracle" it all is, then you can do so by computing polytope volume ratios. One volume being, say, the parameter-vectors yielding a life-friendly universe, To try to do that, I made this collection of physics formulas: https://dl.dropboxusercontent.com/u/3507527/WDSphysicsFormulas.html -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Three inherently aperiodic and fractal tilings
by Warren D Smith 21 Jun '15

21 Jun '15

--Dear delinquents: if you are going to find problems with what I say, I'd appreciate hearing it directly. RWG: Kids. You can't tell them anything these days. Here's Julian trash-talking to some of his fellow young delinquents: Julian Ziegler Hunts wrote: Sure, but first I'd like to point out that he has not proved that any of these tile the plane, and I particularly do not believe that #1 does. He shows that each is the union of scaled copies of itself, but this does not suffice to prove that it tiles the plane—you would also need to prove that the tile has nonempty interior (the negligible-intersection part of being a tiling would follow). It is true that the scaling involved in the constructions would guarantee the tile has dimension 2, but only if the intersection between the scaled copies is negligible (being of smaller dimension would suffice, as would having measure zero if the whole has positive measure (which is not guaranteed by having dimension 2!)). And having dimension 2 does not guarantee that it has non-empty interior or will tile the plane. A simple counterexample is to remove all points with rational coordinates from the unit square; it is then the union of four non-intersecting half-scale copies of itself but does not tile the plane. Julian --In the meantime, I also was developing skepticism for related reasons. Things are not so easy. To reach nirvana, it seems to me we want to prove the tile T has an interior which is both nonempty and arc-connected. If so, then the claim that T is tiled by some suitably-scaled copies, should recursively provide a tiling of the whole plane, as I said. But sadly I did not provide such proof. (All I proved was: T is bounded, and T is tiled by some suitable scaled copies.) Nor did I even compute good pictures. However, Roth's theorem suggests that the points generated by my SUM (if truncated at some finite #digits) are never too close together. http://mathworld.wolfram.com/RothsTheorem.html https://en.wikipedia.org/wiki/Thue%E2%80%93Siegel%E2%80%93Roth_theorem I mean, in plain 1 dimensional binary, T = SUM(j>=0) 2^(-j) b_j, the summands if the SUM is truncated to 0<=j<n are 2^(-n) apart. In certain 2 dimensional binary systems like T = SUM(j>=0) sqrt(-2)^(-j) b_j they are about 2^(-n/2) apart. With an algebraic number radix, those things are again going to happen to within a factor of 2^o(n), note the LITTLE o... is that right? (Also, mine are not merely algebraic, they also arise from Pisot or Salem numbers, which probably provide extra juice for arguments of this kind.) This muttering about Roth seems to me to be inadequate to prove tiling of the plane, but likely *is* adequate to see that the tile I have created, is not a Cantor set or something like that which would obviously have been a failure. If you are skeptical about my tile #1 (which actually was the one I was least skeptical about) then I'd like to know: why is the Rauzy fractal an ok tile, but my tile #1 is not? https://en.wikipedia.org/wiki/Rauzy_fractal (Or maybe Rauzy is not actually ok either. But pictures have been drawn for it and it sure looks ok.) So, bottom line: I do not know if my tiles really tile, but it seems to me that if they have nonempty arc-connected interior, then they must. That seems most likely to me in cases #1 and #3. I'd like to see pictures. Final question: Is it possible to create an "einstein" tile T (which tiles the plane by itself, but only in an aperiodic way) as T = SUM(j>=0) R^(-j) d_j where (1) each d_j is selected from a fixed set of K complex numbers in all possible ways, (2) R is a fixed algebraic complex number with |R|=sqrt(K) got by multiplying the unit-norm root of a "Salem polynomial" by sqrt(K). It has been a long-open question whether an einstein tile exists. This construction might provide one with (say) K=5. But you'd need to understand more than I do about when this construction method works. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Hackenbush applet?
by James Propp 20 Jun '15

20 Jun '15

Does anyone know of a browser applet or smartphone app that lets a human play Red-Blue Hackenbush against a computer? One nice way to convince people that a certain Hackenbush position H has value 1/2 would to give them a chance to play the game consisting of m copies of H plus n copies of a game with value -1, for various values of m and n, sometimes playing Red and sometimes playing Blue. Jim Propp

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[math-fun] Fwd: Three inherently aperiodic and fractal tilings
by Bill Gosper 20 Jun '15

20 Jun '15

WDS wrote: ********Fractal Tile #1:********* Let R obey R^3=R+1, R=-0.662358978622373+0.562279512062301*i arg(R)=139.671923191730 degrees |R|=0.86883696183271 |R|^2=0.75487766625 |R|^(-2)=1.32471795724474602596091 which is the real root of the original cubic, and wikipedia calls this the "plastic number." Because of the theorem that every non-integer algebraic integer is irrational, we know all roots of this cubic are irrational. Hence |R| and |R|^2 are irrational. Hence also any integer linear combination of the roots of the cubic, is irrational or zero. Then the "linear forms in logarithms" theorem by Alan Baker shows arg(R) is irrational measured in degrees, also. Then let tile T be defined by T = SUM(j>=0) R^j b_j where the bits b_j each are either 0 or 1, and the allowed bit sequences b0, b1, b2, ... are defined recursively by appending either 10 or 100 to the left of an allowed bit sequence. The number Q(N) of N-bit long allowed sequences then obeys the recurrence Q(N)=Q(N-2)+Q(N-3) and lim Q(N)^(1/N)=1.3247... Obviously T is a bounded set. Then T is tiled by T*R^2 and T*R^3, two rescaled & rotated copies of itself. By recursing forever, it follows that the three tiles T, T*R, T*R^2 together suffice to tile the plane. T is fractal in shape. The irrationality of arg(R) shows this tiling is necessarily aperiodic. ****************<WDS Kids. You can't tell them anything these days. Here's Julian trash-talking to some of his fellow young delinquents: On 2015-06-19 09:55, Julian Ziegler Hunts wrote: Sure, but first I'd like to point out that he has not proved that any of these tile the plane, and I particularly do not believe that #1 does. He shows that each is the union of scaled copies of itself, but this does not suffice to prove that it tiles the plane—you would also need to prove that the tile has nonempty interior (the negligible-intersection part of being a tiling would follow). It is true that the scaling involved in the constructions would guarantee the tile has dimension 2, but only if the intersection between the scaled copies is negligible (being of smaller dimension would suffice, as would having measure zero if the whole has positive measure (which is not guaranteed by having dimension 2!)). And having dimension 2 does not guarantee that it has non-empty interior or will tile the plane. A simple counterexample is to remove all points with rational coordinates from the unit square; it is then the union of four non-intersecting half-scale copies of itself but does not tile the plane. Julian After In[2]:= SetAttributes[Root, Listable] In[3]:= Root[#^3 - # - 1, Range[3]] Out[3]= {Root[-1 - #1 + #1^3 &, 1], Root[-1 - #1 + #1^3 &, 2], Root[-1 - #1 + #1^3 &, 3]} In[4]:= N[%] Out[4]= {1.32471796, -0.662358979 - 0.562279512 I, -0.662358979 + 0.562279512 I} In[49]:= r = %3[[2]] Out[49]= Root[-1 - #1 + #1^3 &, 2] gosper.org/wdstile1.png (with indispensAble coaching from Julian) shows T/r separated by 2 from Tr + Tr^2 (where the actual separation appears to be 1/r) showing perhaps irremediable porosities and overwrites. --rwg

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Re: [math-fun] DIY penrose shower floor
by Bill Gosper 19 Jun '15

19 Jun '15

Questions: Exactly what happens to the iso"perimetric" quotient when you delete the interior faces of a Weaire-Phelan unit cell? (For all distinguishable translations.) What is the exact, analytic quotient for Kelvin's cell? (The ISCs don't even recognize 64*π/3/(1 + 2*Sqrt[3])^3.) (Shudder) ditto for the Weaire-Phelan? --rwg On 2015-06-10 18:37, James Propp wrote: > Where do things stand now? Is it still possible to get pseudo-aperiodic > toilet paper? > > And while we're on the subject: do Weaire and Phelan have any kind of > copyright on the Weaire-Phelan structure that they discovered (subsequently > incorporated into the Water Cube in Beijing)? > > Jim Propp > > On Wed, Jun 10, 2015 at 8:48 PM, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > >> I regret to have to confess was the culprit responsible for designing the >> modified tiling (periodic along one axis) --- ah --- behind this particular >> contribution to the artistic life of the twentieth century. >> >> Not for the first or last time, I failed received so much as a >> letter of thanks >> from the commercial enterprise concerned. [Though I was subsequently >> obliged to --- er --- come clean to Penrose about my part in the affair.] >> >> Moral: if you are approached by an individual with an enquiry of a >> suspiciously industrial nature, get things --- um --- down on paper >> beforehand! >> >> Fred Lunnon >> >> >> >> On 6/11/15, James Propp <jamespropp(a)gmail.com> wrote: >> > Was all the Penrose tiling toilet paper manufactured by Kimberly Clark in >> > the late 90s destroyed, or does some of it still survive? >> > >> > If there are still rolls out there, and the price isn't prohibitive, you >> > could install some (presumably for display purposes only) elsewhere in >> the >> > bathroom. >> > >> > If you don't know[ꞥ] the story of Sir Roger's lawsuit against Kimberly >> Clark, >> > I urge you to look it up. It features one especially memorable quote, >> from >> > David Bradley: "When it comes to the population of Great Britain being >> > invited by a multinational to wipe their bottoms on what appears to be a >> > work of a knight of the realm without his permission, then a last stand >> > must be made." >> > >> > Jim Propp >> > >> > On Wednesday, June 10, 2015, Dirk Lattermann <dlatt(a)alqualonde.de> >> wrote: >> > >> >> Finally, a year after we moved into our new (old) house, I managed to >> >> finish the penrose mosaic for the shower in our ground floor bathroom. >> >> >> >> I put two quick pictures, before the glass wall was installed, at >> >> >> >> http://folgenschwer.de/mosaik/ >> >> >> >> Thought some of you might enjoy, >> >> Dirk >> >>

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[math-fun] aperiodic parquet deformation
by Warren D Smith 18 Jun '15

18 Jun '15

N.Myrhvold: I also have the first (to my knowledge) aperiodic parquet deformation - a continuous transformation of one aperiodic tiling into another. It is on the walls of a room and wraps around the deformation parameter. --It is trivial to do this because the Penrose tile edges can be deformed into any "S curve" and the Penrose tiling still works. Penrose himself made a "birds" version in this way where each tile was a picture of a bird. But perhaps NM had something less trivial in mind (?).

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[math-fun] Three inherently aperiodic and fractal tilings
by Warren D Smith 18 Jun '15

18 Jun '15

Three Fractal Tiles =====Warren D. Smith====18 June 2015======== ********Fractal Tile #1:********* Let R obey R^3=R+1, R=-0.662358978622373+0.562279512062301*i arg(R)=139.671923191730 degrees |R|=0.86883696183271 |R|^2=0.75487766625 |R|^(-2)=1.32471795724474602596091 which is the real root of the original cubic, and wikipedia calls this the "plastic number." Because of the theorem that every non-integer algebraic integer is irrational, we know all roots of this cubic are irrational. Hence |R| and |R|^2 are irrational. Hence also any integer linear combination of the roots of the cubic, is irrational or zero. Then the "linear forms in logarithms" theorem by Alan Baker shows arg(R) is irrational measured in degrees, also. Then let tile T be defined by T = SUM(j>=0) R^j b_j where the bits b_j each are either 0 or 1, and the allowed bit sequences b0, b1, b2, ... are defined recursively by appending either 10 or 100 to the left of an allowed bit sequence. The number Q(N) of N-bit long allowed sequences then obeys the recurrence Q(N)=Q(N-2)+Q(N-3) and lim Q(N)^(1/N)=1.3247... Obviously T is a bounded set. Then T is tiled by T*R^2 and T*R^3, two rescaled & rotated copies of itself. By recursing forever, it follows that the three tiles T, T*R, T*R^2 together suffice to tile the plane. T is fractal in shape. The irrationality of arg(R) shows this tiling is necessarily aperiodic. ********Fractal Tile #2:********* Let R be a complex non-real root of x^8+3*x^6+4*x^4+2*x^2+1 = 0 with |R|^2=0.61803...=1/GoldenRatio and arg(R) is a nontrivial integer multiple of 72 degrees. (I have constructed this polynomial so those two things happen.) Define tile T by T = SUM(j>=0) R^j b_j where the bits b_j each are either 0 or 1, and the allowed bit sequences b0, b1, b2, ... are any binary sequence not containing two consecutive 1s. The number of N-bit long allowed sequences ending with a 1 at the most-significant end then is 1,1,2,3,5,8,13,21,... for N=1,2,3,... (Fibonacci numbers). Incidentally, it seems what I'd been calling the "Fibonaccci number system" is called "Zeckendorf's theorem" by wikipedia and "Zeckendorf Representation" by Weisstein, which is because it was invented way before Zeckendorf 1972, e.g. published by C.G. Lekkerkerker in Simon Stevin 29 (1952) 190-195. The allowed N-long sequences can be got by appending 1 to the left of an (N-1)-long one after replacing the 1 at its left end by 0; and appending 10 to the left of an (N-2)-long one. Then T is tiled by T*R and T*R^2, two rescaled & rotated copies of itself. By recursing forever, it follows that the two tiles T and S=T*R together suffice to tile the plane. T is fractal in shape. Is this tiling necessarily aperiodic? It does not involve any irrational (in degrees) rotations, all rotation angles are multiples of 72 degrees. But the two basic tiles T and S arise in the recursive process in numbers #T #S 1 0 to tile T 1 1 to tile T/R 2 1 to tile T/R^2 3 2 to tile T/R^3 5 3 to tile T/R^4 8 5 to tile T/R^5 etc, each number is sum of two immediately above it; i.e. the ratio of the two tile-counts approaches the golden ratio, which is irrational, hence yes, this tiling IS aperiodic. ********Fractal Tile #3:********* Let R be 1/sqrt(GoldenRatio) times 0.3319277206059928507595641669946106715942093770*i-0.9433048225750305942590992480932547489056220928 where the latter is the nonreal root of "Lehmer's polynomial" x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1 with |x|=1 and arg(x)=160.61417788097273775412563559435163395654262868 degrees. (I have not worked out the minimal polynomial of this R.) Now do the same construction as for Tile #2, just using this R instead of that R. Then T is tiled by T*R and T*R^2, two rescaled & rotated copies of itself. By recursing forever, it follows that the two tiles T and S=T*R together suffice to tile the plane. T is fractal in shape. Is this tiling necessarily aperiodic? Yes because it involves irrational angles of rotation... because it is known that the Lehmer root of unit norm is not a root of unity, we know arg(R) is irrational in degree measure. ********About Joerg Arndt*********** He clarified some of the things he'd said... JA's computer has found thousands of new tilings arising from "weird complex-radix number systems" in as-yet-unpublished work. He has several ways to describe each one. His counts range from thousands to hundreds of thousands depending exactly what you count. Probably some or all of the tilings I just invented, already were in Arndt's collection. Arndt also pointed out the "Rauzy fractal" https://en.wikipedia.org/wiki/Rauzy_fractal is also a tile of this ilk, 3 scaled copies of itself will tile the plane (also will tile itself) similarly in nature to my example #1 above. Wikipedia gives a picture. My examples #2 and #3 are "cooler than Penrose" in the sense they involve only 2 tiles, are inherently aperiodic, and only 1 tile shape. Warning: I have not drawn pictures of these tilings, I merely proved it works; so caveat emptor. Maybe somebody will draw pictures now?

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[math-fun] Decorating a pseudosphere
by James Propp 17 Jun '15

17 Jun '15

Can anyone suggest a good way to decorate a pseudosphere? Just putting a grid on it, as in the image at the top of Paolo Lazzarini's page http://www.paololazzarini.it/geometria_sulla_sfera/modelli_noneu_start.htm, isn't bad, but I'm looking for something snazzier to serve as the visual "brand" for my blog. Tim Hoffman's image http://www.math.kobe-u.ac.jp/DIVERSIONS/TIMH1.JPG appeals to me more, but I don't know what curves are being approximated. Can any of you figure out what's going on in this image? I like Figure 6' in I. Tadao's "Hyperbolic Non-Euclidean World and Figure-8 Knot" site http://web1.kcn.jp/hp28ah77/us20_pseu.htm, but I'm suspicious. Tadao writes "Fig. 6' shows that the red concentric circles are certainly at regular intervals and that all the white circles are surely the same in size and placed at normal, regular intervals." But is this really true? It's unclear to me why the even spacing assertion holds when one crosses the cut-line. In any case, what I'm hoping for ultimately is an animated gif showing a decorated half-pseudosphere rotating about a vertical symmetry axis (looking something like a sorcerer's cap). I imagined at first that one of Escher's circle-limit pictures could be transported from the Poincare disk to the pseudosphere, but I don't think that that can work. The pseudosphere is obtained from the Poincare disk by cutting out a part of it and then gluing parts of the boundary together; this gluing doesn't play nicely with the symmetry groups of reflection tilings of the hyperbolic plane. (Or maybe it does, and I'm not thinking clearly.) Ditto for Apollonian gaskets and the like. Jim Propp

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Re: [math-fun] Mathematical Enchantments
by Hilarie Orman 17 Jun '15

17 Jun '15

"Thrice attenuated shades of things" may mean that mathematics is written as lines (one dimension), but "things" are three dimensions. "Shades" is just poetic. Hilarie

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[math-fun] Mathematical Enchantments
by James Propp 16 Jun '15

16 Jun '15

A couple of days from now, I'll be launching a monthly blog called "Mathematical Enchantments". I'm hoping I'll get support from some folks in my mathematical-social network who can look at preliminary versions of my posts and make suggestions for how to improve them (or at least keep me from embarrassing myself too badly). Anyone who makes helpful comments on the preliminary version of one of my essays will be acknowledged at the end of the posted version. Do you have any general advice for me on the verge of this large undertaking? e.g., things you've especially liked (or disliked) in the work of math-popularizers in the past? One thing I urgently need is people willing to write spiffy demos (JavaScript would be best). Only a few of you do this, but maybe others of you know how to find people who do this sort of thing well? I'm hoping that fame (not fortune) will be an adequate enticement, especially for the compulsive coders among you. I will always welcome suggestions for topics, though for the first year years of the blog I'm all set. Fear No Spam: I don't plan to write to you again unless you reply (except perhaps if I'm posting on a topic that you have special knowledge about). I'd love to have lots of readers of preliminary versions, but if you just want to follow what I publish, go to http://mathenchant.org from time to time (starting this coming Wednesday evening). I may also create a podcast version; tips on starting a podcast series are also welcome. I've decided to put my blog on wordpress.com, at least for now; if any of you have wordpress experience, as relates to inclusion of mathematical equations and diagrams and animations, I'd appreciate your advice. Here's the zeroeth installment, which I hope to post on June 17. [picture of pseudosphere] At some moment in the past five years I realized that math, in addition to being the way I earn my living, is something even more important for my happiness: it's a consolation for living in a world without magic. By "magic" I don't mean make-believe magic, the sort you'll see in a stage-show; I mean _real_ magic -- the kind that you read about in books when you were a child, and hoped to encounter someday, but gradually stopped believing in as, year after year, the world you inhabit failed to deliver any evidence that magic existed. And by "math" I don't mean the inflexible cookbook math that too often is the only kind children learn; I mean the game that research mathematicians get to play, where they break those cookbook rules, adding new spices to old recipes and inventing entirely new ones. Lots of people (most notably Martin Gardner and more recently Arthur Benjamin, Persi Diaconis, Ron Graham, and Colm Mulahy) have talked and written about the links between math and magic tricks, but hardly anyone talks about the way that math, for many people who do research in it, satisfies a craving for the fantastic that most of us haven't outgrown (even if we've persuaded ourselves that we have). Indeed, I think that most children get glimpses, all too easily forgotten, of math as a wondrous ticket to other worlds. My goal in Mathematical Enchantments is to reawaken in my readers this childlike relationship to the subject, and to make this view of math enticing and even natural. And if you are an actual child, or an actual mathematician, and your sense of mathematical wonder is already awake and active, all the better! There'll be lots of new games you can play. These things are fun, and fun is good. I'll do my best not to assume that you have much mastery of math beyond number-sense and basic algebra. But I'll make heavy demands on your imagination, and your willingness to bend and stretch your mind into new shapes, or even to try to split your mind into two parts (more on this soon!). And I'll assume that you're curious about new ways of looking at familiar things, and bold enough to dance with new ideas that initially seem quite crazy, temporarily letting go of old ideas that get in the way of the dance (the letting go is usually harder than the dancing). Do you want to see a geometry in which there's a single point at infinity? Or another geometry that has infinitely many of them? Do you want to see a number system in which infinity is equal to its own negative? And another number system in which plus infinity and minus infinity are two different numbers? Are you intrigued by the prospect of a logic in which "It is false that P is false" is subtly different from the proposition "P is true"? Do you want to see a way of thinking about turn-taking games in which it makes sense to say you're ahead by exactly half a move? Or a geometry in which you have to make two full turns to be back in the orientation you started in? Or a geometry in which you change size, Alice-like, as you move through space? Do you want to learn how to count the elements of infinitely large sets, and learn why some of them, in a certain sense, have a negative number of elements? Then you'll probably enjoy this blog. I may occasionally touch on some ways in which math is relevant to our day-to-day world, such as the shape of snowflakes (why do the symmetrical ones all have six arms, never eight?), or beer cans (why did Budweiser redesign their can?), or traffic jams (what simple step could all of us take to make traffic jams evaporate more quickly?), or converting between miles and kilometers (how can memorizing the first few Fibonacci numbers help you do these conversions in your head?) --- or the way that some of the mind-bending possibilities I mentioned in the last paragraph are actually covertly at work in our world. But really, my true passion is not the ways mathematics helps us in this world, but the ways mathematics takes us out of it, expanding our imaginations beyond the pull of gravity and even the shackles of space and time. If you want to know why math is useful, read the recent books of Jordan Ellenberg and Steven Strogatz, and the many fine books by authors who preceded them. But if you ever get tired of living in just one world, and our world in particular, and secretly pine for a passport to other, weirder realms, I hope to be your monthly tour-planner. In this blog you'll meet some kids' games that are as fun (and challenging) for grownups as they are for children, a forgotten fractal from the nineteenth century, a simple model of sand piles that will probably perplex researchers for decades to come, high-dimensional spaces and the things that live there, and dozens of other fruits of the mathematical enterprise. And if you've ever wondered what the deal is with .999… (with infinitely many 9's after the decimal point, whatever that means!), well, I've got a whole bunch of ways to look at that one to offer you. I'll also tackle some deeper issues, such as, what is the nature of this wondrous mathematical realm? If it's "out there", why can't we see it? Or, if the facts of mathematics are all in our heads, why are they so obdurately unaffected by our wishes? And why do so many mathematical heads contain the same mathematical facts, if these facts are just figments? What do mathematicians mean by "proof", and why do they care so much about it? How does the human mind engage in mathematics? How should mathematics be taught? How do researchers in pure mathematics decide what projects to spend their time on, and when they are engaged in those projects, what the heck are they actually DOING? One model for what I'm trying to accomplish is the writings of Martin Gardner. Some other models are … well, actually, I'm not going tell you; I'd much rather imitate these writers and see if anyone notices what I'm up to. That's a game I'll be playing with you over the next few years. In the meantime, I'll mention that not all of my role-models wrote about mathematics or even about science. I'll close with a poem by Clarence Wylie whose final line compresses into ten syllables a surprisingly large chunk of what I'll be trying to convey in these many dozens of blog posts: PARADOX Not truth, nor certainty. These I forswore In my novitiate, as young men called To holy orders must abjure the world. “If …, then …,” this only I assert; And my successes are but pretty chains Linking twin doubts, for it is vain to ask If what I postulate be justified Or what I prove possess the stamp of fact. Yet bridges stand, and men no longer crawl In two dimensions. And such triumphs stem In no small measure from the power this game, Played with the thrice attenuated shades Of things, has over their originals. How frail the wand, but how profound the spell! Jim Propp

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