February 2017 - mathfun@mailman.xmission.com

Re: [math-fun] discrete curve shortening flow
by Dan Asimov 20 Feb '17

20 Feb '17

Thing is, the discrete approximation to a polygon's "integral curvature" (w.r.t. arclength) at a vertex is independent of which sequence of smooth curves is used to approach the polygon near the vertex. (The integral is [see below].)* But for the integral of *squared* curvature, that is not the case. On Feb. 19, 2017, at 8:55 PM, Veit Elser <ve10(a)cornell.edu> wrote: ----- If you’re mostly interested in discrete approximations to continuous curves, I would use the integral of the squared curvature as the energy. When applied to 2D surfaces it is called the “Willmore energy”, and topological spheres always flow to the round sphere. I suspect the same happens to curves, although the curve will expand while it becomes round (the energy is not scale invariant in 1D as it is in 2D). There has been quite a lot of research on finding good approximations of curvature for triangulated surfaces. Brakke’s Surface Evolver does a nice job for 2D surfaces and I believe it also can be used for curves. > On Feb 18, 2017, at 9:09 PM, Mike Speciner <ms(a)alum.mit.edu> wrote: ----- > If one considers a polygon as a discrete approximation to a continuous closed curve, then each vertex has influence over a whole section of the curve. So it is not unreasonable to be less local when assigning a direction and 'curvature' at the vertex, dependent on the neighboring vertices. Whether there's a natural way to do this (and perhaps determine a 'best" continuous curve being approximated) might be an interesting question. ----- ----- —Dan —————————————————————————— * It's the exterior angle.

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Re: [math-fun] discrete curve shortening flow
by Mike Speciner 19 Feb '17

19 Feb '17

If one considers a polygon as a discrete approximation to a continuous closed curve, then each vertex has influence over a whole section of the curve. So it is not unreasonable to be less local when assigning a direction and 'curvature' at the vertex, dependent on the neighboring vertices. Whether there's a natural way to do this (and perhaps determine a 'best" continuous curve being approximated) might be an interesting question. Not that I disagree with your notion of fairness. --ms On 18-Feb-17 20:23, Warren D Smith wrote: > Your rectangle example seems damning. > ANY vector "curvature" > which always has the direction of the angle-bisector, > is going to be killed by your rectangle example. > > And I really do not think it is fair to use any other direction! -- > Unless you abandon all pretense this has anything to do with > "motion of the boundary in the 'inward normal direction' > (which is a locally-defined notion) > with velocity proportional to 'curvature' > (which is a locally-defined notion)." > > So that's kind of cute, actually. I mean, the thing works for smooth > curves, but depends upon smoothness, and it works for simplices, > which are not smooth, but that is seen to be special. > >

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[math-fun] Catalan's remains mysterious
by Bill Gosper 19 Feb '17

19 Feb '17

Date: 2017-02-19 00:30 From: Joerg Arndt <arndt(a)jjj.de> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> Just put a 1 into the upper parameters to cancel that lower hidden 1: (arcsin(z))^2 = z^2 * F([1, 1, 1], [3/2, 2], z^2) At least that's what my notes say (fxtbook, p.700). Best regards, jj Sorry, I was too cryptic. I was simply asking where this is in DLMF. It "belongs" in http://dlmf.nist.gov/15.4, but that bogus 1 disqualifies it. Look how trivial it is as a product matrix: 2 2 k z 1 ------- - 1 + 2 k k 0 1 But it doesn't have the obligatory k+1 in the 1,1 denominator. So it gets shunned. --rwg * Bill Gosper <billgosper(a)gmail.com> [Feb 19. 2017 08:50]: > I just fixed an egregious paste-o in gosper.org/pathi.pdf giving a second, > spurious value for > ζ(2,¼) instead of the intended ζ(2,¾), leading to an absurd "closed form" > for Catalan's constant. > --rwg > (I also wasted > ½ an hr trying to figure out why the new pdf didn't > replace the old one at > gosper.org. Eventually it did.) > > DLMF question: Due to the braindead hidden 1 in ₂F₁ notation, it can't > express arcsin². So > where in DLMF is the ₃F₂?

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[math-fun] Catalan's remains mysterious
by Bill Gosper 19 Feb '17

19 Feb '17

I just fixed an egregious paste-o in gosper.org/pathi.pdf giving a second, spurious value for ζ(2,¼) instead of the intended ζ(2,¾), leading to an absurd "closed form" for Catalan's constant. --rwg (I also wasted > ½ an hr trying to figure out why the new pdf didn't replace the old one at gosper.org. Eventually it did.) DLMF question: Due to the braindead hidden 1 in ₂F₁ notation, it can't express arcsin². So where in DLMF is the ₃F₂?

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Re: [math-fun] discrete curve shortening flow
by Mike Speciner 19 Feb '17

19 Feb '17

Warren, nice analysis for the triangle case. But we'd like something that also worked for polygons with more than 3 vertices. My first email stated that using csc(alpha/2) for 'curvature' at a vertex would keep polygons self-similar as they shrunk, but that is false. Consider a rectangle--the shorter sides would collapse to a point before the longer sides. We could get self-similarity (at least for the rectangle case) if we defined the vector curvature at a vertex as the sum of the two vector sides from that vertex. Clearly a nonconvex polygon would not remain self-similar. Is there some definition of [vector] curvature that would cause arbitrary polygons to circularize? Maybe a sum of the two vectors weighted by their relative lengths, or some convex function of their relative lengths? I hacked up a little demo at https://ms0.github.io/polycirc.html --ms On 17-Feb-17 12:19, Warren D Smith wrote: > The distance D from a triangle vertex to the triangle's incenter is > D = csc(alpha/2) > if the inradius=1. > > So I presume you meant that alpha is > the vertex angle, not pi-alpha, > in your claim about csc(alpha/2) being the velocity > of the vertex's infall? > > Then the most natural candidate functions would clearly > seem to be csc(alpha/2)^2 > and exp(csc(alpha/2)). > > Any F(csc(alpha/2)) > where F(x) is a continuous concave-U monotonically increasing function > will cause the triangle to become continually "rounder" during the flow, > where "roundness" is the variance of the vertex-incenter distances > if triangle scaling normalized so inradius=1 always, and is 0 > (roundest possible) for an equilateral triangle, else positive. > > With this normalization the process continues forever, > and roundness continues to improve monotonically forever. > Can the process ever get "hung up" approaching from above, but > never reaching, some positive roundness value? > > No. Because by continuity there would have to be a limit-triangle > in that case, which would be a fixed point of the flow. > It is easy to see that to be such a limit-triangle you > need to have all three incenter-vertex distances equal. > The only triangle for which that is true is equilateral. > > So ANY such flow, for any such F, always equilateralizes > triangles. > QED > > >

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[math-fun] multimagie.com update
by Christian Boyer 17 Feb '17

17 Feb '17

www.multimagie.com updated, with more info on the latest solved enigma. There remain eight unsolved enigmas, open since 2010, for winning €6,500 (~$6,900) and eight bottles of champagne. All of them are on small squares, from 3x3 to 6x6. Christian.

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[math-fun] Fwd: Draft of my February 2017 blog post
by Bill Gosper 16 Feb '17

16 Feb '17

Date: 2017-02-15 01:31 From: Hans Havermann <gladhobo(a)bell.net> To: wclark(a)mail.usf.edu, math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> W. Edwin Clark: "It would be interesting to see a reference to a publication that denies that 1 is a factor of every integer." The 19th century arithmetic texts that I had occasion to look at all agreed that both the multiplicand and multiplier of a product were factors and generally gave examples in pairs that excluded the number itself multiplied by one. Edward Liddell in his Arithmetic for the Use of Schools (1860) made it explicit: "We have already learnt that when 6 and 8 are multiplied together they make the product 48. For this reason 6 and 8 are called factors of 48, from the Latin facio, which signifies to make. We may therefore separate or resolve 48 into the two factors 6 and 8, or into the two factors 4 and 12. The figure 1 is not regarded as a factor; consequently, in whole numbers, all factors are greater than unity." Eric Weisstein cites Ore's 1988 Number Theory and its History and Burton's 1989 Elementary Number Theory for his number theoretic usage that "a factor of a number n is equivalent to a divisor of n". He adds: "In elementary education, the term 'factor' is sometimes used to mean proper divisor, i.e., a factor of n other than the number itself. However, as a result of the confusion this practice creates and its inconsistency with the mathematical literature, it should be discouraged." _____________ Gary Snethen (privately) wrote: Bill, Google has 13,900 hit for "prime numbers have two factors" and only 3 hits for "prime numbers have one factor". Two factors: https://www.google.com/webhp?sourceid=chrome-instant&ion=1& espv=2&ie=UTF-8#q=%22prime+numbers+have+two+factors%22 One factor: https://www.google.com/webhp?sourceid=chrome-instant&ion=1& espv=2&ie=UTF-8#q=%22prime+numbers+have+one+factor%22 Dictionaries are rapidly conceding defeat to the vernacular. In a post-truth world, perhaps mathematics will ultimately do the same. ;) On Tue, Feb 14, 2017 at 9:16 AM, Bill Gosper <billgosper(a)gmail.com> (privately) wrote: > There's a crucial distinction between *factor* and *divisor*! 1 can not > be a factor of anything, not even 1! > In the multiplicative world, 1 is the empty product--a 0th power--the > product of zero things. How many > factors of 1 go into 6? 1 is a divisor of everything and a factor of > nothing. That's why it's not a prime. --Bill > It looks too late to turn this ship around. But by the reasoning that 1 is a multiplicative factor, then 0 is an additive part, and the function that counts unrestricted partitions needs to be increased everywhere by 1. --rwg

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[math-fun] discrete curve shortening flow
by Mike Speciner 16 Feb '17

16 Feb '17

So, I was judging a science fair the other day, and one of the entrants was essentially this [ http://math.mit.edu/research/highschool/primes/materials/2016/Cohen-Rowley.… ], describing a discrete analog of curve shortening flow, where points on a closed curve are moved normal to the curve and proportional to their curvature, with the [known] theorem being that the curves tend toward a circle and shrink to a point. The discrete version just moves the vertices of a polygon. The discrete analog they chose for curvature was pi - alpha, where alpha is the vertex angle, and they showed that the result, starting with a triangle, was either a line or a point, depending on certain conditions about the triangle. It [almost] immediately occurred to me that, for triangles, if you used the distance to the incenter as the curvature, the triangle would remain similar to itself and shrink to a point. Essentially, the "curvature" would be csc alpha/2. And, in fact, this would work for convex polygons as well. But of course, an asymmetric polygon would remain self-similar and so asymmetric. So, is there a discrete analog of curvature which would cause a polygon to "circularize" as it shrunk to a point, and if so, is it in any sense unique? Maybe something like (csc alpha/2)**2

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[math-fun] Draft of my February 2017 blog post
by Bill Gosper 15 Feb '17

15 Feb '17

I agree with Gene: Say more about the continued fraction. How it appears to obey Gauss's lg(1+R) distribution, with infinite expected arithmetic mean but geometric mean = Khinchin's constant. Vs the ancient Chinese value of √10 with arithmetic mean = geometric mean = 6. But if we could only prove lg(1+R), or even much less than that, we would establish the "obvious" facts that π+e, π-e, π e, etc are all irrational. For e, the arithmetic mean and geometric mean are *both* infinite, as would be a e + b for any rational a and b. --rwg The Beckmann book uses "circle ratio" to unambiguously specify π. Date: 2017-02-11 19:23 From: Eugene Salamin via math-fun <math-fun(a)mailman.xmission.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: Eugene Salamin <gene_salamin(a)yahoo.com>, math-fun < math-fun(a)mailman.xmission.com> My comments on this draft. 1. There is no such thing as a pi that varies with the choice of geometry or the size of a circle. Pi is the mathematical constant 3.14..., and that's it by definition. Sure, the ratio circumference/diameter can be different, but pi is the ratio in Euclidean geometry. You do a disservice to your readers by going against the standard usage of the Mathematical community. 2. I agree with the unnamed mathematician that calculating a gazillion decimal digits of pi is a waste of time. There's nothing special about radix 10. Far more interesting would be the continued fraction. 3. "But thanks to Einstein, we now know that the universe we live in is curved at galactic scales, and there’s no natural way to view it as a 3-dimensional curved hypersurface sitting inside an uncurved 4-dimensional space." Actually, general relativity posits a 4-dimensional curved spacetime, and not a curved 3-dimensional space. Furthermore, the invariant metric on spacetime has a (3,1) signature rather than the Euclidean (4,0). 4. Another pi reference: https://smile.amazon.com/History-Pi-Petr-Beckmann/dp/0312381859/ref=sr_1_1?… From: James Propp <jamespropp(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Saturday, February 11, 2017 3:10 PM Subject: [math-fun] Draft of my February 2017 blog post Hi, I started writing a new draft titled "Three-point-one cheers for pi !" and would love to get your feedback. I plan on publishing it on the 17th. I can always use more links to content (images or videos) that are likely to interest readers of my essay. I am particularly interested in a link that would present Archimedes' method of calculating the volume of a ball in an accessible way. For that matter, if anyone has risen to Wigner's tacit challenge of explaining to a lay audience why pi is relevant to statistics, I'd love to know about it. Keep in mind that all math-fun feedback goes into one mail-feed, so I won't know whose feedback is whose unless you sign your comment. Also, all substantive suggestions that I use will be acknowledged (unless you specifically ask me not to do this). Please leave your feedback here: https://mathenchant.wordpress.com?p=1471&shareadraft=589f987f95cf7 Title: Three-point-one cheers for pi ! Beginning: Pi, that most celebrated of mathematical constants, leads a curiously double life. On the one hand, we define it as the ratio of the circumference of a circle to that circle's diameter, and, to the extent that we can imagine a world draped over the armature of a different geometry, we can conceive ... Read more: https://mathenchant.wordpress.com?p=1471&shareadraft=589f987f95cf7 Thanks, Jim Propp

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[math-fun] Draft of my February 2017 blog post
by Bill Gosper 15 Feb '17

15 Feb '17

Date: 2017-02-14 09:16 From: Nick Baxter <nickb(a)baxterweb.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> >ok, someone has to ask...factor and divisor are the same thing; Argh! I had no idea how many people believe this! >why are you implying otherwise? Because they're different! > I know there is potential confusion with prime factor and factorization, but they're different issues. >Nick Primality isn't the issue. How many factors of 6 are there in 12? How many factors of 1? 1 is not a goddam factor of anything. --rwg

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