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Re: [math-fun] discrete curve shortening flow
by Mike Speciner 18 Feb '17

18 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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[math-fun] Draft of my February 2017 blog post
by Bill Gosper 14 Feb '17

14 Feb '17

Almost(?) as bad: The prestigious Nueva School is using "factor" to mean "divisor" (to avoid a three syllable word?) so that "1 is a factor of 6". I actually saw a TV quiz show screw a contestant out of $1M by switching whether 1 could be a factor. --rwg Date: 2017-02-13 18:38 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> It is unfortunate for someone knowledgeable in mathematics to lower himself to the level of the uninformed, instead of making an attempt to raise them up to a higher level. This reminds me of something I was told by a friend who volunteers in the Santa Cruz CA Public School. Apparently it's school doctrine that 1 is a prime number. The teacher told the class that 1 is prime, but added the qualification that mathematicians consider 1 to not be prime. Anyhow, James, it's your First Amendment right to say that π varies with the situation, and it's my First Amendment right to disagree with you. -- Gene

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[math-fun] Magic squares -- and surfaces
by Eric Angelini 13 Feb '17

13 Feb '17

This paper is interesting, I guess, for some of us: https://carresmagiques.blogspot.be/2017/01/area-magic-squares-and-tori-of-or der-3.html Best, É.

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[math-fun] Draft of my February 2017 blog post
by James Propp 11 Feb '17

11 Feb '17

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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Re: [math-fun] math-fun Digest, Vol 168, Issue 9 re: Lifted from Littlewood, hacked from Hacking (Fred Lunnon)
by Stuart Anderson 09 Feb '17

09 Feb '17

> > > > On Tuesday, February 7, 2017, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > > > I had seen this before, long ago, but then forgotten it. > > > > A square can be dissected into finitely many unequal squares, > > but a cube cannot be dissected into finitely many unequal cubes. > > > > Why not? > > > > > this is spot on. *From:* dmoews(a)xraysgi.ims.uconn.edu (David Moews)*Date:* 11 Mar 1998 19:29:42 -0800*Newsgroups:* rec.puzzles,sci.math*Subject:* Re: No CUBE of cubes. In article <6e7eb1$2am$2(a)cantuc.canterbury.ac.nz>, Bill Taylor <mathwft(a)math.canterbury.ac.nz> wrote: |There is a standard puzzle - show it is impossible to have a cube of integer |side partitioned into a finite number of other cubes, also of integer sides, |and all of DIFFERENT sizes. ... |Why doesn't the same proof extend to "no square of unequal smaller squares", |which we know IS possible. Why can't we just look at the top edge |and the smallest square there, go across, then carry on as before? ... |What's specially different between 2D and 3D here? In this posting, by an Xd X I will mean an X partitioned into a finite number of Xs of unequal sizes. The proof is as follows: Look at the bottom of the cubed cube; you will see a squared square. *The smallest square, S, in this squared square cannot be on the boundary.* Therefore, the cube with S for a face is surrounded by four larger cubes, so its opposite face abuts another squared square. We can now look at the smallest square in this squared square. It is then the face of some cube surrounded by four larger cubes, which we can look at the opposite face of, and so on: since this process can be continued indefinitely, there can be no cubed cube. The difference between 2D and 3D is in the starred statement. The smallest line in a segmented segment can be on the boundary, but the smallest square in a squared square cannot be. -- David Moews dmoews(a)xraysgi.ims.uconn.edu

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