June 2017 - mathfun@mailman.xmission.com

Re: [math-fun] More Minsky-Trinsky questions
by Henry Baker 06 Jun '17

06 Jun '17

This question might be related to my question re other algebras -- e.g., complex numbers (Gaussian integers). Speaking of which, the computer graphics folks have done wonderful things by using *quaternions* instead of *complex numbers* in their *fractal pix*. Perhaps a *quaternion* Minsky/Trinsky would provide some really fun & pretty pix, as well? At 12:02 PM 6/6/2017, James Propp wrote: >What if anything is known about Minsky-ish maps whose orbits approximate >hyperbolas rather than circles and ellipses?

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[math-fun] More Minsky-Trinsky questions
by James Propp 06 Jun '17

06 Jun '17

Is it ever possible to predict that two far-apart points will belong to the same orbit? Is it ever possible to predict that an orbit will exhibit 180-degree rotational symmetry? (This is related to the first question, since the orbit of (i,j) has 180-degree rotational symmetry iff (-i,-j) is in the same orbit as (i,j).) What if anything is known about Minsky-ish maps whose orbits approximate hyperbolas rather than circles and ellipses? Jim Propp

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Re: [math-fun] Minsky/Trinsky questions
by Henry Baker 06 Jun '17

06 Jun '17

Now you've lost me. The the video starts by talking about the Minsky algorithm over the integers. Where did the reals come in? Also, if you're calculating with floating point numbers, how can you be so sure that the algorithm is exactly invertible? At 01:15 AM 6/6/2017, Bill Gosper wrote: >Important: > >The algorithm described in the talk and the book is over the reals, not the integers.

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[math-fun] Defending infinity
by James Propp 05 Jun '17

05 Jun '17

In recent email conversation with a polite but insistent .999... dissenter, I've been made aware of strands of thought that rejects the mathematical concept of infinity as being unrealistic, incoherent, and repugnant (three different charges requiring three different sorts of defense). There are books and articles and on-line videos that approach the topic of infinity from different angles, but are there any that address these attacks head-on? It's possible that what's really being rejected (at least by some dissenters) is not the concept of infinity in isolation but the whole enterprise of making deductions in imaginary worlds (like the world of Euclidean geometry) to draw conclusions that are applicable in our world. A convincing defense of "unrealistic" concepts like a Euclidean point or the set of all counting numbers should in my opinion include an admission that we don't know why the mathematical enterprise has been so successful (the "unreasonable effectiveness" phenomenon). What's been done along these lines? One thing that's relevant, sort of, is an essay by Isaac Asimov in which young Isaac gets into an argument with a philosophy professor, challenging him to produce half a piece of chalk and then pointing out that it's probably not exactly HALF. But I think the philosopher was onto something when he got the whole in-class argument started by insisting that mathematicians are idealists because they believe in imaginary numbers. Old Isaac's point in his essaywas that complex numbers, properly understood, are no more fanciful than ordinary fractions; but you can turn this around and say that ordinary fractions, properly understood, are no less fanciful than complex numbers. Jim Propp

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Re: [math-fun] Defending infinity
by Henry Baker 05 Jun '17

05 Jun '17

Perhaps I'm a bit warped, but having had a fair amount of mathematical logic, what with axioms, theories, models and such, I no longer believe in numbers, infinities, etc., in ways that seem to tie many people up in knots. There are usually many different models for the same theory, so it becomes kind of pointless to worry about which one is "best", because different models produce different insights. If one person thinks of +oo as the limit "point" of the positive integers, that's great; if someone else wants to deny pointhood to +oo, but rather talk about some sort of a limit, that's also great. WRT .999... v. 1.000..., there are often congruence classes of representations that can't be distinguished with a certain class of operations -- this happens all the time in computer science when we have to choose a digital representation for an object, and keeping the object in a "canonical" form is (or seems to be) too much work -- e.g., a rational fraction representation that doesn't always "reduce to lowest common terms". At 07:45 AM 6/5/2017, James Propp wrote: >In recent email conversation with a polite but insistent .999... dissenter, >I've been made aware of strands of thought that rejects the mathematical >concept of infinity as being unrealistic, incoherent, and repugnant (three >different charges requiring three different sorts of defense). > >There are books and articles and on-line videos that approach the topic of >infinity from different angles, but are there any that address these >attacks head-on? > >It's possible that what's really being rejected (at least by some >dissenters) is not the concept of infinity in isolation but the whole >enterprise of making deductions in imaginary worlds (like the world of >Euclidean geometry) to draw conclusions that are applicable in our world. A >convincing defense of "unrealistic" concepts like a Euclidean point or the >set of all counting numbers should in my opinion include an admission that >we don't know why the mathematical enterprise has been so successful (the >"unreasonable effectiveness" phenomenon). > >What's been done along these lines? > >One thing that's relevant, sort of, is an essay by Isaac Asimov in which >young Isaac gets into an argument with a philosophy professor, challenging >him to produce half a piece of chalk and then pointing out that it's >probably not exactly HALF. But I think the philosopher was onto something >when he got the whole in-class argument started by insisting that >mathematicians are idealists because they believe in imaginary numbers. Old >Isaac's point in his essaywas that complex numbers, properly understood, >are no more fanciful than ordinary fractions; but you can turn this around >and say that ordinary fractions, properly understood, are no less fanciful >than complex numbers. > >Jim Propp

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[math-fun] Minsky/Trinsky questions
by Henry Baker 05 Jun '17

05 Jun '17

What is known about which orbit sizes are possible? What is known about which orbit sizes are impossible? How many bits of state are required for an orbit of size N ? (non-trivial max ?, non-trivial min ?) The video talks about parallelizing; is there any kind of "speedup" -- e.g., repeated squaring of matrices or some such? Suppose that a Minsky orbit of size N is known; how hard is it to map the integers 1..N to the elements of the orbit, and how hard is it to map back? Or is a table lookup the fastest method in both directions? More generally, can I use Minsky orbits for addition mod N ? Are there any Minsky orbits which "decompose" into direct products of Minsky orbits? (I'm not entirely certain how such a direct product would work, but perhaps the peculiarities of particular numbers might allow the separation of a "high order" part and a "low order" part that happen not to interfere with one another.) Does Minsky/Trinsky work on any other algebraic objects other than standard integers -- e.g., polynomials over Q in a single variable ? Perhaps I didn't understand the video well enough, but are there elegant cellular automata that perform the Minsky/Trinsky computation?

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[math-fun] Random thought
by David Wilson 05 Jun '17

05 Jun '17

It's surprisingly difficult to formulate natural-sounding sentences whose acronyms are words, e.g. "Have a nice day" -> hand.

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[math-fun] Climb to a Prime (counterexample)
by Hans Havermann 05 Jun '17

05 Jun '17

A few years ago John H. Conway offered five $1000 problems: http://www.cheswick.com/ches/conway1000.pdf I'll transcribe the fifth one (Climb to a Prime): _____ Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 2^2*3*5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat. So, for example, f(60) = f(2^2*3*5) = 2235. Next, because 2235 = 3*5*149, it maps, under f, to 35149, and since 35149 is prime, it maps to itself. Thus 60 -> 2235 -> 35149 -> 35149 -> ..., so we have climbed to a prime, and we stop there forever. The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 -> 225 -> 3252 -> 223271 -> ..., eventually getting to more than one hundred digits without yet reaching a prime! _____ A James Davis contacted me yesterday with word of a counterexample to Conway's conjecture (that every number eventually climbs to a prime) which makes him eligible (I think) for the prize. I told him to write Conway a letter. In the meantime, I'll publish his find here: 13532385396179 -> 13532385396179 -> composite forever James Davis writes: "I'm not a mathematician by any stretch - the search that happened to work was hoping n = x*p=f(x)*10^y+p, where p is the largest prime factor of n. That requires that f(x)/(x-1) terminate and it's decimal expansion be a prime (p). That motivates looking for x of the form x=m*10^y+1 and hoping some common factors cancel between f(x) and (x-1). Turned out to be enough: m=1407 fell out immediately."

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Re: [math-fun] Bill & Rohan's G12 talk
by Henry Baker 05 Jun '17

05 Jun '17

Agreed! Where can I get Minsky's & Trinsky's today ? (Digital version preferred) Will there be a Minsky's & Trinsky's II ? At 09:34 PM 6/1/2017, Robert Smith wrote: >Great talk! > >On Thu, Jun 1, 2017 at 19:56 Bill Gosper <billgosper(a)gmail.com> wrote: > >> https://youtu.be/lXsVWwPa7bc --rwg >> http://www.gathering4gardner.org/g4g12-videos/

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[math-fun] "harmonic-like" series
by Henry Baker 05 Jun '17

05 Jun '17

I've been playing with spreadsheets that have formulae with logs in them. Certain of these spreadsheets effectively approximate logs by summing the inverses of something being counted. So, what I'm *really* doing in these spreadsheets is computing harmonic series. The problem is, the approximation of the standard harmonic series to a log function isn't very good. So, still using a spreadsheet, how can we fiddle with the formulae to better approximate a log function? Idea #1. sum(1/(i+fudge),i,1,n), where fudge is some small constant to be determined. Is there an "optimal" value of fudge that minimizes the error under some error model ? Idea #2. sum(1/(i+fudge[i]),i,1,n), where fudge[i] is an array of small (|fudge[i]|<1) constants that "sample" the function 1/x at somewhat more optimum points ? Any other ideas?

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