August 2017 - mathfun@mailman.xmission.com

[math-fun] Pascal and his hexagons
by Bill Gosper 03 Aug '17

03 Aug '17

One of my favorite teacher trollings is: Construct the tangent to a given point on a circle, without a compass. The first step is to choose four more points on the circumference at random. gosper.org/tanimation.gif This was shown me by PDP-1 hacker Dick Howell. It's a consequence of https://en.wikipedia.org/wiki/Pascal%27s_theorem about hexagons inscribed in conics. A completely other connection is with hexagons in Pascal's Triangle (GKP 5.1, p155): For example, there's a curious "hexagon property," illustrated by the six numbers 56, 28, 36, 120, 210, 126 that surround 84 in the lower right portion of Table 155. Both ways of multiplying alternate numbers from this hexagon give the same product: 56·36·210 = 28·120·126 = 423360. The same thing holds if we extract such a hexagon from any other part of Pascal's triangle. Talk about understatement. It even holds for hexagons of any integer "radius" r, anywhere in q-Pascal's Triangle! QBinomial[n, k + r, q] QBinomial[n - r, k - r, q] QBinomial[n + r, k,q] = QBinomial[n, k - r, q] QBinomial[n - r, k, q] QBinomial[n + r, k + r, q] Furthermore, if QBinomial is defined to return its q (Laurent) polynomials in factored form, this identity is syntactic. E.g., for radius 2 around {9,6}, In[521]:= pashexq[{9, 6}, 2] (1-q+q^2) (1+q+q^2) (1+q^4) (1-q+q^2-q^3+q^4) (1+q+q^2+q^3+q^4) (1+q^3+q^6)^2 (1+q+q^2+q^3+q^4+q^5+q^6)^2 (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10) = (1-q+q^2) (1+q+q^2) (1+q^4) (1-q+q^2-q^3+q^4) (1+q+q^2+q^3+q^4) (1+q^3+q^6)^2 (1+q+q^2+q^3+q^4+q^5+q^6)^2 (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10) Likewise for the formal expression in Γ functions for complex n, r, and k. (E.g., Fred's <https://www.dropbox.com/s/9vvl6l6ym1zkret/binomial.pdf?dl=1> crazy surfaces.) The property even holds for Mathematica's (and Maple's) broken "Bermuda" definition of Binomial. --rwg

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[math-fun] Algorithm for matrix 'companionization'
by Henry Baker 03 Aug '17

03 Aug '17

Given a square matrix M (over the integers, e.g.), is there a standard algorithm for 'companionization' -- i.e., rational reduction to companion matrix form? Clearly, the initial and final states are representable, because p(x)=|M-xI|, so I would imagine that such a procedure is possible, but I've never encountered it.

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Re: [math-fun] Differential geometry in *finite* fields!
by Henry Baker 03 Aug '17

03 Aug '17

Be careful! Simon Peyton Jones is a friend of mine! The problem with a number of these folks is that they're trying to reduce *too much* of math to a single idea or representation. While it's important to see how far these ideas can be pushed, the true beauty of math IMHO is the ability to look at the same thing from many different angles & many different representations. For example, I'm so upset with the usual discussions of Galois Theory, that I'm contemplating writing a small paper about representing all polynomials by means of square matrices -- i.e., p(x) is represented by a matrix M, such that |M-x*I|=p(x). Most people cringe at using a representation that uses O(n^2) elements for O(n) data, but since we're mathematicians, insight is more important than efficiency, nicht wahr? Perhaps such a paper has already been written; please let me know if you're aware of such a paper. At 05:43 AM 8/3/2017, Fred Lunnon wrote: > Wildberger does himself no favours with the professional community, >because he takes no interest in situating his self-invented methods in >the broader scheme of established mathematics. I have a good deal >of sympathy with his attitude, as it happens; but there does come a >point where one must down tools and make an effort to get connected, >or else risk wasting further time and temper re-inventing the wheel. > > My impression of "Divine Proportions" for example was that it went >some of the way towards rediscovering Clifford algebra; but given how >that has failed to gain acceptance over the past 144 years, maybe >a fresh approach might be no bad thing! > > His exuberance and blackboard technique in the video were wondrous >to behold. [ I was reminded of another supercharged eccentric, >Simon Peyton Jones on Haskell programming --- > https://www.youtube.com/watch?v=6COvD8oynmI >--- and several more ... ] > >WFL

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[math-fun] ISO a perfect pangram
by Keith F. Lynch 03 Aug '17

03 Aug '17

Bill Gosper's recent anagrams reminded me of my search for a perfect pangram, i.e. a sentence in English containing every letter once and only once. For instance, "Cwm fjord bank glyphs vext quiz" and "Mr. Jock, TV quiz PhD, bags few lynx," neither of which is really satisfactory. My approach was to find a complete list of words, discard any words containing duplicate letters (e.g. "containing," as it contains two "n"s (and also two "i"s)), turn each of them into a 32-bit integer which indicates which letters are present ("of" becomes 2^4 + 2^13 as it contains the 5th and 14th letters). Anagrams are merged (e.g. the numbers for "opts," "post," "pots," "spot," "stop," and "tops" are the same). A depth-first tree search is then done during which the numbers are ORed together. If an AND gets a non-zero result, the branch is immediately abandoned. If 2^26-1 is reached, the result is logged, so that I can manually try to arrange the words (or anagrams of them) into a coherent sentence. I tried this about 30 years ago, on a 286. Needless to say, I didn't get anywhere. It probably would have run for eons. I'm considering trying again, on the same general plan, but only with the most common ten thousand words. Better yet, the most common thousand words containing "a", the most common thousand words containing "b", ..., and the most common thousand words containing "z," with the many duplicates merged, for a total of perhaps ten thousand. I might also include the most common thousand words *not* containing "a", the most common thousand words not containing "b", ..., and the most common thousand words not containing "z." Again, duplicates would be merged. And words that are anagrams would move up in the rankings as the usage of each anagram would be summed as if they were all the same word. With memory as cheap as it is, I might also include every *pair* of words in the list as a word, even though that would increase memory usage from 40 kilobytes to 400 megabytes. Has anyone tried this already? Does anyone have a better algorithm? Thanks.

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Re: [math-fun] Differential geometry in *finite* fields!
by Henry Baker 03 Aug '17

03 Aug '17

Perhaps I'm just weird, but I found Wildberger's intro far more motivating than most math presentations. I would imagine that proper motivation gets you 90% of the way home. I'm really sick to death of math presentations that squeeze all of the fun out of math. Yes, it's fun to abstract away all of the inessential details when trying to generalize a proof; but that's only after you got to the proof in the first place. Yes, the view from the top of Kilimanjaro is wonderful, but some of us want to appreciate it by climbing it, rather than simply looking at a photo. For example, I *never* had a decent presentation of Galois theory, even though it's perhaps the most taught piece of undergraduate algebra. I might have thought that after several hundred years, someone would have figured out how to teach these concepts to high schoolers, but apparently not. At 07:16 AM 8/1/2017, Andy Latto wrote: >While he refers to this as "differential geometry", it's the very >beginning of a very large and well-studied field generally described >as "algebraic geometry".. While I wouldn't discourage interested >amateurs from further exploration, because I don't see any benefit in >discouraging people from exploring any sort of math that interests >them, it's a very well-studied field where it's very unlikely that >interested amateurs will find something new. > >Andy > >On Tue, Aug 1, 2017 at 9:32 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> FYI -- >> >> I ran across this really cool video about how to do differential geometry in *finite* fields. >> >> Suppose you have a polynomial in a finite field. >> >> You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc. >> >> https://www.youtube.com/watch?v=2yGuKIz2wfE >> >> DiffGeom7: Differential geometry with finite fields >> >> njwildberger >> >> https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ >> >> With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus! >> >> In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture. >> >> After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics. >> >> One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made. >> >> This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations. > > Andy.Latto(a)pobox.com

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[math-fun] Perfect pangrams
by Guy Haworth 03 Aug '17

03 Aug '17

The solutions could be improved with a little permutation and punctuation. What a pity the Summer Olympics have not yet been held in New York. Guy

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Re: [math-fun] ISO a perfect pangram
by Dan Asimov 03 Aug '17

03 Aug '17

P.S. Another web page explains a couple of previous ones as well as a few new ones: ----- Squdgy fez, blank jimp crwth vox (by Claude Shannon) What it means: Squat, pudgy (squdgy) brimless hat (fez), obstruct (blank) the voice (vox) of a slender (jimp) Celtic violin (crwth). Note: Vox is Latin for voice. It is not found in Merriam-Webster, except as part of the phrase vox populi (voice of the people). Bortz waqf glyphs vex muck djin (by Ed Spargo) What it means: Symbols (glyphs) indicating an Islamic endowment (waqf) of inferior diamonds (bortz) annoy (vex) dirt genies (muck djin). Jink cwm, zag veldt, fob qursh pyx (by Stephen Wagner) What it means: Move quickly with sudden turns (jink and zag) across mountain hollow (cwm) and African grassland (veldt) to deceitfully obtain (fob) a box (pyx) of Saudi coins (qursh). ----- —D. A noble endeavor. By all means please report what you find! You can get pretty far by hand if you fix the difficult letters first, like Q, J, F, V, W, ... roughly in reverse order of Scrabble tile score. The web page http://www.fun-with-words.com/pang_example.html lists these "perfect pangrams': (26) Mr. Jock, TV quiz PhD, bags few lynx. (26) Cwm fjord-bank glyphs vext quiz. (26) Blowzy night-frumps vex'd Jack Q. (26) Squdgy fez, blank jimp crwth vox! (26) TV quiz drag nymphs blew JFK cox. (26) Q-kelt vug dwarf combs jynx phiz. Based on the length of time people have sought perfect pangrams, what I know about wordplay records, the large number of computer science people who enjoy word recreations, and the 6 pangrams mentioned above, I will go out on a limb and predict that if Keith finds any new perfect pangrams, they won't be much more satisfactory than those are, if at all. —Dan "Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: ----- Bill Gosper's recent anagrams reminded me of my search for a perfect pangram, i.e. a sentence in English containing every letter once and only once. For instance, "Cwm fjord bank glyphs vext quiz" and "Mr. Jock, TV quiz PhD, bags few lynx," neither of which is really satisfactory. ... ... ... -----

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Re: [math-fun] ISO a perfect pangram
by Dan Asimov 03 Aug '17

03 Aug '17

A noble endeavor. By all means please report what you find! You can get pretty far by hand if you fix the difficult letters first, like Q, J, F, V, W, ... roughly in reverse order of Scrabble tile score. The web page http://www.fun-with-words.com/pang_example.html lists these "perfect pangrams': (26) Mr. Jock, TV quiz PhD, bags few lynx. (26) Cwm fjord-bank glyphs vext quiz. (26) Blowzy night-frumps vex'd Jack Q. (26) Squdgy fez, blank jimp crwth vox! (26) TV quiz drag nymphs blew JFK cox. (26) Q-kelt vug dwarf combs jynx phiz. Based on the length of time people have sought perfect pangrams, what I know about wordplay records, the large number of computer science people who enjoy word recreations, and the 6 pangrams mentioned above, I will go out on a limb and predict that if Keith finds any new perfect pangrams, they won't be much more satisfactory than those are, if at all. —Dan ----- Bill Gosper's recent anagrams reminded me of my search for a perfect pangram, i.e. a sentence in English containing every letter once and only once. For instance, "Cwm fjord bank glyphs vext quiz" and "Mr. Jock, TV quiz PhD, bags few lynx," neither of which is really satisfactory. My approach was to find a complete list of words, discard any words containing duplicate letters (e.g. "containing," as it contains two "n"s (and also two "i"s)), turn each of them into a 32-bit integer which indicates which letters are present ("of" becomes 2^4 + 2^13 as it contains the 5th and 14th letters). Anagrams are merged (e.g. the numbers for "opts," "post," "pots," "spot," "stop," and "tops" are the same). A depth-first tree search is then done during which the numbers are ORed together. If an AND gets a non-zero result, the branch is immediately abandoned. If 2^26-1 is reached, the result is logged, so that I can manually try to arrange the words (or anagrams of them) into a coherent sentence. I tried this about 30 years ago, on a 286. Needless to say, I didn't get anywhere. It probably would have run for eons. I'm considering trying again, on the same general plan, but only with the most common ten thousand words. Better yet, the most common thousand words containing "a", the most common thousand words containing "b", ..., and the most common thousand words containing "z," with the many duplicates merged, for a total of perhaps ten thousand. I might also include the most common thousand words *not* containing "a", the most common thousand words not containing "b", ..., and the most common thousand words not containing "z." Again, duplicates would be merged. And words that are anagrams would move up in the rankings as the usage of each anagram would be summed as if they were all the same word. With memory as cheap as it is, I might also include every *pair* of words in the list as a word, even though that would increase memory usage from 40 kilobytes to 400 megabytes. Has anyone tried this already? Does anyone have a better algorithm? Thanks. -----

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[math-fun] Nice threesome
by Bill Gosper 01 Aug '17

01 Aug '17

In[429]:= Normal@anagrams@"centesimal" Out[429]= {"CENTESIMAL", "CLEMASTINE" (* Oh my darlin? Antihistamine *), "LEMNISCATE"} --rwg

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[math-fun] Differential geometry in *finite* fields!
by Henry Baker 01 Aug '17

01 Aug '17

FYI -- I ran across this really cool video about how to do differential geometry in *finite* fields. Suppose you have a polynomial in a finite field. You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc. https://www.youtube.com/watch?v=2yGuKIz2wfE DiffGeom7: Differential geometry with finite fields njwildberger https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus! In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture. After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics. One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made. This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.

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