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Re: [math-fun] next breakthrough?
by Henry Baker 02 Mar '18

02 Mar '18

By "consequential", I think that you may mean "consequential" to a large fraction of the population -- e.g., electrical engineers and physicists are required to work with complex numbers. Consequential enough to include in high school curricula. I'm still waiting for undecidable/non-Turing-computable to make its way into the bulk of society; a lot of the incredibly stupid comments by politicians might be avoided if they had had any education in the *limits* of logical systems and mathematics. For example, no amount of "nerding harder" will enable computer scientists to design a computer such that an arbitrary input won't cause the computer do something "bad". It took physicists many centuries to convince a large fraction of the population about the impossibility of perpetual motion machines, so perhaps we may be facing the same thing here. Curiously, we have *more* trust in undecidability than we have in the impossibility of perpetual motion, because the "laws" of physics are still empirical facts, which might be destroyed with the very next experiment, while undecidability is a function of logic itself, and requires no reference to the "real world". Regarding breakthroughs that are not yet known even to the best of mathematicians, I would say that things related to P=NP and cryptography are pretty high on the list. For example, we are close to trusting cryptography hash functions with all the money in the world (literally!), yet we have no *proof* that these functions are hard to invert or hard to find collisions for. With this much crypto currency in circulation, there's obviously no need for a "Millennium Prize", since a constructive counter-example would produce the richest person in the world! With the recent advances in AI, we might also ponder the "end of mathematics", but not for the usually suggested reason. AI could end mathematics -- not by doing mathematics better, but by doing enough other things better that no one needs mathematicians anymore. This is where we seem to be headed with chess and Go; once machines are so much better, everyone loses interest in the games so no one plays anymore. At 03:47 PM 3/1/2018, Bernie Cosell wrote: >i know this is vague, but my wife was reading about the history of >i and wondered what the next great fundamental conceptual breakthrough >might be. it took something like 1500 years for complex numbers to >get understood and now they're used everywhere. it's now been >about 250 years and she was wondering what might be the next >breakthrough be that could be as consequential as complex numbers are.

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[math-fun] Everybody needs a conversation stopper.
by Bill Gosper 02 Mar '18

02 Mar '18

Sum[ArcTanh[√3Tan[t/(-2)^n]], {n,∞}] == -ArcTanh[Tan[t]/√3] Actually, this clearly must telescope, with a presumably easy inductive verification. It might even be worthwhile for Mathematica to check for summands and prodands of the form f(index+freevariable). --rwg

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[math-fun] packing polyominoes?
by Mike Stay 01 Mar '18

01 Mar '18

My family recently gave me some packing puzzles for my birthday. One is a rectangle packing puzzle, where the rectangles' edges remain parallel to the sides of the tray; there are some pretty obvious algorithms for solving that one. The other is "Stewart's Coffin", which Martin Gardner pronounced "...the finest dissection puzzle of all time. It looks easy but is fiendishly difficult." It involves four laser-cut polyominoes and a tray that is just slightly too small in either direction for easy placement of the polynomials. Clearly the puzzle involves placing the pieces at an angle, but exactly which angle and how to arrange the pieces is the hard problem. Are there algorithms for solving a puzzle like this other than brute force? -- Mike Stay - metaweta(a)gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com

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[math-fun] next breakthrough?
by Bernie Cosell 01 Mar '18

01 Mar '18

i know this is vague, but my wife was reading about the history of i and wondered what the next great fundamental conceptual breakthrough might be. it took something like 1500 years for complex numbers to get understood and now they're used everywhere. it's now been about 250 years and she was wondering what might be the next breakthrough be that could be as consequential as complex numbers are. /bernie\ -- Bernie Cosell bernie@fantasyfarm. com -- Too many people, too few sheep --

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Re: [math-fun] next breakthrough?
by Hilarie Orman 01 Mar '18

01 Mar '18

What things do we know about today but don't realize how important they will be in the future? A guessing game. Conway numbers. Quantum automata. Leave everything to computers; ask AlphaZero to explain it to you. DNA customization melded with silicon and the creation of the Superhuman. Fractals. Continued fractions. Probablistic arithmetic. Catastrophe theory. Dystopian theory. The end of time. Hilarie

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[math-fun] Math camps...help!
by Marnie Kanarek 28 Feb '18

28 Feb '18

Hi all, I don't know if you can help me but does anyone have or know anyone that has any experience with CTY summer camps? Gabe really wants to do a math camp and CTY has a campus like 10 mins away (as opposed to Epsilon or C&amp which are very far away). They are all so expensive so we want to know that they are good before we commit. Also, if anyone has any summer camp options that would be...appropriate for a super advanced 8 year old that AREN'T super expensive, I'm all ears! (I tried to convince Gray to bring Gabe into work for a week at STSCI and just let him bug the astronomers...but that is probably not a realistic option :P) Thanks in advance! Marnie

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Re: [math-fun] packing polyominoes?
by Dan Asimov 27 Feb '18

27 Feb '18

I have a puzzle of 4 purple plastic pieces that sounds vaguely like what you're describing. They are arranged in a square, and each piece interlocks the next one around in a keyhole-shaped hole. (http://www.cs.brandeis.edu/~storer/JimPuzzles/SHAPE/JigSawFourPiece/JigSawF…) That's not it, is it? It's available in pretty much all game stores. —Dan Mike Stay wrote: ----- My family recently gave me some packing puzzles for my birthday. One is a rectangle packing puzzle, where the rectangles' edges remain parallel to the sides of the tray; there are some pretty obvious algorithms for solving that one. The other is "Stewart's Coffin", which Martin Gardner pronounced "...the finest dissection puzzle of all time. It looks easy but is fiendishly difficult." It involves four laser-cut polyominoes and a tray that is just slightly too small in either direction for easy placement of the polynomials. Clearly the puzzle involves placing the pieces at an angle, but exactly which angle and how to arrange the pieces is the hard problem. >Are there algorithms for solving a puzzle like this other than brute force? -----

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Re: [math-fun] packing polyominoes?
by Bill Gosper 27 Feb '18

27 Feb '18

Veit, isn't this your cue? https://en.wikipedia.org/wiki/Difference-map_algorithm I wouldn't be surprised if it could _design_ the puzzle (a.k.a Four Fit, Martin's Menace): Given the pentominoes, find the minimal box. --rwg Quibble: I wouldn't call this a dissection puzzle. On 2018-02-26 22:31, Mike Stay wrote: > My family recently gave me some packing puzzles for my birthday. One is a > rectangle packing puzzle, where the rectangles' edges remain parallel to > the sides of the tray; there are some pretty obvious algorithms for solving > that one. > > The other is "Stewart's Coffin", which Martin Gardner pronounced "...the > finest dissection puzzle of all time. It looks easy but is fiendishly > difficult." It involves four laser-cut polyominoes and a tray that is just > slightly too small in either direction for easy placement of the > poly[ominoe]s. Clearly the puzzle involves placing the pieces at an angle, > but exactly which angle and how to arrange the pieces is the hard problem. > Are there algorithms for solving a puzzle like this other than brute force?

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[math-fun] student faces expulsion for joking that square-root symbol looks like a gun
by Bill Gosper 26 Feb '18

26 Feb '18

https://www.truthrevolt.org/news/student-faces-expulsion-joking-square-root… --rwg

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[math-fun] dragun[2x]/dragun[x] = {1 - I}
by Bill Gosper 26 Feb '18

26 Feb '18

(0≤x≤½) What about dragun[5x]? In[347]:= dragun[5/6]/dragun[1/6] Out[347]= {-1 - 2 I} In[348]:= dragun[5/12]/dragun[1/12] Out[348]= {-1 - 2 I} In[349]:= dragun[5/8]/dragun[1/8] Out[349]= {-1 - 2 I} In[350]:= dragun[5/7]/dragun[1/7] Out[350]= {-I} Dang. In[351]:= dragun[2/7]/dragun[1/7] Out[351]= {1 - I} --rwg

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