June 2008 - mathfun@mailman.xmission.com

Re: [math-fun] Tiling the Plane with 5-, 6- and 7-gons only
by Dan Asimov 18 Jun '08

18 Jun '08

If convexity of the tiles is not an issue (big if), then here's a way to do it in a periodic pattern: Start with three adjacent infinite columns of squares, with the rightmost column half a side out of phase with the other two. Alter the leftmost column by replacing each horizontal side with two side making a V shape. This converts each square to a non-convex hexagon. For the last two columns, alter each interior vertical side (i.e., where they meet out of phase) to a V shape, pointing alternately right and left. This results in converting all squares of the last two columns to (convex) house-shaped pentagons. Finally, replace all the horizontal sides of the rightmost column by a pair of sides making a V shape. This converts each of its pentagons to a non-convex heptagon. The three altered columns still occupy the same space in the plane, so they can be repeated indefinitely to the right and left. --Dan P.S. In case it's desired that adjacent edges not line up, the angles of the three types of V's can be easily chosen to avoid this. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

1 0

[math-fun] Tiling the Plane with 5-, 6- and 7-gons only
by Guy Haworth 18 Jun '08

18 Jun '08

A colleague wants a computer to define a 2D-graph, a tiling of the plane if you will, which only has pentagons, hexagons and heptagons. He's a chemist, which may give someone a clue what this is about, but that someone is not me. I don't suppose the n-gons can all be convex (but don't have a proof of that) but concave n-gons are not necessarily an issue. Anyone got an algorithm or even better, a program. My colleague did have the suggestion of just tiling the plane with triangles and then knocking sides out, but the problem then becomes one of inspecting what is left to see that it meets the requirements. Thanks - Guy

3 3

Re: [math-fun] Proof (was: Missing Catalan identity)
by Fred lunnon 18 Jun '08

18 Jun '08

On 6/14/08, Steve Witham <sw(a)tiac.net> wrote: > >James Buddenhagen wrote: > > > >. . . something that has been bothering me for a long time: > >when should I accept that something is true? > ... > Nobody mentioned truly formalizing math and using proof- > checking programs. Doing that would break the problem down. > ... There was some discussion of this on math-fun a year or so ago ... Certainly, there are plenty of people working on it. One example is the Hilbert II project at www.qedeq.org/ WFL

1 0

Re: [math-fun] Autobiographical numbers revisited
by Eric Angelini 17 Jun '08

17 Jun '08

Hello Maximilian, > - why just stop at 11 ?? ... because this is how A046043 works ! A046043 starts with array of length 1 (no integer) then 2 (no integer) then 3 (no integer) then 4 (2 integers, 1210, 2020) then 5 (1 integer, 21200) etc. > go to other bases ... ah yes, good idea. Best, É. -----Message d'origine----- De : Maximilian Hasler [mailto:maximilian.hasler@gmail.com] Envoyé : mardi 17 juin 2008 16:06 À : Eric Angelini Cc : SeqFan Objet : Re: Autobiographical numbers revisited > Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 | > Occurrences | 5 | 3 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 2 | --> 53110100002 > More such numbers: > Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 | > Occurrences | 5 | 4 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 1 | --> 541011000021 this seems not very natural to me, since to be consistent, you should then include all possible substrings, e.g. also "54", "41", "21" etc. (at least up to a given length, say 2 digits) - why just stop at 11 ?? And also, you silently ignore the substrings "01", ... so in fact you should not talk about "strings" since you consider numbers, not strings. A better justified alternative would be, IMHO, to go to other bases (e.g. count hexadecimal digits using hexadecimal digits, etc.) One maybe "interesting" non-base sequence would be: a(n)=number of autobiographical numbers [in the original strict sense] in base n Maximilian PS: A047841 should have kw "full" (I suppose, the b-file justifies this) A138480 - the 3 additional terms could have been just a remark in A046043, or vice versa ("3 last terms not allowed if "digit 10 etc" is considered as nonsense")....

1 0

Re: [math-fun] Autobiographical numbers revisited
by Eric Angelini 17 Jun '08

17 Jun '08

read : -- only method saying the truth about N: "there are 11 zeros in N". >How could the reader know that he has to link the first >two "1"'s of N -- only method saying the truth about N: >"there are 11 one's in N".

1 0

[math-fun] Autobiographical numbers revisited
by Eric Angelini 17 Jun '08

17 Jun '08

[a private mail from Neil gives me the opportunity to ask the same unanswered questions I posted a few months ago] ----- Autobiographical numbers revisited ----- 2020 is an autobiographical number because 2020 describes its own "digit content" like this: Digit: 0 | 1 | 2 | 3 | Occurrences: 2 | 0 | 2 | 0 | ("In 2020 there are 2 zeros 0 one 2 twos 0 three") This method gives the traditionnal (finite) list of autobio- graphical numbers: [http://www.research.att.com/~njas/sequences/A046043] 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000. Should we count substrings instead of digits, then we could prolong the seq with a few new terms. Example: Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 | Occurrences | 5 | 3 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 2 | --> 53110100002 Two "10" substrings: ---> ^^^^^ ^^^^^ Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 | Occurrences | 6 | 2 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | --> 62200010001 The "10" substring: -------------> ^^^^^ Sequence A046043 becomes: 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001. More such numbers: Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 | Occurrences | 5 | 4 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 1 | --> 541011000021 ^^^^^^^^^ (one "10" substring and the "11" substring are interleaved in "110") Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 | Occurrences | 6 | 4 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 3 | 1 | 0 | --> 6401101000310 Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 | Occurrences | 7 | 4 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 3 | 1 | 0 | 0 | --> 74011001003100 Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 |14 | Occurrences | 8 | 4 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 3 | 1 | 0 | 0 | 0 | --> 840110001031000 Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 |14 |15 | Occurrences | 9 | 3 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 0 | 1 | 0 | 0 | 0 | --> 9321000001201000 Sequence A046043 becomes: 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 53110100002, 62200010001, 541011000021, 6401101000310, 74011001003100, 840110001031000, 9321000001201000. Can one go further? I guess not. Let's explain why with this example: Substring | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 |12 |13 |14 |15 |16 |17 | Occurrences 11 6 0 0 1 0 1 0 0 0 4 1 0 0 0 0 1 0 The array says the truth -- but how can one read the number N it produces? This number N is: 1160010100041000010. With the above adopted rule the first "1" of N means that "there is only 1 zero in N" -- which is false. How could the reader know that he has to link the first two "1"'s of N -- only method saying the truth about N: "there are 11 one's in N". Is all this worth a new entry in the OEIS or a new "comment" in A046043? Is this way of counting substrings well-defined? If yes, are there numbers (having less than 15 digits) which were forgotten? Best, É. (the same in french: http://www.cetteadressecomportecinquantesignes.com/SubStrings.htm)

1 0

Re: [math-fun] becoming a 21st century mathematician
by James Propp 16 Jun '08

16 Jun '08

Other thoughts about things I'm thinking I might want to get: Dedicated server: I can see how it might be useful to have a server sitting in my office linked to the Internet via a fixed IP address, but I'd have to administer it myself, and this is not something that interests me. (Unless one of you knows of a no-brain-required way of doing this.) Hardware and software that would enable me to work during my 40-minutes-each-way, 4-days-a-week commute to campus (e.g., to read and reply to emails): I've asked around, and the consensus seems to be that there's no good way to do this yet. I could probably kludge something together with existing tools for speech recognition and speech synthesis, but it wouldn't work too well, so the cognitive load would interfere with my driving. I might try something like this for my treadmill workout routine, where the risks are lower (falling off a treadmill can hurt, but not as much as getting into a car accident!). One complicating problem is noise: whether you're running on a treadmill or driving a car, even the best speech recognition software is going to have problems filtering out the noise. Voice recorder: Well, at least I should be able to record my thoughts while I drive. My Treo 700p has a Voice Memo function, but it's not very versatile. Does anyone have a voice recorder that they like, that's suited to use as a digital dictaphone? Tablet: I think tablet entry would be fun and useful, though I'm worried that existing tablet systems will be too slow to do other things I want my PC to do (like run Dragon Naturally Speaking), so I might just go with an external tablet device. Or maybe one of those "smart pens" that turns any surface into an external tablet (as long as you write on special paper). Anyone know what they're called and where one can get them? Extended Apple technical support: This is a must. Apple tech support is great. Data projector: I'll get one of my own, so that when I'm giving a talk I don't face the problem that suddenly my laptop doesn't understand what resolution I want, or doesn't know whether the screen image should be on my laptop or the display device or both, etc. Steven G. Krantz's "A Primer of Mathematical Writing": I'll probably never find the time to read it, but I've always liked Krantz's essays, so I'd probably enjoy what he has to say about writing. Laptop bags: a friend told me I should be sure to get ones with airbags, because it's so easy to damage a hard drive. Wireless signal booster: I once bought one from a company called Quickertek, but they were never able to get it to work for me (in part because Max OS X 10.3.9 was already ancient by the time I bought the Quickertek device). Anyone out there know of a good way to enhance my laptop's ability to pick up distant signals? (This was a handier thing back before so many people used encryption; back then, it was easier to find some person with a Netgear router who hadn't set up security.) Mini surge protectors: Are these a good idea? Postscript Viewer for the Mac: The one I've got is okay but not great. ispell: Is a free version available? Polydron and Zome kits: Seem like they'd be good for the math club. Then there's a whole bunch of Treo software I'd like, such as: * Good way to view PDF files * A better web-browser than the one Sprint gave me * A C compiler * Notepad/scribble software (BugMe has never worked for me on the Treo, and I suspect that there's no way to get it to work well on a non-Graffiti platform) Jim

2 1

[math-fun] Usenet sci.math & .research
by Schroeppel, Richard 16 Jun '08

16 Jun '08

<asked with some trepidation> Is anyone still reading or contributing to the Usenet groups sci.math or sci.math.research? Rich

3 2

[math-fun] System of 2 diophantine equations
by Christian Boyer 15 Jun '08

15 Jun '08

I am looking for at least one integer solution of this system: a^3 - d^3 = b^3 - e^3 = c^3 - f^3 a^3 - g^3 = b^3 - h^3 = c^3 - i^3 Quite easy to find near solutions, for example: 165^3 - 72^3 = 178^3 - 115^3 = 162^3 - 51^3 165^3 - 618^3 = 178^3 - 619^3 = 162^3 - 235788435 Unfortunately 235788435 is not a cube... It seems that there is no solution with integers < 300000^3. Any idea? Christian.

3 5

Re: [math-fun] Proof (was: Missing Catalan identity)
by Steve Witham 13 Jun '08

13 Jun '08

>James Buddenhagen wrote: > >. . . something that has been bothering me for a long time: >when should I accept that something is true? It's always been true that you had to trust your own brain, that it hadn't been fooled. A proof is just an argument that's supposed to be convincing, that hopefully gets you a lot closer to believing. Math proofs convinced in a way that was (still can be) so impressive that it seemed to transcend human limitation. But proofs always run on a substrate. Nobody mentioned truly formalizing math and using proof- checking programs. Doing that would break the problem down. 1) Most proofs aren't perfectly formal. Someone would have to fill in the tedious gaps, and yet, 2) You want at least the statement of what is proven to be still human readable, because unless you think you know what is claimed, what good is a proof? 3) The proof-checker should be a simple program, easy enough for many people to understand in detail, and many many people to understand in principle. It should be compilable and runnable on anybody's computer. 4) Once you're convinced you know what it says and that it's true, you still have to worry about what it means. Consider all the baloney smart people say about Goedel's proof. I couldn't remember what it means exactly or how it works, a month after reading GEB. I think if such a movement could get started, it would fuel itself, as a huge backlog of proofs were debugged. Enough evidence of sloppiness and interesting questions and there would be a moral momentum. I think this would air out some of the psychological, sociological or institutional problems others have mentioned here. It wouldn't be so easy to mystify, or to brush over things. I'm still stirred by Leibniz's nutty "Calculemus!" ideal: "For it would suffice to take their pencils in their hands, and say to each other: Let us calculate." http://www.tiac.net/~sw/2003/08/calculemus.html On the other hand it could reduce the pressure to make proofs clear to humans ("Just run the checker!"). Item 2 above amounts to a user-interface or programming-language design problem. I would like to see usability engineers, especially, critiquing math. For use by other mathematicians, mainly. --Steve

1 0