[math-fun] seven of three
yet another pretty matrix: k = 4; vec = IntegerDigits[Range[0, 3^k - 1], 3, k] - 1; or, all (-1,0,1) vectors of length k, sev = Outer[Plus, vec, vec, 1] /. (q : {__Integer}) /; (Min[q] < -1 || Max[q] > 1) :> 0 /. (q : {__Integer}) :> 1 + FromDigits[q + 1, 3]; .. added in all possible ways, and replacing (-1,0,1) vecs with 1, the others with 0 (those containing -2 or 2). ListDensityPlot[sev /. i_?Positive -> 1, Mesh -> False] .. results in a fractal plot of 7 hexagons, made of 7 hexagons, made of ... and its determinant is 1. The '1''s are of course counted as 7^k. Beauty is in the eye of the beholder. W.
This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag? There are nice arrangements for 48, 49, 50 stars: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , but although I've considered this problem many times, I never came up with any arrangement I considered even remotely satisfactory before last night. Since I'd like to hear other people's ideas, I won't post last night's discovery until some of you have had a go at the problem, although I can hardly think any of you will come up with a pattern as good as mine! It would also be interesting to know whose job it is do settle such things, whether some 51-star design has already been provisionally decided upon, and of course, whether that's as good as mine. Regards, John Conway
The Puerto Rico flag design is nice. http://flagspot.net/flags/us51star.html Erik Demaine (http://theory.lcs.mit.edu/~edemaine/ ) told us at g4g5 that the US flags' stars are due to Betsy Ross knowing how to fold and cut 5-pointed stars. Otherwise, the flag would have 6-pointed stars. --Ed --- John Conway <conway@Math.Princeton.EDU> wrote:
This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag?
There are nice arrangements for 48, 49, 50 stars:
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ,
but although I've considered this problem many times, I never came up with any arrangement I considered even remotely satisfactory before last night.
Since I'd like to hear other people's ideas, I won't post last night's discovery until some of you have had a go at the problem, although I can hardly think any of you will come up with a pattern as good as mine!
It would also be interesting to know whose job it is do settle such things, whether some 51-star design has already been provisionally decided upon, and of course, whether that's as good as mine.
Regards, John Conway
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I rather like the "circular" version favored by the Puerto Rico folks. But it's looking more like the next flag will have the stars arranged in a crescent shape, if the talk against Iran continues to heat up... At 08:14 AM 9/5/03 -0700, Ed Pegg Jr wrote:
The Puerto Rico flag design is nice. http://flagspot.net/flags/us51star.html
Erik Demaine (http://theory.lcs.mit.edu/~edemaine/ ) told us at g4g5 that the US flags' stars are due to Betsy Ross knowing how to fold and cut 5-pointed stars. Otherwise, the flag would have 6-pointed stars.
--Ed
--- John Conway <conway@Math.Princeton.EDU> wrote:
This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag?
There are nice arrangements for 48, 49, 50 stars:
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ,
but although I've considered this problem many times, I never came up with any arrangement I considered even remotely satisfactory before last night.
Since I'd like to hear other people's ideas, I won't post last night's discovery until some of you have had a go at the problem, although I can hardly think any of you will come up with a pattern as good as mine!
It would also be interesting to know whose job it is do settle such things, whether some 51-star design has already been provisionally decided upon, and of course, whether that's as good as mine.
Regards, John Conway
There is a very nice display of historical US flags, with a particular emphasis on the arrangement of the stars over time, in the San Francisco airport international terminal, or at least there was, as recently as 4 weeks ago. I wanted to take pictures of it but forgot my digital camera. I got the impression from the display that for a large fraction of US history, the arrangement of stars has been left to the flag maker. There were flags with circular symmetry, for example, involving 23-29 stars or so I think. I also remember that all the flags were from one person's collection in San Francisco, or somewhere nearby. I've thought the 46 star flag shown at the URL below must not be the best possible. http://www.law.ou.edu/hist/flags/fedflag.shtml Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm ----- Original Message ----- From: "John Conway" <conway@Math.Princeton.EDU> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Friday, September 05, 2003 7:57 AM Subject: [math-fun] The problem of the 51st state.
This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag?
There are nice arrangements for 48, 49, 50 stars:
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ,
but although I've considered this problem many times, I never came up with any arrangement I considered even remotely satisfactory before last night.
Since I'd like to hear other people's ideas, I won't post last night's discovery until some of you have had a go at the problem, although I can hardly think any of you will come up with a pattern as good as mine!
It would also be interesting to know whose job it is do settle such things, whether some 51-star design has already been provisionally decided upon, and of course, whether that's as good as mine.
Regards, John Conway
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Fri, 5 Sep 2003, Thane Plambeck wrote:
There is a very nice display of historical US flags, with a particular emphasis on the arrangement of the stars over time, in the San Francisco airport international terminal, or at least there was, as recently as 4 weeks ago.
I wanted to take pictures of it but forgot my digital camera. I got the impression from the display that for a large fraction of US history, the arrangement of stars has been left to the flag maker. There were flags with circular symmetry, for example, involving 23-29 stars or so I think.
I also remember that all the flags were from one person's collection in San Francisco, or somewhere nearby.
I've thought the 46 star flag shown at the URL below must not be the best possible.
I don't know how to read that - can you roughly draw or describe it for me? JHC
On 5 Sep 2003 at 11:37, John Conway wrote:
On Fri, 5 Sep 2003, Thane Plambeck wrote:
I've thought the 46 star flag shown at the URL below must not be the best possible.
I don't know how to read that - can you roughly draw or describe it for me? JHC
8 7 8 8 7 8 The double-row of '8's in the middle is really ugly..:o( /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
The 46 star flag was a 8 7 8 8 7 8 arrangement * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I think it looks clunky Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm ----- Original Message ----- From: "John Conway" <conway@Math.Princeton.EDU> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Friday, September 05, 2003 8:37 AM Subject: Re: [math-fun] The problem of the 51st state.
On Fri, 5 Sep 2003, Thane Plambeck wrote:
There is a very nice display of historical US flags, with a particular emphasis on the arrangement of the stars over time, in the San Francisco airport international terminal, or at least there was, as recently as 4 weeks ago.
I wanted to take pictures of it but forgot my digital camera. I got the impression from the display that for a large fraction of US history, the arrangement of stars has been left to the flag maker. There were flags with circular symmetry, for example, involving 23-29 stars or so I think.
I also remember that all the flags were from one person's collection in San Francisco, or somewhere nearby.
I've thought the 46 star flag shown at the URL below must not be the best possible.
I don't know how to read that - can you roughly draw or describe it for me? JHC
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Fri, 5 Sep 2003, Thane Plambeck wrote:
The 46 star flag was a 8 7 8 8 7 8 arrangement
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
I think it looks clunky
Then you probably won't like my 51-star design! JHC
Barry Cipra is visiting. His first solution was (issued without his permission) *************************************************** His second was * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Here's mine: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Mweanwhile, back at the ... Barry comes up with (if my asciigraphy is up to it) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Ah, well, I'll leave you to edit it .... R. ---------- On Fri, 5 Sep 2003, John Conway wrote:
This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag?
There are nice arrangements for 48, 49, 50 stars:
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ,
but although I've considered this problem many times, I never came up with any arrangement I considered even remotely satisfactory before last night.
Since I'd like to hear other people's ideas, I won't post last night's discovery until some of you have had a go at the problem, although I can hardly think any of you will come up with a pattern as good as mine!
It would also be interesting to know whose job it is do settle such things, whether some 51-star design has already been provisionally decided upon, and of course, whether that's as good as mine.
Regards, John Conway
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Fri, 5 Sep 2003, Richard Guy wrote:
Barry Cipra is visiting. His first solution was (issued without his permission)
***************************************************
His second was * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Say hello to him for me. Yes, this was one of the ones I rejected.
Here's mine: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
That's really nice! I prefer it to mine, if indeed its non-rectangular shape is acceptable.
Mweanwhile, back at the ... Barry comes up with (if my asciigraphy is up to it)
*
* * * * * *
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Ah, well, I'll leave you to edit it .... R.
No need - it comes out fine! But although it looks very nice, I think it wastes too much space to be acceptable. JHC
51: x x xxxx xxxx x x x x x x x x x x x x xxxx xxxx x x x x x x x x x x x xxxx xxxx x x Although the extra x doesn't really belong, so perhaps this is only a new design for the current flag. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com
participants (8)
-
Bernie Cosell -
Ed Pegg Jr -
Henry Baker -
John Conway -
Jon Perry -
Richard Guy -
Thane Plambeck -
wouter meeussen