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[math-fun] NY Times obituary for Ron Graham
by Tom Duff 24 Jul '20

24 Jul '20

https://www.nytimes.com/2020/07/23/science/ronald-l-graham-who-unlocked-the… Probably requires an NY Times account.

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Re: [math-fun] Advances in Squared Squares
by Dan Asimov 24 Jul '20

24 Jul '20

Thanks for clarifying, Tom — I read Allan's post too quickly. That's good to know. (Did we ever decide how many different-size squares could tile the plane periodically?) But I'm pretty sure I've never seen a squared square torus with all different size squares aligned orthogonally (meaning, parallel to the sides of the "square" of the torus) and no 4-way intersections (which excludes squared squares). (As I mentioned, I first wondered about this regarding whether the 24 squares of sides 1 through 24 could tile a 70x70 square torus.) —Dan ----- Allan mentioned Pythagorean tilings, which satisfy the conditions outlined: Any two different sized squares tile the torus (with only one instance of each square). There are no 4-way intersections. The sides are not parallel to the sides of the square torus. Here's an image from the wiki page: https://en.wikipedia.org/wiki/Pythagorean_tiling#/media/File:Pythagorean_di… Clearly this is a minimal solution. If you want to add the constraint that the sides must be parallel to the sides of the square torus, then this solution wouldn't qualify. Tom Dan Asimov writes: > I mean the same thing that a "squared square" usually means: a tiling of > a square torus by squares of distinct sizes. > > Ideally this would be without any 4-way intersections: > > _|_ > | > > (and I don't want to exclude the possibility of such tilings having sides > that are not parallel to the "sides" of the square torus, in case any > such may exist). > > But even forgetting oblique tilings, the possibility exists that if there are > such things as squared square tori, they may beat the records for squared squares, > since they involve fewer constraints. > > —Dan > > ----- > Dan, do you mean to exclude the near-trivial family related to the > Pythagorean tiling? Any two squares tile a square torus. I don't know of > any *other* examples, though. > > On Thu, Jul 23, 2020 at 8:19 PM Dan Asimov <dasimov(a)earthlink.net> wrote: > > > Has anyone ever found a squared square torus that wasn't a squared square? > ----- > -----

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Re: [math-fun] Rectilinear crossing numbers
by Keith F. Lynch 24 Jul '20

24 Jul '20

Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: > The "theory" link goes to a list of papers that are all online there > on the site -- all those ".ps.gz" links. Thanks. Apparently, if I understand correctly, the key insight is that every triangle, i.e. subset of three of the n vertices, is either clockwise or counterclockwise when drawn in order of the vertex numbers, and that the RCN, number of edges in the convex hull, and everything else relevant is uniquely determined by that n-choose-3 bit number. Of course it has to be such that none of the properties change under any of the n! possible renumberings of the vertices. (Or under reflection, but that's the same as reversing the order of the numbers.) I came close, in that I was thinking about keeping track of the number of vertices interior to each such triangle. But I also had a lot of unrelated ideas, such as treating the edges as unit resistors, or perhaps as resistors whose resistance equaled their lengths, and computing the in-circuit resistances between each pair of vertices. >> I re-ran my program on two billion arrangements of seven points, >> and counted the RCNs of each. ... Anyhow, I got surprising >> results. .... >> I'm not sure this isn't a bug. Can someone come up with a proof >> that there can't be any even-numbered RCN for 7 points? Or that >> there can be? Thanks for the proof that there can't be. > In general, when n is odd, the parity will always be the same > as that of (n choose 4). That is: even when n is 0,1,2,3 mod 8 > and odd when n is 4,5,6,7 mod 8. I did notice that the smallest and largest RCNs were always of the same, vastly more common, parity for every odd n-gon, and I did notice the mod 8 pattern. > I think your isolated 1s and 2s and so forth are occurring when you > have points collinear or nearly enough collinear that numerical > errors can produce inconsistent results. I thought that might be the case, but I also thought that such collinearity should be far too rare to account for my results. And I was right about the latter. I just ran it again, keeping track of slopes and other things, and no two edges were anywhere close to having the same slope. (Three vertices on the same line would cause three edges to have the same slope, the three edges of a degenerate triangle.) Just 21 of the 2 billion sets of 7 points had an even RCN. Here's one of them: The (31-bit positive integer) coordinates of the 7 vertices: 0: x=1656517416, y=413766638 1: x=156845284, y=289993739 2: x=912742874, y=1050551595 3: x=1121769010, y=1049033428 4: x=1512620013, y=1049033428 5: x=996250662, y=1587493622 6: x=1715909114, y=2007618816 The slopes of the 21 edges. Note that they're all quite different: 0,1: dx=-1499672132, dy=-123772899, dy/dx=0.082533 0,2: dx=-743774542, dy=636784957, dy/dx=-0.856153 0,3: dx=-534748406, dy=635266790, dy/dx=-1.187973 0,4: dx=-143897403, dy=635266790, dy/dx=-4.414720 0,5: dx=-660266754, dy=1173726984, dy/dx=-1.777656 0,6: dx=59391698, dy=1593852178, dy/dx=26.836279 1,2: dx=755897590, dy=760557856, dy/dx=1.006165 1,3: dx=964923726, dy=759039689, dy/dx=0.786632 1,4: dx=1355774729, dy=759039689, dy/dx=0.559857 1,5: dx=839405378, dy=1297499883, dy/dx=1.545737 1,6: dx=1559063830, dy=1717625077, dy/dx=1.101703 2,3: dx=209026136, dy=-1518167, dy/dx=-0.007263 2,4: dx=599877139, dy=-1518167, dy/dx=-0.002531 2,5: dx=83507788, dy=536942027, dy/dx=6.429844 2,6: dx=803166240, dy=957067221, dy/dx=1.191618 3,4: dx=390851003, dy=0, dy/dx=0.000000 3,5: dx=-125518348, dy=538460194, dy/dx=-4.289892 3,6: dx=594140104, dy=958585388, dy/dx=1.613400 4,5: dx=-516369351, dy=538460194, dy/dx=-1.042781 4,6: dx=203289101, dy=958585388, dy/dx=4.715380 5,6: dx=719658452, dy=420125194, dy/dx=0.583784 Here are the 12 pairs of edges which allegedly intersect: 0: 0,2;1,3 1: 0,2;1,4 2: 0,3;1,4 3: 0,5;1,4 4: 0,5;2,4 5: 0,5;2,6 6: 0,5;3,6 7: 2,4;3,5 8: 2,4;3,6 9: 2,6;3,5 10: 2,6;4,5 11: 3,6;4,5 Unfortunately, I don't have any way to turn this into a graphic. All I can see are lists of numbers. If you, or anyone else reading this, has a way, the bogus or missing crossing should immediately be evident. With 2 billion sets, 21 edges per set, and all coordinates being 31-bit positive integers, I should get about 20 horizontal lines and about 20 vertical lines if the (pseudo-)random number generator is good. I got 18 horizontal lines and 21 vertical lines, which are in good agreement with this. (I discarded the 21 sets with vertical lines.) One curiosity is that of the 18 horizontal lines in the 2 billion sets, 4 of them were in the 21 even-RCN sets, including the set above. So whatever is going wrong probably has something to do with that. (Odds of that being a coincidence are on the order of 10^-32.) But that can't be the whole of it, since 17 of the 21 even-RCN sets did not have horizontal lines, and 14 of the horizontal lines were in odd-RCN sets. I can email you (or anyone else interested) the other 20 even-RCN sets too. I won't bore the whole of math-fun with them.

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Re: [math-fun] : Advances in Squared Squares (Dan Asimov) math-fun Digest, Vol 209, Issue 32
by Stuart Anderson 23 Jul '20

23 Jul '20

See Gambini's thesis page 79, it's in French http://alain.colmerauer.free.fr/alcol/ArchivesPublications/Gambini/carres.p… On Fri, Jul 24, 2020, 10:19 <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Advances in Squared Squares (ed pegg) > 2. Advances in Squared Squares (ed pegg) > 3. Re: Advances in Squared Squares (ed pegg) > 4. Re: Rectilinear crossing numbers (Keith F. Lynch) > 5. Re: Advances in Squared Squares (Dan Asimov) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Thu, 23 Jul 2020 20:48:59 +0000 (UTC) > From: ed pegg <ed(a)mathpuzzle.com> > To: Math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Advances in Squared Squares > Message-ID: <5165370.6291833.1595537339182(a)mail.yahoo.com> > Content-Type: text/plain; charset=UTF-8 > > https://community.wolfram.com/groups/-/m/t/2044450?? > > Jim Williams recently finished a multi-year search for all simple perfect > squared squares up to order 37. More details will eventually by at > squaring.net. For orders 21 to 37, there are {1, 8, 12, 26, 160, 441, > 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, > 17086918} such squares (A006983).? > > Codes, numbers, and pictures at the link. > --Ed Pegg Jr > > > > ------------------------------ > > Message: 2 > Date: Thu, 23 Jul 2020 20:48:59 +0000 (UTC) > From: ed pegg <ed(a)mathpuzzle.com> > To: Math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Advances in Squared Squares > Message-ID: <5165370.6291833.1595537339182(a)mail.yahoo.com> > Content-Type: text/plain; charset=UTF-8 > > https://community.wolfram.com/groups/-/m/t/2044450?? > > Jim Williams recently finished a multi-year search for all simple perfect > squared squares up to order 37. More details will eventually by at > squaring.net. For orders 21 to 37, there are {1, 8, 12, 26, 160, 441, > 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, > 17086918} such squares (A006983).? > > Codes, numbers, and pictures at the link. > --Ed Pegg Jr > > > > ------------------------------ > > Message: 3 > Date: Thu, 23 Jul 2020 20:56:11 +0000 (UTC) > From: ed pegg <ed(a)mathpuzzle.com> > To: Math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Advances in Squared Squares > Message-ID: <947050002.6300228.1595537771779(a)mail.yahoo.com> > Content-Type: text/plain; charset=UTF-8 > > https://community.wolfram.com/groups/-/m/t/2044450 > > > > > On Thursday, July 23, 2020, 03:49:42 PM CDT, ed pegg < > ed(a)mathpuzzle.com> wrote: > > https://community.wolfram.com/groups/-/m/t/2044450?? > > Jim Williams recently finished a multi-year search for all simple perfect > squared squares up to order 37. More details will eventually by at > squaring.net. For orders 21 to 37, there are {1, 8, 12, 26, 160, 441, > 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, > 17086918} such squares (A006983).? > > Codes, numbers, and pictures at the link. > --Ed Pegg Jr > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > Message: 4 > Date: Thu, 23 Jul 2020 19:27:11 -0400 (EDT) > From: "Keith F. Lynch" <kfl(a)KeithLynch.net> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Rectilinear crossing numbers > Message-ID: <4BCT4v0dCqz1ZWL(a)panix2.panix.com> > > Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: > > The "theory" link goes to a list of papers that are all online there > > on the site -- all those ".ps.gz" links. > > Thanks. Apparently, if I understand correctly, the key insight is > that every triangle, i.e. subset of three of the n vertices, is either > clockwise or counterclockwise when drawn in order of the vertex > numbers, and that the RCN, number of edges in the convex hull, and > everything else relevant is uniquely determined by that n-choose-3 > bit number. > > Of course it has to be such that none of the properties change under > any of the n! possible renumberings of the vertices. (Or under > reflection, but that's the same as reversing the order of the numbers.) > > I came close, in that I was thinking about keeping track of the number > of vertices interior to each such triangle. But I also had a lot of > unrelated ideas, such as treating the edges as unit resistors, or > perhaps as resistors whose resistance equaled their lengths, and > computing the in-circuit resistances between each pair of vertices. > > >> I re-ran my program on two billion arrangements of seven points, > >> and counted the RCNs of each. ... Anyhow, I got surprising > >> results. .... > > >> I'm not sure this isn't a bug. Can someone come up with a proof > >> that there can't be any even-numbered RCN for 7 points? Or that > >> there can be? > > Thanks for the proof that there can't be. > > > In general, when n is odd, the parity will always be the same > > as that of (n choose 4). That is: even when n is 0,1,2,3 mod 8 > > and odd when n is 4,5,6,7 mod 8. > > I did notice that the smallest and largest RCNs were always of the > same, vastly more common, parity for every odd n-gon, and I did > notice the mod 8 pattern. > > > I think your isolated 1s and 2s and so forth are occurring when you > > have points collinear or nearly enough collinear that numerical > > errors can produce inconsistent results. > > I thought that might be the case, but I also thought that such > collinearity should be far too rare to account for my results. > > And I was right about the latter. I just ran it again, keeping track > of slopes and other things, and no two edges were anywhere close to > having the same slope. (Three vertices on the same line would cause > three edges to have the same slope, the three edges of a degenerate > triangle.) > > Just 21 of the 2 billion sets of 7 points had an even RCN. Here's one > of them: > > The (31-bit positive integer) coordinates of the 7 vertices: > > 0: x=1656517416, y=413766638 > 1: x=156845284, y=289993739 > 2: x=912742874, y=1050551595 > 3: x=1121769010, y=1049033428 > 4: x=1512620013, y=1049033428 > 5: x=996250662, y=1587493622 > 6: x=1715909114, y=2007618816 > > The slopes of the 21 edges. Note that they're all > quite different: > > 0,1: dx=-1499672132, dy=-123772899, dy/dx=0.082533 > 0,2: dx=-743774542, dy=636784957, dy/dx=-0.856153 > 0,3: dx=-534748406, dy=635266790, dy/dx=-1.187973 > 0,4: dx=-143897403, dy=635266790, dy/dx=-4.414720 > 0,5: dx=-660266754, dy=1173726984, dy/dx=-1.777656 > 0,6: dx=59391698, dy=1593852178, dy/dx=26.836279 > 1,2: dx=755897590, dy=760557856, dy/dx=1.006165 > 1,3: dx=964923726, dy=759039689, dy/dx=0.786632 > 1,4: dx=1355774729, dy=759039689, dy/dx=0.559857 > 1,5: dx=839405378, dy=1297499883, dy/dx=1.545737 > 1,6: dx=1559063830, dy=1717625077, dy/dx=1.101703 > 2,3: dx=209026136, dy=-1518167, dy/dx=-0.007263 > 2,4: dx=599877139, dy=-1518167, dy/dx=-0.002531 > 2,5: dx=83507788, dy=536942027, dy/dx=6.429844 > 2,6: dx=803166240, dy=957067221, dy/dx=1.191618 > 3,4: dx=390851003, dy=0, dy/dx=0.000000 > 3,5: dx=-125518348, dy=538460194, dy/dx=-4.289892 > 3,6: dx=594140104, dy=958585388, dy/dx=1.613400 > 4,5: dx=-516369351, dy=538460194, dy/dx=-1.042781 > 4,6: dx=203289101, dy=958585388, dy/dx=4.715380 > 5,6: dx=719658452, dy=420125194, dy/dx=0.583784 > > Here are the 12 pairs of edges which allegedly intersect: > > 0: 0,2;1,3 > 1: 0,2;1,4 > 2: 0,3;1,4 > 3: 0,5;1,4 > 4: 0,5;2,4 > 5: 0,5;2,6 > 6: 0,5;3,6 > 7: 2,4;3,5 > 8: 2,4;3,6 > 9: 2,6;3,5 > 10: 2,6;4,5 > 11: 3,6;4,5 > > Unfortunately, I don't have any way to turn this into a graphic. All > I can see are lists of numbers. If you, or anyone else reading this, > has a way, the bogus or missing crossing should immediately be evident. > > With 2 billion sets, 21 edges per set, and all coordinates being > 31-bit positive integers, I should get about 20 horizontal lines and > about 20 vertical lines if the (pseudo-)random number generator is > good. I got 18 horizontal lines and 21 vertical lines, which are in > good agreement with this. (I discarded the 21 sets with vertical > lines.) > > One curiosity is that of the 18 horizontal lines in the 2 billion > sets, 4 of them were in the 21 even-RCN sets, including the set above. > So whatever is going wrong probably has something to do with that. > (Odds of that being a coincidence are on the order of 10^-32.) But > that can't be the whole of it, since 17 of the 21 even-RCN sets did > not have horizontal lines, and 14 of the horizontal lines were in > odd-RCN sets. > > I can email you (or anyone else interested) the other 20 even-RCN sets > too. I won't bore the whole of math-fun with them. > > > > ------------------------------ > > Message: 5 > Date: Thu, 23 Jul 2020 20:18:55 -0400 (GMT-04:00) > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Advances in Squared Squares > Message-ID: > <1216132882.10493.1595549935567(a)wamui-moana.atl.sa.earthlink.net> > Content-Type: text/plain; charset=UTF-8 > > Has anyone ever found a squared square torus that wasn't a squared square? > > ?Dan > > ----- > https://community.wolfram.com/groups/-/m/t/2044450 > > Jim Williams recently finished a multi-year search for all simple perfect > squared squares up to order 37. More details will eventually by at > squaring.net. For orders 21 to 37, there are {1, 8, 12, 26, 160, 441, > 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, > 17086918} such squares (A006983). > > Codes, numbers, and pictures at the link. > ----- > > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > End of math-fun Digest, Vol 209, Issue 32 > ***************************************** >

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Re: [math-fun] Advances in Squared Squares
by Dan Asimov 23 Jul '20

23 Jul '20

I mean the same thing that a "squared square" usually means: a tiling of a square torus by squares of distinct sizes. Ideally this would be without any 4-way intersections: _|_ | (and I don't want to exclude the possibility of such tilings having sides that are not parallel to the "sides" of the square torus, in case any such may exist). But even forgetting oblique tilings, the possibility exists that if there are such things as squared square tori, they may beat the records for squared squares, since they involve fewer constraints. —Dan ----- Dan, do you mean to exclude the near-trivial family related to the Pythagorean tiling? Any two squares tile a square torus. I don't know of any *other* examples, though. On Thu, Jul 23, 2020 at 8:19 PM Dan Asimov <dasimov(a)earthlink.net> wrote: > Has anyone ever found a squared square torus that wasn't a squared square? -----

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Re: [math-fun] Advances in Squared Squares
by Dan Asimov 23 Jul '20

23 Jul '20

Has anyone ever found a squared square torus that wasn't a squared square? —Dan ----- https://community.wolfram.com/groups/-/m/t/2044450 Jim Williams recently finished a multi-year search for all simple perfect squared squares up to order 37. More details will eventually by at squaring.net. For orders 21 to 37, there are {1, 8, 12, 26, 160, 441, 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, 17086918} such squares (A006983). Codes, numbers, and pictures at the link. -----

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[math-fun] Advances in Squared Squares
by ed pegg 23 Jul '20

23 Jul '20

https://community.wolfram.com/groups/-/m/t/2044450 Jim Williams recently finished a multi-year search for all simple perfect squared squares up to order 37. More details will eventually by at squaring.net. For orders 21 to 37, there are {1, 8, 12, 26, 160, 441, 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, 17086918} such squares (A006983). Codes, numbers, and pictures at the link. --Ed Pegg Jr

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Re: [math-fun] Rectilinear crossing numbers
by Keith F. Lynch 22 Jul '20

22 Jul '20

Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: > "Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: >> What's needed is a way to look at every possible arrangement of n >> points. By "arrangement" I don't mean the uncountable infinity of >> possible locations of points in the unit square, I mean a difference >> that makes a difference. >> How would one go about enumerating all arrangements? > This looks relevant: > http://www.ist.tugraz.at/staff/aichholzer/research/rp/triangulations/ordert… Thanks. But that apparently just points to abstracts of papers that aren't online, and binary data files in undocumented formats. I re-ran my program on two billion arrangements of seven points, and counted the RCNs of each. This time, I discarded any arrangements that contained any vertical line. Earlier I had just not worried about the resulting divide-by-zero errors, as those would cause errors only in the ninth digit or later. I also checked for the other possible source of division by zero, parallel lines, but as always those never happened. I did not check for three points in a line. Anyhow, I got surprising results. There are no longer just 13 possible RCNs for n=7, there are 24: 9: 2854081 10: 1 11: 11027816 12: 1 13: 23633438 14: 6 15: 57311057 16: 2 17: 146814179 18: 4 19: 239174223 20: 6 21: 275889258 22: 4 23: 384422525 24: 1 25: 309111456 26: 2 27: 205481998 28: 2 29: 180328085 30: 1 31: 116777008 35: 47174846 I'm not sure this isn't a bug. Can someone come up with a proof that there can't be any even-numbered RCN for 7 points? Or that there can be? I got similar results for n=9, except that this time it's the even numbers that are very common and the odd numbers that are very rare: 36: 13927 38: 66278 40: 180469 42: 327642 44: 533071 46: 890830 47: 1 48: 1463260 49: 1 50: 2562426 51: 7 52: 4240140 53: 8 54: 6419703 55: 2 56: 10012353 57: 12 58: 15368762 59: 22 60: 21885908 61: 25 62: 31885941 63: 50 64: 44747612 65: 28 66: 57233553 67: 74 68: 75238774 69: 76 70: 93957198 71: 67 72: 105470193 73: 86 74: 121665340 75: 78 76: 132425970 77: 73 78: 133795238 79: 69 80: 142405957 81: 61 82: 139308013 83: 64 84: 126803508 85: 45 86: 125471727 87: 32 88: 110246476 89: 25 90: 93769089 91: 23 92: 89460437 93: 20 94: 67815191 95: 7 96: 54064073 97: 19 98: 54720289 99: 5 100: 33644869 101: 3 102: 26704075 103: 6 104: 26087965 105: 4 106: 11087442 107: 1 108: 13027353 110: 11485129 112: 1437325 113: 1 114: 5655861 116: 3348265 120: 2333611 126: 737762 I haven't done the full run for higher n yet, but there are preliminary signs that n=11 and n=17 will be almost but not quite all evens, and that n=13 and n=15 will be almost but not quite all odds. There's no sign of these biases for n=even.

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[math-fun] rectangular roots
by Marc LeBrun 21 Jul '20

21 Jul '20

I'm puzzled by the following; maybe someone here can wrangle the asymptotics? Forrest Gump fills n unit chocolate squares into the squarest possible rectangular box (ie the sides are A033676 and A033677). Sometimes this works well (when n is square) and sometimes not so well (when n is prime). How can he best approximate the perimeter-to-area ratio as n increases? It's a slowly decelerating decreasing function. The average for n up to 8000 is about 0.23728, to 10000 it's around 0.23004. Does it vanish or approach some non-zero limit or what? For more fun we might generalize to d>2 dimensions, seeking the average minimal hypersurface-to-hypervolume ratio. (For example in 3d cubes are nice, but primes remain a pain...) Be well and prosper!

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[math-fun] Torus question
by Dan Asimov 21 Jul '20

21 Jul '20

Here's a "puzzle" I don't know the answer to offhand: Call two points X, Y' of the plane "equivalent": X ~ X' if they differ by an integer point: X' = X + v for some v in Z^2. Likewise, call any two circles C, C' in the plane "equivalent": C' ~ C if one is a translation of the other by an integer translation: C' = C + w for some w in Z^2. Now let A, B, C be any three distinct points in R^2. Among all possibilities for A, B, C, what is the maximum number of inequivalent circles that pass through 3 points A', B', C' with A' ~ A, B' ~ B, C' ~ C such that no points equivalent to A, B, or C lie in the interior of such a circle ??? —Dan --------- What is the maximum number of circles Unrolling the torus to obtain the plane results in all the points of the form A + (K,L), B + (K,L), C + (K,L) appearing on the plane

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