http://gosper.org/x=1, y=1, dmin=emin=3o2, dmax=emax=8o3, dg=eg=1o2574, de<=49o12.png , corrects and refines the "δ-ℇ plot" on p23 of http://www.blurb.com/books/2172660 (also Neil's blog, http://nbickford.files.wordpress.com/2011/03/x1-y1-dminemin3o2-dmaxemax8o3-d... ). It's a map of the periods of the Minsky circle algorithm, starting with x0=1, y0=1, with multipliers 3/2 ≤ δ,ℇ ≤ 8/3. With a zoomable, nonblurring viewer, I find it fascinating and surprising. The (δ,ℇ) which blow up exponentially (because δ ℇ>4) are shown in white. But note that some colored rectangles transgress δ ℇ = 4 ! The floor function in the Minsky iteration can actually tame an infinite sequence. Contrariwise, imposing the floor operation on x0=1, y0=1/2, δ 3^(n+1), ℇ = 3^-n changes a mere period 3 into infinite linear growth! Any such growers in the graphic appear black, along with any orbits longer than 10000, of which there are >57000 (in this plot). Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. Zooming at a crack between large rectangles often shows a single layer of variegated pixels, due to the sampler landing exactly on the crack, which, remarkably, is often composed of numerous truly 1-dimensional line segment subregions. Now zoom at the top of the upper largish pink rectangle. Euclid's orchard? Other high-frequency patches resemble peering into factory windows or out of skyscrapers. Also note the apparent, possibly provable accumulation point at (2,2). See book, p 23, for Julian's proof why this would guarantee unbounded periods, despite the lack of thin black rectangles at this sampling density (1/2574)). Also, note the optically illusory reverse curvature when you zoom at (2,2). The plot was computed by Corey's old program, and took 14 hrs, and produced a blank screen due to a bug in Mathematica. Exporting it took several more minutes, returning the filename, but writing no file! I spent a day fighting back creeping insanity, until Julian found an ingenious workaround. --rwg I have proofread this message, and disavow it. --Julian
For those having trouble parsing the link, it's http://gosper.org/x=1,%20y=1,%20dmin=emin=3o2,%20dmax=emax=8o3,%20dg=eg=1o25... On Sun, Apr 22, 2012 at 2:52 AM, Bill Gosper <billgosper@gmail.com> wrote:
http://gosper.org/x=1, y=1, dmin=emin=3o2, dmax=emax=8o3, dg=eg=1o2574, de<=49o12.png , corrects and refines the "δ-ℇ plot" on p23 of http://www.blurb.com/books/2172660 (also Neil's blog, http://nbickford.files.wordpress.com/2011/03/x1-y1-dminemin3o2-dmaxemax8o3-d... ). It's a map of the periods of the Minsky circle algorithm, starting with x0=1, y0=1, with multipliers 3/2 ≤ δ,ℇ ≤ 8/3. With a zoomable, nonblurring viewer, I find it fascinating and surprising. The (δ,ℇ) which blow up exponentially (because δ ℇ>4) are shown in white. But note that some colored rectangles transgress δ ℇ = 4 ! The floor function in the Minsky iteration can actually tame an infinite sequence. Contrariwise, imposing the floor operation on x0=1, y0=1/2, δ 3^(n+1), ℇ = 3^-n changes a mere period 3 into infinite linear growth! Any such growers in the graphic appear black, along with any orbits longer than 10000, of which there are >57000 (in this plot).
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four.
Zooming at a crack between large rectangles often shows a single layer of variegated pixels, due to the sampler landing exactly on the crack, which, remarkably, is often composed of numerous truly 1-dimensional line segment subregions.
Now zoom at the top of the upper largish pink rectangle. Euclid's orchard? Other high-frequency patches resemble peering into factory windows or out of skyscrapers.
Also note the apparent, possibly provable accumulation point at (2,2). See book, p 23, for Julian's proof why this would guarantee unbounded periods, despite the lack of thin black rectangles at this sampling density (1/2574)).
Also, note the optically illusory reverse curvature when you zoom at (2,2).
The plot was computed by Corey's old program, and took 14 hrs, and produced a blank screen due to a bug in Mathematica. Exporting it took several more minutes, returning the filename, but writing no file! I spent a day fighting back creeping insanity, until Julian found an ingenious workaround. --rwg I have proofread this message, and disavow it. --Julian _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I would like to see a version of this plot where period is mapped to grayscale, so that I could see which areas had larger periods. On Sun, Apr 22, 2012 at 3:29 AM, Mike Stay <metaweta@gmail.com> wrote:
For those having trouble parsing the link, it's
http://gosper.org/x=1,%20y=1,%20dmin=emin=3o2,%20dmax=emax=8o3,%20dg=eg=1o25...
On Sun, Apr 22, 2012 at 2:52 AM, Bill Gosper <billgosper@gmail.com> wrote:
http://gosper.org/x=1, y=1, dmin=emin=3o2, dmax=emax=8o3, dg=eg=1o2574, de<=49o12.png , corrects and refines the "δ-ℇ plot" on p23 of http://www.blurb.com/books/2172660 (also Neil's blog,
http://nbickford.files.wordpress.com/2011/03/x1-y1-dminemin3o2-dmaxemax8o3-d...
). It's a map of the periods of the Minsky circle algorithm, starting with x0=1, y0=1, with multipliers 3/2 ≤ δ,ℇ ≤ 8/3. With a zoomable, nonblurring viewer, I find it fascinating and surprising. The (δ,ℇ) which blow up exponentially (because δ ℇ>4) are shown in white. But note that some colored rectangles transgress δ ℇ = 4 ! The floor function in the Minsky iteration can actually tame an infinite sequence. Contrariwise, imposing the floor operation on x0=1, y0=1/2, δ 3^(n+1), ℇ = 3^-n changes a mere period 3 into infinite linear growth! Any such growers in the graphic appear black, along with any orbits longer than 10000, of which there are >57000 (in this plot).
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four.
Zooming at a crack between large rectangles often shows a single layer of variegated pixels, due to the sampler landing exactly on the crack, which, remarkably, is often composed of numerous truly 1-dimensional line segment subregions.
Now zoom at the top of the upper largish pink rectangle. Euclid's orchard? Other high-frequency patches resemble peering into factory windows or out of skyscrapers.
Also note the apparent, possibly provable accumulation point at (2,2). See book, p 23, for Julian's proof why this would guarantee unbounded periods, despite the lack of thin black rectangles at this sampling density (1/2574)).
Also, note the optically illusory reverse curvature when you zoom at (2,2).
The plot was computed by Corey's old program, and took 14 hrs, and produced a blank screen due to a bug in Mathematica. Exporting it took several more minutes, returning the filename, but writing no file! I spent a day fighting back creeping insanity, until Julian found an ingenious workaround. --rwg I have proofread this message, and disavow it. --Julian _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Sat, Apr 21, 2012 at 6:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
http://gosper.org/x=1, y=1, dmin=emin=3o2, dmax=emax=8o3, dg=eg=1o2574, de<=49o12.png , corrects and refines the "δ-ℇ plot" on p23 of http://www.blurb.com/books/2172660 (also Neil's blog,
http://nbickford.files.wordpress.com/2011/03/x1-y1-dminemin3o2-dmaxemax8o3-d... ). It's a map of the periods of the Minsky circle algorithm, starting with x0=1, y0=1, with multipliers 3/2 ≤ δ,ℇ ≤ 8/3. With a zoomable, nonblurring viewer, I find it fascinating and surprising. The (δ,ℇ) which blow up exponentially (because δ ℇ>4) are shown in white. But note that some colored rectangles transgress δ ℇ = 4 ! The floor function in the Minsky iteration can actually tame an infinite sequence. Contrariwise, imposing the floor operation on x0=1, y0=1/2, δ 3^(n+1), ℇ = 3^-n changes a mere period 3 into infinite linear growth! Any such growers in the graphic appear black, along with any orbits longer than 10000, of which there are >57000 (in this plot).
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. [...]
Talk about hanging by a thread! This is Corey's region finder: In[164]:= ExpandAll[ RiskyCongruenceRegion[1, 2, 38/29 + 1/9999, 58/19 + 1/9999]] Out[164]= {MatrixForm[{(Inequality[55/42, Less, \[Delta], LessEqual, 38/29] && Inequality[-53 + 72/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (Inequality[38/29, Less, \[Delta], LessEqual, 97/74] && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (97/74 < \[Delta] < 101/77 && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, -43 + 59/\[Delta]]), (Inequality[58/19, Less, \[Epsilon], LessEqual, 171/56] && Inequality[-18 + 58/\[Epsilon], LessEqual, x0, Less, 20 - 58/\[Epsilon]]) || (171/56 < \[Epsilon] < 113/37 && Inequality[38 - 113/\[Epsilon], LessEqual, x0, Less, -36 + 113/\[Epsilon]])}], 72} In[163]:= Simplify[% /. x0 -> 1 /. y0 -> 2] Out[163]= {MatrixForm[{38/29 < \[Delta] < 59/45, 58/19 < \[Epsilon] < 113/37}], 72} I.e., three corners out and one corner *on* the hyperbola, but it's and open rectangle, and thus entirely out! --rwg PS, a wider view of that "cave" plot shows a remarkable 1D transgression<http://gosper.org/x=1,y=2,1o4led,ele4,dg=eg=1o720.png> at (4/3,3).
On Mon, Apr 23, 2012 at 6:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Sat, Apr 21, 2012 at 6:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
http://gosper.org/x=1, y=1, dmin=emin=3o2, dmax=emax=8o3, dg=eg=1o2574, de<=49o12.png , corrects and refines the "δ-ℇ plot" on p23 of http://www.blurb.com/books/2172660 (also Neil's blog,
http://nbickford.files.wordpress.com/2011/03/x1-y1-dminemin3o2-dmaxemax8o3-d... ). It's a map of the periods of the Minsky circle algorithm, starting with x0=1, y0=1, with multipliers 3/2 ≤ δ,ℇ ≤ 8/3. With a zoomable, nonblurring viewer, I find it fascinating and surprising. The (δ,ℇ) which blow up exponentially (because δ ℇ>4) are shown in white. But note that some colored rectangles transgress δ ℇ = 4 ! The floor function in the Minsky iteration can actually tame an infinite sequence. Contrariwise, imposing the floor operation on x0=1, y0=1/2, δ 3^(n+1), ℇ = 3^-n changes a mere period 3 into infinite linear growth! Any such growers in the graphic appear black, along with any orbits longer than 10000, of which there are >57000 (in this plot).
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. [...]
Talk about hanging by a thread! This is Corey's region finder: In[164]:= ExpandAll[ RiskyCongruenceRegion[1, 2, 38/29 + 1/9999, 58/19 + 1/9999]]
Out[164]= {MatrixForm[{(Inequality[55/42, Less, \[Delta], LessEqual, 38/29] && Inequality[-53 + 72/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (Inequality[38/29, Less, \[Delta], LessEqual, 97/74] && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (97/74 < \[Delta] < 101/77 && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, -43 + 59/\[Delta]]), (Inequality[58/19, Less, \[Epsilon], LessEqual, 171/56] && Inequality[-18 + 58/\[Epsilon], LessEqual, x0, Less, 20 - 58/\[Epsilon]]) || (171/56 < \[Epsilon] < 113/37 && Inequality[38 - 113/\[Epsilon], LessEqual, x0, Less, -36 + 113/\[Epsilon]])}], 72}
In[163]:= Simplify[% /. x0 -> 1 /. y0 -> 2]
Out[163]= {MatrixForm[{38/29 < \[Delta] < 59/45, 58/19 < \[Epsilon] < 113/37}], 72}
I.e., three corners out and one corner *on* the hyperbola, but it's and open rectangle, and thus entirely out! --rwg PS, a wider view of that "cave" plot shows a remarkable 1D transgression<http://gosper.org/x=1,y=2,1o4led,ele4,dg=eg=1o720.png> at (4/3,3).
There appears to be an infinite sequence of these teaser transgressors, with left edge 4 (1 + 3 n + 3 n^2)/(4 + 9 n + 9 n^2), bottom edge (4 + 9 n + 9 n^2)/(1 + 3 n + 3 n^2), and top edge (5 + 18 n + 18 n^2)/(1 + 6 n + 6 n^2), n≥0, with a limit point smack on (4/3,3), where that freak segment protrudes. The right edges go 59/45, 115/87, 126/95, 93/70, 129/97, 342/257, 438/329, 273/205, 333/250, 798/599, 942/707, 549/412, 633/475, 1446/1085, 1638/1229, 921/691, 1029/772, 2286/1715, 2526/1895, 1389/1042, 1521/1141, 3318/2489, ... for which Mma gives a recurrence it can't solve!
On Tue, Apr 24, 2012 at 1:16 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Mon, Apr 23, 2012 at 6:52 PM, Bill Gosper <billgosper@gmail.com> wrote:
[...]
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the [δ*ℇ=4] hyperbola. Julian completed my humiliation by finding the period 159 region {x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. [...]
Talk about hanging by a thread! This is Corey's region finder: In[164]:= ExpandAll[ RiskyCongruenceRegion[1, 2, 38/29 + 1/9999, 58/19 + 1/9999]]
Out[164]= {MatrixForm[{(Inequality[55/42, Less, \[Delta], LessEqual, 38/29] && Inequality[-53 + 72/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (Inequality[38/29, Less, \[Delta], LessEqual, 97/74] && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (97/74 < \[Delta] < 101/77 && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, -43 + 59/\[Delta]]), (Inequality[58/19, Less, \[Epsilon], LessEqual, 171/56] && Inequality[-18 + 58/\[Epsilon], LessEqual, x0, Less, 20 - 58/\[Epsilon]]) || (171/56 < \[Epsilon] < 113/37 && Inequality[38 - 113/\[Epsilon], LessEqual, x0, Less, -36 + 113/\[Epsilon]])}], 72}
In[163]:= Simplify[% /. x0 -> 1 /. y0 -> 2]
Out[163]= {MatrixForm[{38/29 < \[Delta] < 59/45, 58/19 < \[Epsilon] < 113/37}], 72}
I.e., three corners out and one corner *on* the hyperbola, but it's and open rectangle, and thus entirely out! --rwg PS, a wider view of that "cave" plot shows a remarkable 1D transgression<http://gosper.org/x=1,y=2,1o4led,ele4,dg=eg=1o720.png> at (4/3,3).
There appears to be an infinite sequence of these teaser transgressors, with left edge 4 (1 + 3 n + 3 n^2)/(4 + 9 n + 9 n^2), bottom edge (4 + 9 n + 9 n^2)/(1 + 3 n + 3 n^2), and top edge (5 + 18 n + 18 n^2)/(1 + 6 n + 6 n^2), n≥0,
Sorry, that's n≥2. And the period is 36n. --rwg with a limit point smack on (4/3,3), where that freak segment protrudes.
The right edges go 59/45, 115/87, 126/95, 93/70, 129/97, 342/257, 438/329, 273/205, 333/250, 798/599, 942/707, 549/412, 633/475, 1446/1085, 1638/1229, 921/691, 1029/772, 2286/1715, 2526/1895, 1389/1042, 1521/1141, 3318/2489, ... for which Mma gives a recurrence it can't solve!
On Tue, Apr 24, 2012 at 2:33 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Tue, Apr 24, 2012 at 1:16 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Mon, Apr 23, 2012 at 6:52 PM, Bill Gosper <billgosper@gmail.com>wrote:
[...]
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the [δ*ℇ=4] hyperbola. Julian completed my humiliation by finding the period 159 region
{x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. [...]
Talk about hanging by a thread! This is Corey's region finder: In[164]:= ExpandAll[ RiskyCongruenceRegion[1, 2, 38/29 + 1/9999, 58/19 + 1/9999]]
Out[164]= {MatrixForm[{(Inequality[55/42, Less, \[Delta], LessEqual, 38/29] && Inequality[-53 + 72/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (Inequality[38/29, Less, \[Delta], LessEqual, 97/74] && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (97/74 < \[Delta] < 101/77 && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, -43 + 59/\[Delta]]), (Inequality[58/19, Less, \[Epsilon], LessEqual, 171/56] && Inequality[-18 + 58/\[Epsilon], LessEqual, x0, Less, 20 - 58/\[Epsilon]]) || (171/56 < \[Epsilon] < 113/37 && Inequality[38 - 113/\[Epsilon], LessEqual, x0, Less, -36 + 113/\[Epsilon]])}], 72}
In[163]:= Simplify[% /. x0 -> 1 /. y0 -> 2]
Out[163]= {MatrixForm[{38/29 < \[Delta] < 59/45, 58/19 < \[Epsilon] < 113/37}], 72}
I.e., three corners out and one corner *on* the hyperbola, but it's and open rectangle, and thus entirely out! --rwg
PS, a wider view of that "cave" plot shows a remarkable 1D transgression<http://gosper.org/x=1,y=2,1o4led,ele4,dg=eg=1o720.png> at (4/3,3).
There appears to be an infinite sequence of these teaser transgressors, with left edge 4 (1 + 3 n + 3 n^2)/(4 + 9 n + 9 n^2), bottom edge (4 + 9 n + 9 n^2)/(1 + 3 n + 3 n^2), and top edge (5 + 18 n + 18 n^2)/(1 + 6 n + 6 n^2), n≥0,
Sorry, that's n≥2. And the period is 36n. --rwg
with a limit point smack on (4/3,3), where that freak segment protrudes.
The right edges go 59/45, 115/87, 126/95, 93/70, 129/97, 342/257, 438/329, 273/205, 333/250, 798/599, 942/707, 549/412, 633/475, 1446/1085, 1638/1229, 921/691, 1029/772, 2286/1715, 2526/1895, 1389/1042, 1521/1141, 3318/2489, ... for which Mma gives a recurrence it can't solve!
Unless you're Julian and delete the first two terms. The rest are merely
(12 (1 + n + n^2))/(10 + 9 n + 9 n^2). I'm a little surprised that there's a fairly low order recurrence with polynomial coefficients whose solution is a rational function except for two glitches at the beginning.
Using the four-dimensional region corresponding to the first "glitch" Julian constructed a totally offshore 37820000000000/28517500000031 < δ < 39060000000000/29452499999969 1010160000000000/334880000000549 < ℇ < 2003850000000000/664299999999451, with In[420]:= 37820000000000./28517500000031*1010160000000000/334880000000549 Out[420]= 4.00047 destroying my conjecture once and for all. The only consolation is that all the 4D regions in the sequence contain points with δ*ℇ < 4. Is there anywhere in Minskyspace a 4D region that doesn't? --rwg
On Sun, Apr 29, 2012 at 2:57 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Tue, Apr 24, 2012 at 2:33 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Tue, Apr 24, 2012 at 1:16 AM, Bill Gosper <billgosper@gmail.com>wrote:
On Mon, Apr 23, 2012 at 6:52 PM, Bill Gosper <billgosper@gmail.com>wrote:
[...]
Prior to this hi-res plot, I had conjectured that no rectangle had two corners transgressing the [δ*ℇ=4] hyperbola. Julian completed my humiliation by finding the period 159 region
{x=1, y=1, 83/49 ≤ δ < 61/36, 85/36 ≤ ℇ < 111/47} with *three* transgressing corners. I still conjecture the impossibility of four. [...]
Talk about hanging by a thread! This is Corey's region finder: In[164]:= ExpandAll[ RiskyCongruenceRegion[1, 2, 38/29 + 1/9999, 58/19 + 1/9999]]
Out[164]= {MatrixForm[{(Inequality[55/42, Less, \[Delta], LessEqual, 38/29] && Inequality[-53 + 72/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (Inequality[38/29, Less, \[Delta], LessEqual, 97/74] && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, 31 - 38/\[Delta]]) || (97/74 < \[Delta] < 101/77 && Inequality[34 - 42/\[Delta], LessEqual, y0, Less, -43 + 59/\[Delta]]), (Inequality[58/19, Less, \[Epsilon], LessEqual, 171/56] && Inequality[-18 + 58/\[Epsilon], LessEqual, x0, Less, 20 - 58/\[Epsilon]]) || (171/56 < \[Epsilon] < 113/37 && Inequality[38 - 113/\[Epsilon], LessEqual, x0, Less, -36 + 113/\[Epsilon]])}], 72}
In[163]:= Simplify[% /. x0 -> 1 /. y0 -> 2]
Out[163]= {MatrixForm[{38/29 < \[Delta] < 59/45, 58/19 < \[Epsilon] < 113/37}], 72}
I.e., three corners out and one corner *on* the hyperbola, but it's and open rectangle, and thus entirely out! --rwg
PS, a wider view of that "cave" plot shows a remarkable 1D transgression <http://gosper.org/x=1,y=2,1o4led,ele4,dg=eg=1o720.png> at (4/3,3).
There appears to be an infinite sequence of these teaser transgressors, with left edge 4 (1 + 3 n + 3 n^2)/(4 + 9 n + 9 n^2), bottom edge (4 + 9 n + 9 n^2)/(1 + 3 n + 3 n^2), and top edge (5 + 18 n + 18 n^2)/(1 + 6 n + 6 n^2), n≥0,
Sorry, that's n≥2. And the period is 36n. --rwg
with a limit point smack on (4/3,3), where that freak segment protrudes.
The right edges go 59/45, 115/87, 126/95, 93/70, 129/97, 342/257, 438/329, 273/205, 333/250, 798/599, 942/707, 549/412, 633/475, 1446/1085, 1638/1229, 921/691, 1029/772, 2286/1715, 2526/1895, 1389/1042, 1521/1141, 3318/2489, ... for which Mma gives a recurrence it can't solve!
Unless you're Julian and delete the first two terms. The rest are merely
(12 (1 + n + n^2))/(10 + 9 n + 9 n^2). I'm a little surprised that there's a fairly low order recurrence with polynomial coefficients whose solution is a rational function except for two glitches at the beginning.
Using the four-dimensional region corresponding to the first "glitch" Julian constructed a totally offshore 37820000000000/28517500000031 < δ < 39060000000000/29452499999969 1010160000000000/334880000000549 < ℇ < 2003850000000000/664299999999451, with In[420]:= 37820000000000./28517500000031*1010160000000000/334880000000549
Out[420]= 4.00047
destroying my conjecture once and for all. The only consolation is that all the 4D regions in the sequence contain points with δ*ℇ < 4. Is there anywhere in Minskyspace a 4D region that doesn't? --rwg
Oops, sorry, I neglected to mention that this is no longer for (x0,y0) =
(1,2), but rather (550/549 - 10^-10, 249/124 - 10^-10). A somewhat less outlandish solution:
x0 = 551/550, y0 = 251/125, 15250/11499 < δ < 7875/5938, 101200/33549 < ℇ < 200750/66551, with In[99]:= 15250/11499.*101200/33549 Out[99]= 4.00047 --rwg
participants (3)
-
Allan Wechsler -
Bill Gosper -
Mike Stay