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