On Sun, Nov 20, 2011 at 9:32 PM, Bill Gosper <billgosper@gmail.com> wrote:
ACW>
What about thirds and sevenths?
http://gosper.org/drag3rds7ths.png
The palindromic nature of the dragon will make the end segments congruent,
but only steps of 1/5/2^n will coincide with the single-point "necks" connecting
the lobes.
I enlarged the graphic to show that several of the k/3 and k/7 phase shifts naturally self-avoid: http://gosper.org/drag3rds7thsbig.png
Is this dragon-subdivision all old news?
You don't need a magic, exact Drag function to predict that the lobes have area
1/5/2^n, n>-2, nor Julian's triple point {13/30, 21/30, 23/30}. But it's sure nice to be able
to check! I would be interested to hear if anyone else has bothered to implement an
exact rational dragon function. And especially an inverse.
The popular, Koch construction treatments don't conceive of a dragon function--they just play connect-the-dots on a sequence of finite sets. This is like restricting the domain to a small set of dyadic rationals.
One mystery that bears investigating: Divide the domain [0,1] into 2^n equal subintervals,
chaining 2^n little subdragons with necks at <integer>/5/2^(n+k). Many of these necks
will lie in the interior of the big [0,1] dragon. Why aren't they triple points?
I just checked denominators thru 5*2^8. Nada. I'm really puzzled.
--rwg
On Thu, Nov 17, 2011 at 6:00 AM, Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:
In[135]:= $RecursionLimit = 9999; Clear[Drag];> Drag[t_, a1_: 1, a0_: 0] := Drag[t, b1_: 1,> b0_: 0] = (Drag[t, s1_: 1, s0_: 0] = ((a0 - s0)/(s1 - a1));> Module[{t2 = 2*t, n}, n = Floor[t2]; t2 -= n; Switch[n,> 0, (I + 1)/2*Drag[t2, a1*(I + 1)/2, a0],> 1, 1 + (I - 1)/2*Drag[1 - t2, (I - 1)/2*a1, a1 + a0], 2, 1]])> E.g.,> In[136]:= Drag[7/22]>> Out[136]= 23/82 + 39 I/82>> In[137]:= % - Drag[113/355]>> Out[137]= 517517964757026532787/96808512898827726880686 +> 650828352727980554837 I/96808512898827726880686>> In[138]:= N[%]>> Out[138]= 0.00534579 + 0.00672284 I> (The Dragon function is continuous.)>> Dividing the domain [0,1] into equal fifths,> ListLinePlot[> Partition[Table[{Re@#, Im@#} &@Drag[t/4/2048], {t, 8190}],1638],> AspectRatio -> Automatic, Axes -> None]>> may surprise you:> http://gosper.org/dragon5ths.png>> The obligatory continuous texture drift:> ListLinePlot[Table[{Re@#, Im@#} &@Drag[t], {t, 0, 1, 1/513}],> AspectRatio -> Automatic, Axes -> None]>> and> ListLinePlot[Table[{Re@#, Im@#} &@Drag[t], {t, 1/1024, 1, 1/513}],> AspectRatio -> Automatic, Axes -> None]>> http://gosper.org/driftdrags.png> --rwg