[math-fun] Oddity: What is the average value of the Dragon Curve?
Which we can now define as the restriction of our Dragon Function to dyadic rationals. Successively higher averages look a little crazy: In[370]:= Table[Mean[First@dragun@# & /@ Range[0, 1, 2^-n]], {n, 11}] Out[370]= {1/2 + I/6, 2/5 + I/5, 7/18 + I/6, 7/17 + (3 I)/17, 9/22 + (13 I)/66, 2/5 + I/5, 103/258 + (17 I)/86, 103/257 + (51 I)/257, 137/342 + (205 I)/1026, 2/5 + I/5, 1639/4098 + (273 I)/1366} Note the three occurrences of 2/5+i/5. Remarkably, In[393]:= FindSequenceFunction[%370, n] Out[393]= ((1/5 + I/10) ((2 - I) - I I^n + 2^(1 + n)))/(1 + 2^n) In[373]:= Limit[%, n -> ∞] Out[373]= 2/5 + I/5 How often do we see sequences tipping off their limit like this? In fact, every fourth value is 2/5 + I/5. Perhaps more remarkably, 2/5+i/5 is an endpoint of the common boundary where two half-size Dragon images self-similarly unite to form the Dragon joining 0+0i to 1+0i via 1/2+i/2. This is not obvious without the Dragon Function. (Hey, wait a minute. Why aren't the averages in Out[370] dyadic rationals, like the actual Dragon values?-) —rwg
I'm not sure I understand the issue that is disturbing Gosper here. I got a mean of (2+i)/5 in about four lines of algebra, by cutting the whole curve into two congruent parts and using the fact that the mean of each part must be situated similarly to the mean of the whole. I suppose one can be surprised that successive approximations visit the actual limit periodically, but this doesn't actually seem all that gobsmacking to me. I assume they tip off the limit less and less as the order of the approximation increases. I think the denominators are coming from the number of points being sampled, and as I'd expect they involve 3, 5, 9, 17, ... the incremented powers of 2. This is a classic fencepost situation; your fence has 2^n panels but 2^n+1 posts. If you leave off one endpoint, you will get dyadic rationals, you will lose the repeated visits to (2+i)/5, but it will converge to the same limit. On Wed, Mar 4, 2020 at 10:15 AM Bill Gosper <billgosper@gmail.com> wrote:
Which we can now define as the restriction of our Dragon Function to dyadic rationals. Successively higher averages look a little crazy: In[370]:= Table[Mean[First@dragun@# & /@ Range[0, 1, 2^-n]], {n, 11}]
Out[370]= {1/2 + I/6, 2/5 + I/5, 7/18 + I/6, 7/17 + (3 I)/17, 9/22 + (13 I)/66, 2/5 + I/5, 103/258 + (17 I)/86, 103/257 + (51 I)/257, 137/342 + (205 I)/1026, 2/5 + I/5, 1639/4098 + (273 I)/1366}
Note the three occurrences of 2/5+i/5.
Remarkably, In[393]:= FindSequenceFunction[%370, n]
Out[393]= ((1/5 + I/10) ((2 - I) - I I^n + 2^(1 + n)))/(1 + 2^n)
In[373]:= Limit[%, n -> ∞]
Out[373]= 2/5 + I/5
How often do we see sequences tipping off their limit like this? In fact, every fourth value is 2/5 + I/5.
Perhaps more remarkably, 2/5+i/5 is an endpoint of the common boundary where two half-size Dragon images self-similarly unite to form the Dragon joining 0+0i to 1+0i via 1/2+i/2. This is not obvious without the Dragon Function.
(Hey, wait a minute. Why aren't the averages in Out[370] dyadic rationals, like the actual Dragon values?-) —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This may be premature, but it's ripening way too slowly to tell: Repeating the averaging process for the Flowsnake (= "Gosper Curve") seems to be converging to an irrational, nonquadratic quantity, ~ 0.53689 + 0.29105 I, where the curve is positioned, like the Dragon, to join 0+0i to 1+0i. This is probably consistent with APG's multiprecision evidence suggesting that the inradius and circumradius of the France Flake are transcendental. —rwg Have others besides NeiB and me observed prohibitively bad Mathematica performance if you attempt a lot of memoizing? The above 0.53689 + 0.29105 I (6th order) took > 4 hrs! The recomputation time you save is lost to bad hash performance. This seems to have been true forever. I wonder if WRI is working on it. On Wed, Mar 4, 2020 at 7:14 AM Bill Gosper <billgosper@gmail.com> wrote:
Which we can now define as the restriction of our Dragon Function to dyadic rationals. Successively higher averages look a little crazy: In[370]:= Table[Mean[First@dragun@# & /@ Range[0, 1, 2^-n]], {n, 11}]
Out[370]= {1/2 + I/6, 2/5 + I/5, 7/18 + I/6, 7/17 + (3 I)/17, 9/22 + (13 I)/66, 2/5 + I/5, 103/258 + (17 I)/86, 103/257 + (51 I)/257, 137/342 + (205 I)/1026, 2/5 + I/5, 1639/4098 + (273 I)/1366}
Note the three occurrences of 2/5+i/5.
(= 1/HoldForm @@ {1/(2/5 + I/5)} = 1/(2 - I) —RCS)
Remarkably,
In[393]:= FindSequenceFunction[%370, n]
Out[393]= ((1/5 + I/10) ((2 - I) - I I^n + 2^(1 + n)))/(1 + 2^n)
In[373]:= Limit[%, n -> ∞]
Out[373]= 2/5 + I/5
How often do we see sequences tipping off their limit like this? In fact, every fourth value is 2/5 + I/5.
Perhaps more remarkably, 2/5+i/5 is an endpoint of the common boundary where two half-size Dragon images self-similarly unite to form the Dragon joining 0+0i to 1+0i via 1/2+i/2. This is not obvious without the Dragon Function.
(Hey, wait a minute. Why aren't the averages in Out[370] dyadic rationals, like the actual Dragon values?-) —rwg
participants (2)
-
Allan Wechsler -
Bill Gosper