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