With neither permission nor prohibition from Julian: These two families give all of the triple points. Call a double/triple point "primitive" if two of its preimages are contained in different halves of the dragon (more generally, different pieces of the first replacement). All double/triple points are primitive in some subdivision. A primitive double point lies on the boundary between the two halves. A primitive triple point lies on this boundary, and is a double point in one of the halves. Thus a primitive triple point comes from a (primitive) double point on the boundary. Primitive double points on the boundary can be found visually, and consist of the first contact between the halves (preimages 3/7, 5/7), and an infinite family of points where the solid components of the first half connect (preimages 1/2-1/20/2^k, 1/2+11/60/2^k). The former gives the family with 7's in the denominators, the latter the family with 5's. -------- On Tue, Feb 20, 2018 at 6:30 PM, Bill Gosper <billgosper@gmail.com> wrote:
I thought OEIS A260482, A260747..A260750 completely described the (initial) Dragon triple points, all of which have preimages a(n)/15/2^k. But I just noticed an editor's remark that there is another infinite family of the form a(n)/7/2^k. These are much less "photogenic" than the old set (e.g. gosper.org/dragtrip!.png) <http://gosper.org/dragtrip!.png> because two preimages always(?) differ by 2/7/2^k, and ~75% of them have the
3rd preimage within 4/7/2^k. But they can get "arbitrarily" lopsided:
In[508]:= undrag[(69 + 27 I)/160]
Out[508]= {1539/3584, 2551/3584, 2553/3584}
(e.g., gosper.org/dragtripj1.png; note the green diamond)
(Differences presumably limited to 40%.) Also, Julian has privately remarked that there are no irrational triple points. --rwg
Note that neither grid in the latter png is axis-aligned. All the vertices are true Dragon points, and all the vertices of a given color have preimages in arithmetic progression. Thus every vertex of order 4 with a (necessarily retraced) edge that is or leads to a dead-end is a triple point! Likewise vertices of order >4. Likewise in gosper.org/dragtripcoarse.png. (The edges are not part of the Dragon. They're drawn only to order the vertices.) All continuous planefills are dense with triple points. --rwg