In answer to the question "But how do you spot a triple point?": The vertices of the curve I linked to fall on a square grid. If you look at the 4 points of any one of the atomic squares in that grid, any two of them that are *not* connected through the perimeter of the square lie on different loops of the curve. If the square contains three distinct loops, then it corresponds to a triple point. There are *lots* of them. Here's an example, taken from the curve I posted before: http://www.karzes.com/dragon-triple.gif If you look at the square in the exact center, inside the red circle, you can see that three loops of the curve converge there: One loop passes through the two bottom vertices, a second passes through the upper-left vertex, and the third passes through the upper-right vertex. So it corresponds to a triple point. There are lots and lots of these. Tom Bill Gosper writes:
Nice. style=rect is the "median curve" of style=basic. I.e. join the midpoints of consecutive line segments of the latter. I suspect these are images of equally spaced smallish rationals in the preimage. But how do you spot a triple point? (I should plot a Dragon connecting only small-rational triple points.) Note https://oeis.org/A260482 & its many friends. Of course, all spacefills are provably dense with triple points. You may recall Julian's triangle filler dense with sextuple points:
*http://howwords.com/triangles/?a_=sexttri&radiusPct=50&nRotors=1476&dtPct=20... <http://howwords.com/triangles/?a_=sexttri&radiusPct=50&nRotors=1476&dtPct=20&rotorSort=1&cmn_=32&cmt_=highcontrast&cmabs_=false&stepsPerCycle_=5&delayMS_=0>* ------------------------------
© 2019 by Howard I. Cannon —rwg
On 2019-05-28 23:07, Tom Karzes wrote:
Bill, you aren't kidding about the Heighway Dragon being dense with triple points! Here's a self-avoiding rendering that makes them really easy to spot:
http://www.karzes.com/dragon/dragon.html?style=rect&form=h
Tom
Bill Gosper writes:
to accompany should've been "sampHilbert" <http://www.tweedledum.com/rwg/sampeano.htm> Hilbert's [spacefilling] function is dense with quadruple points, i.e. points in [0,*i*] with four distinct preimage points in [0,1]: gosper.org/hilbquad.png . E.g., the preimage of 1/2 + *i*/4 is {5/48, 7/48, 41/48, 43/48}. This picture connects, in the order they were swept, all the points in [0,*i*] with preimages having denominators ≤ 3⨉4096. (All quadruple point preimages have denominators 3⨉2ⁿ.) E.g., the lower-leftmost vertex is 1/32 + *i*/64, the image of 5/12288, 7/12288, 41/12288, and 43/12288. The white segments are retraced boundary, retroflexed in the middle, (making quadruple points). (Spacefilling functions map closed intervals to closed sets.)
The Heighway Dragon is dense with mere triple points. Here's one: Hi res Heighway <http://gosper.org/dragtrip!.png> . —rwg