Sweet! Doesn't that mean that the area of all those "dragons", when looked upon as fundamental tile of the corresponding numeration system is = 1 ? Anyway, here is Dekking morphism: a |--> ab A |--> BA b |--> Ab B |--> Ba where I wrote A for a^(-1) and B for b^(-1). In the image https://jjj.de/tmp-math-fun/twindragon-boundary-dekking.pdf I have not done the deletions aA = Aa = bB = Bb = empty word, thus the "excursions" into and out off the twindragon. Best regards, jj Note to myself: zrender -d=4 -m='1 12 2 32 3 43 4 41' -e=0.1 -w=2 -a='1234' -k=-1 -i=5 --arrows --letters where 1 = a, 2 = b, 3 = A, 4 = B * Tom Karzes <karzes@sonic.net> [Nov 09. 2019 16:53]:
The Heighway area should be .5 and the twindragon area should be 1. You can get them directly from the tiling polygons.
Heighway:
https://www.karzes.com/xfract/img/dragon.html
Polygon: (0, 0) (.6, -.2) (1, 0) (.8, .4) (.2, .6) Area: .5
Twin:
https://www.karzes.com/xfract/img/twindragon.html
Polygon: (.2, -.4) (.8, -.6) (1.2, -.4) (.8, .4) (.2, .6), (-.2, .4) Area: 1
Tom
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