As far as I am aware that is an open question ...apart from special cases where automata have been given (don't recall the papers/authors, J-P Allouche should know more). Usually one first tries to determine how to increment a number. This can already be tricky. Given that such algorithms may lead to more efficient CPUs I am surprised this field isn't much more active. The numeration system corresponding to gosper.org/base2+w.bmp may be a "bad" one: no disc around zero is contained in the (shown approximation of the) fundamental tile. The (essentially unique when requiring rotational symmetry) numeration systems on the Eisenstein grid using three or four digits don't have this problem. On the Gaussian grid the numeration systems with 4-symmetric fundamental tile never have this problem, the smallest one having digit set {0, +1, -1, +i, -i} and basis 2+i. Section 3.3 of https://arxiv.org/abs/1607.02433 mumbles about this and may contain a useful reference. Best regards, jj * James Propp <jamespropp@gmail.com> [Apr 22. 2018 15:47]:
Is there a nice algorithm for adding two complex numbers via their expansions in this base, using some sort of carrying process?
Jim Propp
On Sunday, April 22, 2018, Bill Gosper <billgosper@gmail.com> wrote:
When the digits are 0 and 1 and the other two cuberoots of 1. See gosper.org/gaskettalk.pdf pp 11 and 13. E.g., i/√3 = red.cyangreencyangreencyangreen... = cyan.greenmagentagreenmagentagreen... . Presumably we can expect a lot of these multiple representations when the unit patch has a fractal boundary. PUZZLE: Here of dimension what? We need |base|² digits to cover a positive area in the complex plane. The "France fractal" bounds the unit patch of the base 2+i^(2/3) with digits zero and the 6th roots of 1. |2+i^(2/3)| = √7. Base 2+i^(2/3) with three digits moved outward gosper.org/base2+w.bmp has a unit patch with a boundary of large dimension. (PUZZLE: D=?) --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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