On 2019-04-17 16:27, Adam P. Goucher wrote:

Interesting! Are there any reasons to prefer quater-imaginary over base i-1? Rectangular vs fractal pure fractions region? Julian's magic piecewiserecursivefractal function, seems to solve membership questions. E.g., it can give the exact real preimages of any complex value of the Heighway Dragon function.Heighway triple point <http://gosper.org/dragtrip!.png>>

As mentioned before, base 3 + omega (with digits 0 and the 6 roots of unity) is my favourite positional number system for the complex numbers. The reason being that:

(a) the sum of any 3 one-digit numbers is a two-digit number; (b) the product of any 2 one-digit numbers is a one-digit number;

which together make it possible to build a ripple carry adder out of 'full adders'. (The same is, of course, true for binary.)

But balanced ternary, which Knuth himself describes as the most beautiful of all positional number systems (which I agree with), satisfies a strengthened version, where '3' is replaced with '4' in (a).

So a balanced ternary adder can sum *three* numbers together, instead of (as is the case in binary) requiring sums to be performed pairwise. (That's actually quite cute: in binary, the adder is a binary function; in balanced ternary, it's a ternary function.)

The magicd function in gasket Fourier <http://gosper.org/gaskettalk.pdf> is a fairly exotic application of balanced ternary. balter[x_Integer] := Sign[x]* (IntegerDigits[Abs[x] + (1/2)* (3^Ceiling[Log[3, 1 + Abs[2*x]]] - 1), 3] - 1) That paper also illustrates an interesting alternative to convergent sums, products, and continued fractions: Infinite products of idempotent matrices with vanishing perturbations. —rwg

Best wishes,

Adam P. Goucher

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Sent: Thursday, April 18, 2019 at 12:14 AM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] One base, two digit sets

Knuth has proposed two positional base + digits schemes for representing complex numbers. Base i-1, digits {0,1}:

https://en.wikipedia.org/wiki/Complex-base_system#/media/File:ComplexTwindra...

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, gosper.org/basei-1.png and Base 2i, digits {0,1,2,3}: https://en.wikipedia.org/wiki/Quater-imaginary_base These systems are closely related, because (i-1)² = -2i . So in base 2i, you can get the mirror image of the twindragon system by using the digits {0, 1, - i - 1, -i} instead of {0,1,2,3}! —rwg ______