[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... , 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
Some related fractals: https://www.khanacademy.org/computer-programming/eisenstein-flowsnake/617014... https://www.khanacademy.org/computer-programming/gaussian/6554929830723584 https://www.khanacademy.org/computer-programming/eisenstein/5069082270793728 On Wed, Apr 17, 2019 at 5:15 PM Bill Gosper <billgosper@gmail.com> wrote:
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... , 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
Interesting! Are there any reasons to prefer quater-imaginary over base i-1? 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.) Best wishes, Adam P. Goucher
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... , 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Adam P. Goucher -
Bill Gosper -
Mike Stay