Talk:Riffs and Rotes

Theme One Program

• Theme One Program • Overview

PARI Script

I have written a PARI script:

encode(p)=if(p!=1, if( type(p)=="t_INT", p=factor(p)~; p[1,]=apply(x->primepi(x),p[1,])); apply(encode,Vec(p)))

It gives:

 ? for(i=1,20,print(" "i": "encode(i)))
 1: 0
 2: [[0, 0]]
 3: [[[[0, 0]], 0]]
 4: [[0, [[0, 0]]]]
 5: [[[[[[0, 0]], 0]], 0]]
 6: [[0, 0], [[[0, 0]], 0]]
 7: [[[[0, [[0, 0]]]], 0]]
 8: [[0, [[[[0, 0]], 0]]]]
 9: [[[[0, 0]], [[0, 0]]]]
 10: [[0, 0], [[[[[0, 0]], 0]], 0]]
 11: [[[[[[[[0, 0]], 0]], 0]], 0]]
 12: [[0, [[0, 0]]], [[[0, 0]], 0]]
 13: [[[[0, 0], [[[0, 0]], 0]], 0]]
 14: [[0, 0], [[[0, [[0, 0]]]], 0]]
 15: [[[[0, 0]], 0], [[[[[0, 0]], 0]], 0]]
 16: [[0, [[0, [[0, 0]]]]]]
 17: [[[[[[0, [[0, 0]]]], 0]], 0]]
 18: [[0, 0], [[[0, 0]], [[0, 0]]]]
 19: [[[[0, [[[[0, 0]], 0]]]], 0]]
 20: [[0, [[0, 0]]], [[[[[0, 0]], 0]], 0]]
 ? encode(123456789)
 [[[[0, 0]], [[0, 0]]], [[[0, [[[[0, 0]], 0]]], [[[0, 0]], [[0, 0]]], [[[0, [[0, 0]]]], 0]], 0], [[[[[[[0, 0]], [[0, 0]]]], [[0, 0]]]], 0]]

I think this translates directly to the rotes (I'm not sure if I understand the riffs …); establishing the dictionary is left as an exercise … — M. F. Hasler 22:30, 21 November 2010 (UTC)

I don't know PARI, but the first few cases look okay. Would be nice if there's a way to generate the graphics! The riffs were actually discovered first — the arcs pointing into a node come from the index of a prime power and the arcs pointing out of a node go to its exponents. There should be more pictures around the site, but I do need to expand the "primer" — ha ha — a bit more to explain all that. Jon Awbrey 15:02, 22 November 2010 (UTC)

I did spend a decade or so developing a related species of graph-theoretic data structures for AI applications. I'm still getting around to documenting the program. I'll put a link to the current state of things in the next section. Jon Awbrey 15:44, 7 December 2010 (UTC)

Basics

well, at least for me, I need more - I have never heard of riffs & rotes, why were they derived? How are they used ie how are they useful? I follow it's a way to go from a number to a type of graphical depiction but ...?--Bill McEachen 03:42, 28 January 2011 (UTC)

Hi, Bill. I've been away. The political situation in the Midwest has been sucking … up a lot of time. The Riffs and Rotes article page is probably the most coherent thing I've written on Riffs and Rotes so far. These are things I first started working on in the 70's. The basic line of inquiry began with an interest in place-value notations. I was asking the question, “Can there be a notation for natural numbers that is purely place-value, where all the values are place-values?“ There was during this period a series of columns by Martin Gardner on Zermelo–Von Neumann Integers, Conway's Surreal Numbers, Catalan Numbers and Planted Plane Trees, Peirce–Bell Numbers, Charles Sanders Peirce, and Gödel Numbers that fed into the complex of associations.

I seem to be spending more time these days on Facebook, and I had created a Community Page there on Riffs & Rotes that is more linked in to my other interests. See FB : Riffs & Rotes.

I'll try to write more later as things come to mind. Jon Awbrey 12:54, 21 April 2011 (UTC)

Nice Work!

I think this is a nice piece of work! It could be interesting to have some "binary"(?) numerical representation of the graphics (unless the number n itself is the best way to represent its riffs and rotes...). Also, you should add 2013-2015 in the "misc examples" section! ;-) — MFH 20:42, 5 April 2014 (UTC)

DRF as Sierpinski Gasket

Just a lark: doubly recursive factorizations whose representation looks like the Sierpinski gasket:

--Robert Dickau (talk) 13:57, 15 August 2018 (EDT)

Miscellany

59281

• Persiflage • 59281

• Nth Prime Page

DRF 59281

LaTeX

${\begin{array}{l}59281~{\text{is the}}~5995^{\text{th}}~{\text{prime}}\\\mathrm {pi} (59281)=5995\end{array}}$

${\begin{array}{rcl}59281&=&p_{5995}\\5995&=&5\cdot 11\cdot 109\\5995&=&p_{3}p_{5}p_{29}\\29&=&p_{10}\\10&=&p_{1}p_{3}\\[8pt]59281&=&p_{5995}\\&=&p_{p_{3}p_{5}p_{29}}\\&=&p_{p_{p_{2}}p_{p_{3}}p_{p_{10}}}\\&=&p_{p_{p_{p_{1}}}p_{p_{p_{2}}}p_{p_{p_{1}p_{3}}}}\\&=&p_{p_{p_{p_{1}}}p_{p_{p_{p_{1}}}}p_{p_{p_{1}p_{p_{2}}}}}\\&=&p_{p_{p_{p_{1}}}p_{p_{p_{p_{1}}}}p_{p_{p_{1}p_{p_{p_{1}}}}}}\\&=&p_{p_{p_{p}}p_{p_{p_{p}}}p_{p_{pp_{p_{p}}}}}\\[8pt]59281&=&p_{5995}\\&=&p_{p_{p_{p}}p_{p_{p_{p}}}p_{p_{pp_{p_{p}}}}}\end{array}}$

PNG

Riff 59281

Rote 59281