On Colouring Sequences of Digital Roots
Year: 2014 Authors: Gabriele Gelatti
Core claim
Fibonacci-like digital-root sequences can be classified into five chromatic patterns using RYB colour logic and modulo 9 arithmetic.
Topics
digital roots, Fibonacci-like sequences, chromatic arithmetic, Pisano period
Domains
modular arithmetic, recursive sequences, number theory, colour theory, visual symbolism, data visualization
Methods
colour-number mapping, digital root reduction, sequence table analysis, visual proof
Media
RYB colour model, tables, figures
Paper text
The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture
On Colouring Sequences of Digital Roots
Gabriele Gelatti
Italy
Abstract
In this paper we introduce a logic of transforming integers from 1 to 9 into colours, and present a visual method of seeing the periods of digital roots in Fibonacci-like sequences.
Transforming Numbers into Colours
The RYB (Red-Yellow-Blue) colours model [1] is formed by a triad of primary colours (yellow, red, blue), and a triad of secondary colours (orange, green, purple) deriving from the mix of primary colours. We apply it to describe some properties of positive integers from 1 to 9. The logical model is completed with the addition of a third triad of achromatic colours (black, white and grey). We match colours to numbers as: ; ; ; ; ; ; ; ; .
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|
Figure 1: Numbers and colours.
Digital Root
We use the operation of digital root [2] to reduce any positive integer bigger than 9 to a one cypher number. For example: . This can also be done by dividing any positive integer by 9 and considering its remainder (e.g.: ). Multiples of 9 leave no remainder. All positive integers receive a given colour according to the above chromatic rules. Assigning colours to digital roots is equivalent to assigning colours to the values modulo 9 [3], considering grey = 0, as .
Complementarity and Sequences of Integers
The above Figure 1 shows the triads of colours (1, 4, primary colours; 2, 5, secondary colours; 3, 6, achromatic colours) combined in couples of symmetric complementary colours, except number 9.
The mix of each complementary couple, according to the RYB colours model, produces the neutral colour grey; in the same way the addition of the couples of numbers , , , makes always 9.
The logical rules of RYB colours model and the digital root operation are here combined to create a “chromatic arithmetic”. The pattern in the Figure 2 displays the symmetries of colours complementarity in the digital root reduction of the multiplication table.
Chromatic Analysis of Fibonacci-like Sequences
The rules of “chromatic arithmetic” applied to Fibonacci [4] and Lucas [5] sequences display: Fibonacci = … 2, 3, 5, 8… (secondary colours + 3); Lucas = … 1, 3, 4, 7… (primary colours + 3).
We align the digital roots of Fibonacci and Lucas sequences on the common term “3”:
1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1…;
2, 1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9, 7, 7, 5, 3, 8, 2…
Gelatti
The sum in column of the terms of the two sequences creates a third sequence of digital roots: 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3… entirely made of achromatic colours. The three sequences could also be simplified as “adding 3 to numbers 1, 2 and 3”.
Figure 2: Chromatic multiplication table.
Figure 3: Tables of Fibonacci-like sequences, starting by adding 3 to numbers from 1 to 9.
Figure 4: Tables of Fibonacci-like sequences, starting by adding 9 to numbers from 1 to 9.
We then start any sequence by adding 3 to all numbers from 1 to 9 (Figure 3): a new chromatic sequence appears only in column 7 and 8. Three chromatic columns repeat twice (1 and 5, 2 and 4, 7 and 8 are equivalent). They are made of all colours and have a Pisano period [6] of 24, divided into two complementary periods of 12. The achromatic column repeats with a Pisano period of 8, divided into two complementary periods of 4. For the given equivalence of digital roots with modulo 9, only by adding 9 (Figure 4) to all numbers from 1 to 9, we produce a new column made of grey and Pisano period of 1.
From these observations we can derive a theorem of “chromatic arithmetic”.
Theorem. All the possible Fibonacci-like sequences of digital roots are described by only five sequences of colours: three are made of all colours, one of achromatic colours and one of only grey.
Proof of theorem. The patterns of the chromatic tables of Figures 3 and 4, along with general rules of recursive addition, digital root reduction, and modulo 9 arithmetic, provide a visual mathematical proof.
References
[1] Goethe, Zur Farbenlehre. Cotta, 1810; also Itten, Kunst der Farbe, Verlag, 1961. [2] http://mathworld.wolfram.com/DigitalRoot.htmla, (as of Feb. 2, 2014). [3] http://en.wikipedia.org/wiki/Modular_arithmetic, (as of Apr. 13, 2014). [4] Fibonacci, Liber Abaci, Springer, 2003. [5] http://oeis.org/A000032, (as of Feb. 2, 2014). [6] http://en.wikipedia.org/wiki/Pisano_period, (as of Apr. 13, 2014).