Metallic Number Systems

I’ve often talked about the golden ratio in my blog, this is because it is a large part of what I have been casually researching. Used by artists such as Leonardo Da Vinci in art and linked to the Parthenon and other arcitecture the golden ratio is the first in a series of irrational numbers called the Silver Means.
In this post I shall refer to them as the Metallic Series and denote them as \mho_1, \mho_2, \mho_3, ..., \mho_n. Also note that: \mho_1 = \varphi the golden ratio, and \mho_2 = {\delta_s} the silver ratio.
The letter ℧ is pronounced ‘mho‘ and is an inverted omega ‘ohm’.

1. The Metallic Series (Silver Means)

\mho_n = \frac{1}{2}({n + \sqrt{n^2 +4}}) To generate the Nth metallic mean, this formula is used.
x^2 - nx - 1 = 0 The metallic means are the solutions to this quadratic equation where x = \mho_n
n+\cfrac{1}{n+\cfrac{1}{n+\cfrac{1}{n+\cfrac{1}{\ddots}}}} This is the continued fraction for the Nth metallic mean. It can also be expressed [n; n,n,n,n, ...]

Above is the important information about the series of numbers I have been looking into, alongside finding out about their general numeric and geometric properties I have been looking into their possible uses in computing as numeral systems. Comming into this I already had a lot to read up on because of the exstencive work already done on the golden and silver ratios. Infact there is already a golden ratio base numeral system, which I talk about in this post. The golden ratio and the Fibonacci numbers are deeply intwined as are the Pell numbers with the silver ratio. For neetness I bundled all the number sequences into one function:

2. The Metallic Sequences
\boldsymbol{\mathcal{M}_b^S}(n) = \begin{cases} S & \text{if } n = 0 \\ 1 & \text{if } n = 1 \\ b \mathcal{M}(n-1) + \mathcal{M}(n-2) & \text{otherwise. } \end{cases}

This is the Metallic Sequence Function. The base b of the sequence relates directly to the metallic series. The starting value S affects the numbers that apear in the sequence.
The reason for this function is to encompass all the necessary number sequences, for example: {\mathcal{M}_1^0} = {F} the Fibonacci numbers, {\mathcal{M}_1^2} = {L} the Lucus numbers and {\mathcal{M}_2^0} = {P} the Pell numbers (Look here for a previous post and summary of the Fibonacci, Lucus, Pell numbers). It goes on to make the Pell-Lucus numbers and other related sequences.
A very important attribute is that whatever the start value S is set as – the sequence produces the same ratio, regardless of the numbers of the sequence being different:

\displaystyle \lim_{n\to\infty}\frac{\boldsymbol{\mathcal{M}_b}(n+1)}{\boldsymbol{\mathcal{M}_b}(n)}= \mho_b

3. The Metallic Series as Radix Systems
For the purpose of this post I will refer to the radix as radix and the amount of symbols or logical base as the base type, so Phinary is radix 1.61803.. and a binary system. Here is an example of what we’ll be working with: {1000.1001}_{\mho_1} = 5_{10} you may have seen this before with the phinary notation {1000.1001}_{\varphi} . The subscript denotes the radix in use.
The system we all use is the decimal system, radix 10. Through out the ages many different ones have been used: sexagesimal (Summarians, Babylonians, Ptolemy), vigesimal (Mayans), duodecimal, octal, etc. The positional numeral system:

a_n \dots a_2a_1a_0 = (a_n {\times} b^n) + \dots + (a_2 {\times} b^2) + (a_1 {\times} b^1) + (a_0 {\times} b^0)
Example: 1250 = (1 \times 10^3) + (2 \times 10^2) + (5 \times 10^1) + (0 \times 10^0)

with a fixed radix value hasn’t always been in use. The Romans for example used symbols with fixed values, as did the egyptians. There are lots of unusual number systems (complex, alphabetic, fractional, etc) and I find them genuinly interesting – I recommend “The Universal History of Numbers I” which is a book I’ve enjoyed.
But anyway, where was I? Ah yes- Phinary!

Something to point out from 1000.1001_{\mho_1} = 5_{10} is that the phinary representation is a lot longer a number than the decimal version. But lets level the playing fields and see what 5 looks like in binary, radix 2:

1000.1001_{\mho_1} = 101_2 = 5_{10}

These are both binary systems yet the phinary number is still really quite long compared. This isn’t unexpected because the phinary radix is less than the binary one and generally the larger the radix the shorter the representation. But what about that decimal point? Well the golden ratio (phi) is an irrational number and the rational number 5 needs negative powers of phi to get to the whole number. Infact all integers (bar 1) need them, all multiples of phi ofcourse don’t: 10_{\mho_1} = 1 \times {\varphi} \approx 1.61803 and then 50_{\mho_1} = 5 \times {\varphi} \approx 8.09016

Using the method of converting decimal to phinary (wikipedia link) we can convert decimal to any metallic system by replacing \varphi with any \mho_n, here are some numbers in the first four metallic systems:


4. Metallic Arithmetic
For a number system to be useful it needs to have some rules governing it’s behaviour – arithmetic. After taking the rules for phinary arithmetic and attempting to apply them to the silver ratio base (℧2) it was obvious the adjustments would be minor and infact, much like the metallic sequence function, a general set of rules can govern the whole series of systems. This is akin to standard arithmetic and integer bases.
One important thing that the metallic systems have is a duality of form, they can be standard or non-standard. In phinary no two 1s can be next to each other, so “011” becomes “100”, this equality can be shown via the definition of the Fibonacci sequence: F_{n} = F_{n-1} + F_{n-2}, the general rule to standardize any metallic number is:

0BN + \rightarrow 1\underline{B}\underline{1}

The next two rules govern possible situations after arithmetic, overflows and negatives.

0C00 + \rightarrow 1\underline{C}A1
00\underline{N} + \rightarrow [\underline{1}B1]_{\times N}

Where B is the base, using the rules of beta-expansion it is the floor of the radix. A=(B-1) and unsuprisingly C=(B+1). I also used N to denote any number and underlining to denote negative numbers. The rules are additive and should always be applied from the right (LSD) to the left (MSD).

Standardize examples :

\text{Radix } \mho_1 \text{Radix } \mho_4
\begin{matrix} 1000.1001 + 1 &=& 1001.1001 \\ & & 01.1 \longrightarrow 10.0 \\ \\ &=& 1010.0001 \\ \end{matrix} \begin{matrix} 40.02 + 3 &=& 43.01 \\ & & 043 \longrightarrow 102 \\ \\ &=& 102.01 \\ \end{matrix}

Overflows & standardize examples:

\text{Radix } \mho_2 \text{Radix } \mho_3
\begin{matrix} 200.02 + 10.11 &=& 210.13 \\ & & 210.2011 \\ \\ &=& 1000.2011 \\ \end{matrix} \begin{matrix} 20.11 + 30.01 &=& 50.12\\ \\ \\ &=& 112.22 \\ \end{matrix}

Negatives, overflows & standardize examples:

\text{Radix } \mho_1 \text{Radix } \mho_2
\begin{matrix} 1000.1001 - 100.01 &=& 1\underline{1}00.1\underline{1}01 \\ & & 1\underline{1}0\underline{1}.2001 \\ & & 1\underline{1}00.0011 \\ & & 1\underline{1}00.01 \\ & & \underline{1}2000.01 \\ \\ \\ &=& 10.01 \\ \end{matrix} \begin{matrix} 1000.2011 - 1 &=& 100\underline{1}.2011 \\ & & 1\underline{1}20.2011 \\ & & \underline{1}3020.2011 \\ & & 130.2011 \\ & & 201.3011 \\ & & 202.0121 \\ \\ &=& 202.02 \\ \end{matrix}

Hypothesis:
\text{These rules are applicable to all irrational radix systems}

Another series of radix systems that it will be easy to check against is the N-root series. The √2 as a radix makes for easy conversion from binary, between each digit we just put a zero – so 1011_2 becomes 1000101_{\sqrt{2}} . For the cube root two zeros are placed between each digit. So (N-1) zeros are placed between each digit to convert to an N-root radix.

\begin{matrix} 101_{\sqrt{2}} + 100_{\sqrt{2}} &=& 201 &=& 3_{10} + 2_{10}\\ & & 1002 & & \\ & & 1010.01 & \approx & 4.74264_{10} \\ \end{matrix}

Result:
\text{Nope.}

I’ll be dipping in and out of this page, when I edit or add to this research I will also post about it on the home page..

Related reading:

5 thoughts on “Metallic Number Systems”

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  3. I discovered phinary numbers a few years ago and still find the whole subject pretty fascinating.
    I think, the most interesting thing about them is that they can be read backwards (right to left) while negating each odd place digit. This is the same as interpreting them (left to right) as base negative 1/phi (=-0,618) instead of base phi.
    I haven’t looked into the other metallic bases, but I imagine that they have similar inversions / symmetries. Possibly one would need to convert them into something other than standard form, also allowing for negative “phigits” to make this more apparent.
    The other thing that is rarely mentioned about them is how the sum of positive even powers and their inverses (i.e. phi^8 + phi^-8), as well as the difference of odd powers and their inverses (i.e. phi^7 – phi^-7) always result in a whole (Lucas) number.

    Also, many interesting things can be said about phi and its relation to binomial coefficients… It may seem quite trivial in hindsight – once you realize that formulas such as (phi^2 – phi^1)^x , (phi^2 – phi)^x or (phi^-2 + phi^-1)^x are always equal to 1^x = 1 for any x, but just expanding the binomials yields beautiful identities, such as the ones for the sums over k=0..n of binom(k,n) * phi^-k and binom(k,n) * (-1)^k * phi^k and such.
    If you’re interested in sharing / discussing such findings in more detail, I’d be quite happy to.

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