Numeral System Automata

Cellular Automata
Cellular automata Is made up of a grid of cells which sit in a finite number of states (such as on/off or blue/green/red). Cellular automaton (singular) adhear to the rules of the system and evolve over periodic time intervals. One successful application of CA is Conway’s Game of Life, the universe is an infinite 2D grid and the rules are as follows:

A single Gosper's Glider Gun creating "gliders"

  1. Any live cell with fewer than two live neighbours dies, as if by needs caused by underpopulation.
  2. Any live cell with more than three live neighbours dies, as if by overcrowding.
  3. Any live cell with two or three live neighbours lives, unchanged, to the next generation.
  4. Any dead cell with exactly three live neighbours cells will come to life.

Positional Number Systems
We use these all the time, notably the decimal system in which when a symbol exceeds 9 it “carries” one to the left and negates ten. 9+1=A, then A goes to 10.

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)

The Idea
To take the properties of a number system and create rules to govern individual cells. For our numeral universe we are going to need something different to what Conway used, ours is going to have an intrinsic property, gravity – and thus a floor to gravitate to. The center of our universe will be the radix point (decimal point), you could say the numeral universe is radixocentric. The real-life universe is made up of multiple dimensions, the numeral one is made up of infinite column dimensions.

Although initially I will only being dealing with addition and thus carry, I have put on paper negation in phinary which was quite fun. I talk about the arithmetic of phinary in my research (section 4). The Numeral Universe rules and an example are below:

  1. Positional System – Each dimensional element (cell) is the value of the previous dimension’s capacity.
  2. Gravity – Any cell above an empty cell moves to it.
  3. Carry – When a dimension is full, it moves to the next higher one.

Example in base 3 (ternary). The green and red blocks are the two numbers being added to the universe, the grey block is the carry.

But what does this all mean?
Well, I’m not really sure – it’s nice to show number systems in a different way, it could be useful for teaching. But as for me, I’m using the rules to govern an alternative tetris game, I’ll upload it when I’m finnished.

Also if anyone is wondering how I made those fractal looking blocks, I spammed the hell out of the filters on Pixlr the free online image editor.


5 thoughts on “Numeral System Automata

  1. I’ve always wanted to spend more time learning about number systems and such things that I never properly studied (for many reasons) but have read and tried to self-teach as and when time and resources permit.
    Would you recommend any topics/books/sites (just a couple at most) that are simple and easy ways to get into this kind of mathematical exploration? A good starting point?
    Thanks for sharing.

    • Do you understand binary, octal, decimal and hexidecimal?

      For other ones, if it’s recreational, I’d say start at the interesting ones then span off from there. Some interesting ones are complex number systems like Donald Knuth’s Quater-Imaginary (base 2i) and the dragon curve base (−1±i). The relationship between fractals and complex radix systems is amazing. Also irational bases are interesting and a must is the golden base (1+sqrt{5})/2.
      (Check my Research page if you like the golden ratio base)

      For non-standard systems, interesting ones are:
      The Mayans – First zero to be used arithmetically. Base 20.
      Sumerians (and Babylonians) – Later adapted by Ptolemy for astronomy and the reason we have 60mins/60secs and 360degrees. Base 60.
      Greek (Attic), Etruscan, Roman, Medieval Roman – Emerged in that order from each other. Non-positional, symbol values (V=5, I = 1, etc).

      There are absolutely loads: alphabetic ones like Hebrew, Greek (Ionic), Cryllic. For a book detailing many ancient/old systems try The Universal History of Numbers I by Georges Ifrah.

      I did once write a decimal to many convertor which had information about the history and properties of the systems but I have no idea where it is. Also it’s in Qt and I never learned how to set up for static release..

      Good luck!

      • Wow! Thanks a load, truly. Thats a lot more than I was expecting and I’m sure just perusing it will keep me busy a while!
        I was actually planning on taking up vedic mathematics as an alternate because I believe its remarkably effective and accurate.
        Some stuff that you’ve pointed out here is historical info I was not aware of either, thanks again for all this.

  2. Pingback: Complex Bases | Neural Outlet..

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