Show HN: an extremely simple program shows a 2D structure in the integers

24 pointssage_joch12y ago13 comments

This seems unlikely to be a "new" idea, but I stumbled into it by accident and thought it was cool. Java source code below:

  // This class is in the public domain.
  public class Numbers
  {
    public static void main(String[] args)
    {
      int numRows = 256;
      int numColumns = 128;
      for (int row = 0; row < numRows; row++)
      { 
        StringBuilder rowOutput = new StringBuilder();
        for (int column = 0; column < numColumns; column++)
        {
          if ((row & column) != 0)
            rowOutput.append("1");
          else
            rowOutput.append("0");
        }
        System.out.println(
          String.format("row/col %s:\t%s", row, rowOutput.toString()));
      }
    }
  }

13 comments

psuter12y ago

This was on HN a couple of days ago:

  http://www.oftenpaper.net/sierpinski.htm

It points out that you can interpret the table as the relation "is disjoint" between the bitsets encoded in the row and col indices (CTRL+F for "binary logic"). Following links from there will take you to a rabbit hole of combinatorics that I will not claim to fully understand.

The whole page is a delight, though, and given your enjoyment for your discovery, you'll certainly love browsing it.

duncan_bayne12y ago

Reminds me of Sierpinski triangles:

http://en.wikipedia.org/wiki/Sierpinski_triangle

I'm sure someone with a greater intuition for mathematics than I have will be able to explain why.

daxelrod12y ago

My first thought is that it has something to do with the fact that you get a Sierpenski Triangle from a Pascal's triangle when you assign one color to even numbers and another to odd: http://en.wikipedia.org/wiki/Pascal%27s_triangle#Overall_pat...

1 more reply

modeless12y ago

Here's a C version in less than 80 characters:

  main(c,r){for(r=32;r;)printf(++c>31?c=!r--,"\n":c<r?" ":~c&r?" `":" #");}

I first learned of this phenomenon in my discrete math class, and we had a competition to see who would write the shortest program to draw a Sierpinski triangle. This version would certainly have won if I'd been clever enough to write it at the time, though even shorter versions are possible in other languages.

chch12y ago

http://faculty.msmary.edu/heinold/bitwise_and.pdf

seems to be a paper on this phenomenon, with

http://faculty.msmary.edu/heinold/maa_pres2009-11-14.pdf

being a similar work in presentation form.

I've never seen it before; I like it!

rjbwork12y ago

For any lazy C# folks:

    internal class Program
    {
        public static void Main()
        {
            int numRows = 256;
            int numColumns = 128;
            for (int row = 0; row < numRows; row++)
            {
                StringBuilder rowOutput = new StringBuilder();
                for (int column = 0; column < numColumns; column++)
                {
                    if ((row & column) != 0) rowOutput.Append("1");
                    else rowOutput.Append("0");
                }
                Console.WriteLine("row/col {0}:\t{1}", row, rowOutput);
            }
            Console.Read();
        }
    }

Cool stuff, very interesting to see that triangle structure emerge.

parkovski12y ago

For the even lazier: http://dotnetfiddle.net/gsdfJp

zoba12y ago

Interesting, I've not seen this before. Here is a screenshot of some of the output: http://i.imgur.com/Ro2AA6u.jpg

andrewflnr12y ago

That's cool. You can expand the pattern, presumably arbitrarily, by setting the number of columns to 256 or 512 (or greater?), but on my little 1440px screen I had to redirect to a file and scroll it sideways in an editor to see the whole thing.

vittore12y ago

You will probably like searches for rule NNN on wolfram alpha, like http://www.wolframalpha.com/input/?i=rule+101

LocalMan12y ago

Back when they taught me the number line and the Cartesian plane, it all seemed smooth and straightforward.

In the years since, I've seen primes, rationals, irrationals, Mandelbrot, and all manner of weird structures popping up out of what seemed like nothing.

How did this happen? And who put all that stuff in there?

catmanjan12y ago

Sierpinski's Triangles?

tunesmith12y ago

Is this the same as Rule 90?

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