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

24 points by sage_joch 12 years ago | 13 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()));
      }
    }
  }

psuter 12 years 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_bayne 12 years 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.

daxelrod 12 years 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...

modeless 12 years 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.

chch 12 years 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!

rjbwork 12 years 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.

parkovski 12 years ago | |

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

zoba 12 years ago |

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

andrewflnr 12 years 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.

vittore 12 years ago |

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

LocalMan 12 years 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?

catmanjan 12 years ago |

Sierpinski's Triangles?

tunesmith 12 years ago |

Is this the same as Rule 90?