Found Math — Neolithic Edition

OU Math Logo By now you surely know of the OU Math Club logo (to the left).  You may even know that the non-OU part of the design is Sierpinski’s Triangle (p.s. The entire Triangle is in the logo — you just have to look really, really close :-)).  Dr. Brady recently let us know that the ancient Irish may have had a similar design in mind over 5000 years ago!  Namely, one of the oldest buildings still standing in the world is the Newgrange Tomb which is located here in Ireland.

Besides being over 500 years older than the Great Pyramids, and being of somewhat mysterious origins (If you ask an Irishman, they may say it was built by Dagda), it has beautiful carvings in it’s stonework.  It is most famous for the Triple Spiral image:

However, there are many other images, including one on Kerb 67 which some argue is an early version of Sierpinski’s Triangle:

A closeup of Kerb 67

Here is another look at the same stone:

Another image of Kerb 67 with Sierpinski's Triangle

Now it may be that Dr. Brady has a pro-Irish, pro-Math Club logo, pro-Neolithic bias, but there definitely some similarity.  You’ll have to judge for yourself if an artist 5000+ years ago was the first to think about a pattern of nesting triangles.

However, we can tell you of our two favorite ways of making the Sierpinski Triangle:

  1. Make a very large (20+ rows) Pascal’s Triangle, ideally on a sheet of graph paper with one number per square so everything stays symmetric.  Color in all the even numbers and see what pattern emerges.  Bonus:  Do the same, but use one color for those which are evenly divisible when you divide by 3, and another color for all the numbers which have remainder 1 when you divide by 3.  How can you generalize this?  See the papers of Dr. John Holte.
  2. Start with an equilateral triangle.  Label the corners with 1, 2, 3.  Pick one corner as your starting position.  Randomly (e.g. roll a die and divide by 2, or use a random number generator) choose between 1,2,3.  Go half the distance between your current position and the corner you randomly selected.  Make a dot.  This is your new position.  Randomly choose between 1,2,3 and go half the distance between your current position and the corner you randomly selected.  Make a dot.  Repeat 100+ times.  What pattern emerges? Bonus:  Write a script for a computer to do this for you and save yourself a lot of work.  Then use it to see what you get for a square, pentagon, etc.