The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a fractal named after Wacław Sierpiński who described it in 1915. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.

Given my curiousity for such sets I decided to use my 3D IFS modeling tool of choice: Xenodream, to explore a little deeper and see what I could find.

The basic triangle is usually constructed in the 2D plane with these simple steps
1) Start with any triangle in a plane
2) Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner
3)Repeat step 2 with each of the smaller triangles infinitely (or until you run out of resolution)

The next highest order is to do the same thing in the 3d plane. Xenodream ships with several starter .xep files (xenodream extension) which I used as starting points for these images I started by making this movie which is a seamless loop of a 360 degree rotation of the 3D Sierpinski triangle with some lighting effects. the movie is licensced under creative commons - please credit this page if you use it elsewhere

I started exploring this model from different perspectives and when viewed from the top down (as if it were sitting on a table and you were hovering right above it) I saw some very familiar geometries. The first thing I noticed is that it appeared isometric  (equality in dimensions or measurements) in this case - all the angles were the same (60 degrees) which is not a very large surprise considering the model is composed of equilateral triangles.

Sierpinski’s Triangle (all aspects equilateral) as viewed from above looking straight down. The center is the point.

I also noticed that the geometry was in Octaves and since the hexagons were so prevalent in the structure I started wondering if perhaps the Flower of Life could be mapped on top of this. Without getting too detailed the Flower of Life is a simple rule of repeating geometry by which all five of the platonic solids can be extrapolated - it is the basis on which all of sacred geometry is formed.  The image below shows the standard method of creating the seed of life - which if you continue adding circles after this to any point where a line intersects you end up with the Flower of life.

 

From the Flower of Life you can extrapolate the “Fruit of Life” which if you connect all the center points of these circles together you end up with Metatrons cube - of which you can selectively show lines to create all 5 of the platonic solids as illustrated below. What i find most fascinating is that this geometry is not only fractal but can grow in Octaves. Which makes it very useful for practical applications such as the creation of art work or creating structures that can scale infinitely which is useful for such things as the sand circles we used to do.

So, with that explained, I was curious if Sierpinski’s Triangle followed the same geometry as Metatrons cube. I had some initial trouble with the starter files as the placement of the triangles wasn’t exact, however after some very gracious work from Garth (creator of Xenodream) everything lined up and I was able to lay a Flower of Life grid on top of the render of Sierpinski’s Triangle as seen from above.

You can download this xep file for Xenodream here

As you can see below the teal lines  show that the geometry is also isometric which is also very handy for creative applications. Since the flower of life maps absolutely perfectly that means Metatrons Cube also will and anything derived from that as well. See the images below for some examples.

The Flower of life and Isometric grid laid on top of Sierpinski’s Triangle as seen from the top (click to see full res)

 

Metatrons Cube Overlaid on Sierpinski’s Triangle as viewed from above (click for full res)

 

Octohedral lines extrapolated from Metatrons cube overlaid on top of Sierpinski’s Triangle

What does this mean? Is it anything more than intellectual masturbation? I personally dont know, yet none the less I feel like I just bumped my nerdiness up a notch and will be able to lose people the moment I open my mouth. Most Superheros get some great superpower… I got stuck with the ability to make peoples eyes glaze over and look to the middle distance when I start talking. Oh well.

So if you made it this far and still want more: Be sure to check out some illustrations I made (poorly and not 100% precise due to my lack of skill in Illustrator when they were made several years ago) of Octaves of the platonic solids as derived from Metatrons Cube

http://www.phidelity.com/photos/v/Artwork/Sacred+Geometry/Octaves/

Or the article I wrote about Pythagoras Tree

Dont Forget the Sand Circles derived from some basic Sacred Geometry

I also asked what would Sierpinski’s Triangle look like if it had a square base rather than a triangular one and came up with this image which is viewed from the top. That’s all for this article! thanks for dropping by and be sure to leave your comments and thoughts!

This article was posted on Friday, June 27th, 2008 in the category: Fractal.
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