# Sierpinski Tetrahedron

Numerical Geometry Group

## Rendering with Povray

The Sierpinski tetrahedron has a simple construction principle which can be built impressively
with metal spheres. The raytracing procedure is excellent for the reflections on
spheres.
The poster on the left side was rendered with 300 dpi for an ISO A0 format.
The PDF can be downloaded
here.

Click on the images to see a bigger version.

## Iteration principle

The Sierpinski tetrahedron is the three dimensional version of the Sierpinski triangle
of Waclaw Sierpinski (1915). Halving the sides of an equilateral triangle, you can inscribe
three equilateral triangles of a quarter of the size of the area.
The fractal dimension lies between one and two. The (similarity) dimension D is defined as
logarithm of the multiplicity of the modules (c = 3, because three triangles substitute the original)
divided by the logarithm of the reciprocal scaling factor (r = 2, because two edges fit into the original).
Therefore we get D = log c/log r = log 3/log 2 = 1.58... .

The iteration principle for a tetrahedron is similar: the whole volume will be replaced
by four tetrahedrons with half the length of the edges. The (similarity) dimension of this fractal
is astonishingly an integer and computes to two:
D = log 4/log 2 = 2.

On the right hand side you see a simple tetrahedron built out of four spheres,
the zeroth iteration step.

In the next iteration step a tetrahedron of half the size will be constructed in each
vertex of the original. In this new tetrahedron there is a hole of the size of an
octahedron with triangle surfaces.

## Iterations steps two to seven

Going through the next iteration steps, the number of spheres used will increase
by a factor of four for each step.
The diameter will be halved at the same time.

Proceeding to infinity with this process leads to a fractal structure.

Its dimension is at least two, which can be seen easily when parallel light is
coming in over the middle of one edge and shining towards the middle of the
opposite edge. The resulting shadow then is a filled rhombus.

Last Update: 01.02.2011