Sierpinski TetrahedronProject of the Visualization and Numerical Geometry GroupRendering 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.
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... .
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.
Casting a 2D shadow
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.
