Version 3 of Producing a fractal convex solid

Arjen Markus (7 may 2003) Have you ever seen the set of Pythagorean (regular) solids? Or the Archimedean solids that consist of two types of regular polygons? I find them fascinating - both with plain faces or as an Escher drawing.

Keith Vetter produced a script that helps you create them from paper. So I am not the only one. KPV Thanks, see Polyhedron Nets.

Here is my idea of producing a completely different type of solid. It is convex and it has all the characteristics of a fractal - that is: features that are repeated on ever smaller scales.

This is the procedure:

Take a Pythagorean solid - say a cube with a side of 1.
Cut off all corners, by removing a pyramid of side approximately 1/3 (*) (difficult to draw with plain text and I have not written a script yet to show the process)
This leaves an isosceles triangle as a new face and three new corners for each corner that was removed.
The original squares are now turned into regular octagons with side approximately 1/3. (This polyhedron is called a truncated cube, and you can see a picture of it at [L1 ]. KPV)
In the next round, cut off all the new corners again - by removing a pyramid of side 1/3 of the current side, so 1/9 of the original.
We now end up with a solid that has hexadecagons (16-gons), hexagons and triangles as faces - all regular with a side of approximately 1/9.
We can repeat the process ''ad inifinitum".

When we are done (in maths anything can be done, or at least imagined), we have a solid whose every face is a circle! Admittedly, there will be large circles and smaller ones, but there is no angular corner left.

Unless this kind of solid is already described, I claim the name Markus solid for this construction (or perhaps, to make sound more classic, Adrianic solid).

What I have not done yet, is concoct a script that will show the process step by step ...

(*) The approximate factor 1/3 is actually 1-sqrt(2)/2. Just apply Pythagoras' famous theorem ...

Category Mathematics