[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.

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:

   1. Take a Pythagorean solid - say a cube with a side of 1.  
   1. Cut off all corners, by removing a pyramid of side 1/3 (difficult to draw with plain text and I have not written a script yet to show the process)
   1. This leaves an isosceles triangle as a new face and three new corners for each corner that was removed.
   1. The original squares are now turned into regular octagons with side 1/3.
   1. 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.
   1. We now end up with a solid that has hexadecagons (16-gons), hexagons and triangles as faces - all regular with a side of 1/9.
   1. 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 concot a script that will show the process step by step ...

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[Category Mathematics]