Evolution of the Golden Cantor Comb and Fractal Arc de Triomphe

Hello, My name is Dr. Donald Plante and I have always been interested in visualizing mathematics through the use of models.  With the advent of 3D printing this has recently become much easier to do.  No more nights glueing 400 cubes together to make a level 2 Menger sponge!  Now I can design and print one in much greater detail in a matter of hours instead of days.

I just attended a conference held at Brown Universities ICERM called “Illustrating Mathematics”.  Many topics were discussed that involved cutting edge methods for creating 3D prints of various mathematical models.  This has inspired me to start a blog of the same name to show off some of my own work.

I have been inspired by Henry Segerman’s work with 3D printed fractal formations so I attempted to create my own in Tinkered.  To start off simple I wanted to show the evolution of the classic cantor set.

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This was printed in PLA to show the continuous transformation of the line segment [0,1] into the 4th level of the Cantor set.  I also made the the dimensions of the overall rectangle match the golden ratio and printed it in gold to give the comb some style.

Next, I wanted to see if I could model something a little more difficult so I set my eyes on the evolution of the Cantor Dust fractal.

Evolution of the Cantor Dust fractal
Evolution of the Cantor Dust fractal

This was also made in Tinkered but printed in ABS on a Lulzbot Taz 5.  The evolution from the original square goes down to the 4th level of the fractal.  It starts with one cube and ends with 4^4=256 cubes.

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All 256 cubes

I also included arches in the transformation to give it the look of the Arc de Triomphe.  Like the golden Cantor Comb I scaled this fractal’s main arch to have similar dimensions to the real Arc de Triomphe.

Fractal Arc de Triomphe
A look inside the Fractal Arc de Triomphe

 

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