In my writing, I've come into contact with tons of amazing mathematicians, one of them being a world leading expert in the Cosmati floors. Meet Laura. She knows my mentor, which is super cool. Laura lives in Rome.. in my search for the medieval techniques used by the Roman marble workers (Marmorari Romani), she has suggested that I organize a group construction project to recreate a Sierpinski triangle using these old techniques. Because DOC would probably not let me build using tile, I settled for quilting. That was a long story made short.
Ad triangulum is a rule of subdivision. It's used in many old buildings, especially in medieval times. Timber framers still use it as well. Basically, begin with the construction of an equilateral triangle (all sides equal). If you inscribe a circle, you will see that there are precisely three spots where the circle and the triangle intersect. These are called points of tangency. By connecting the points of tangency with straight lines, you will see that you have just subdivided the triangle into four identical triangles.. except, well I'll be damned, there is an upside down triangle right in the middle! Throw that out, remove it.. whatever. With the three remaining upright triangles (again they are all equilateral), repeat the process. Now we will have 9 upright triangles. continue applying ad triangulum on all the upright triangles infinitely.. that is, ad triangulum ad infinitum. Thus, in this manner, ad triangulum ad infinitum becomes the Sierpinski triangle. That is the medieval method, a bit toned down and completely devoid of any witch burnings.
This project we're working on is a recreation of a Cosmati mosaic found at the Cloister of St. John at Lateran. The original piece is in ruins... we are recreating the piece via quilt. The old geometry will be shown in the actual quilting pattern. So, not only will the quilt show the mosaic, but we are mapping the process used by the early Marmorari Romani in our thread design. This will only be noticed by historians or mathematicians. The casual observer will be very impressed, but might not know what they are looking at. We wanted to capture the old technique into the construction of our piece, originally done almost 1000 years ago by the Vasaletto family of artisans.
Finally, the back is a cellular automaton called Rule 90... aka the Sierpinski automaton... Google it if you must. Also, here's what I want the world to realize. Waclaw Sierpinski is credited the fractal called Sierpinski's gasket (or the Sierpinski triangle) from 1915. He called it "a curve, every point of which is a point of ramification". But how could he have discovered the Sierpinski triangle when it was seen on the floors of Italian churches for many hundreds of years? I propose that Sierpinski gained inspiration from the Cosmati floors, either directly or indirectly (Janizewski), prior to 1915.