Wednesday, October 19, 2011

Toroidal Symmetries and Fractal Divisions

Noodling around with a protractor, bisecting circles into smaller circles to make them into toroids. Along the way, the 2-D drawing displays beautiful symmetries of various kinds.
I drew this toroidal pattern using a pencil compass on paper. The lines you see at the circles' centers is where the compass dug into the paper. Pardon the mathematical imperfections, as this was done freehand.



I imported the drawing into Paint and airbrushed out the compass scratches. I can see toroids all over this drawing, but you may only be seeing the circles.

Here I've used the Paint program to highlight the largest torus in this drawing, which shows up if you imagine this as a cross-section of a sphere. The blue lines show the cross section of the torus cutaway. This inner torus has been subdivided again into two smaller tori (darker blue circles). The vertical yellow lines are the poles; the longest pole is for the central torus; the two shorter lines are the poles of the two subdivided tori. You can go on subdividing each torus this way, each time dividing the cutaway of the torus into half-sized tori.
This torus and its poles was used to illustrate "the great square within the torus." Same view as the blue lined cutaway above.

The darkest blue circles represent my clumsy eyeball method of illustrating how the torus is dividing fractally. Each torus cross section can divide into two more.
I wish I had a program that would draw these things more accurately. Any volunteers?
Notice the interesting way this fractal division works. It will go on forever, larger and larger or smaller and smaller, in true fractal manner.

Usually when I've thought about multiply-linked tori, they appear to nest at a single center pole, like Russian dolls. But these new drawings show how the tori can divide fractally along different lines. 

Concentric tori courtesy of http://www.multidimensionalmusic.com/review2.html
The Simple Explanation has also used nesting toroids to illustrate chakras.
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A little more fooling around in Paint gives us a "quasi torus" in the vertical direction (yellow). Look for more on this tie-in to quasi-particles in a future article.
Here I've used the poles to define a quasi torus perpendicular to the blue set. Following the logic of the Simple Explanation, I don't really think the tori divide up and down like this, along the poles, since the poles represent time and motion rather than space. I think the tori only divide in the horizontal (blue) space. But here we can see that this structure has room for a fully symmetrical set of divisions in the vertical (yellow) dimension.