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
The Simple Explanation has also used nesting toroids to illustrate chakras.
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.


  1. Hi, Cyd
    What you've discovered is known as the flower of life - which has significant and central spiritual meaning for universal consciousness at macro levels and human consciousness at micro level. Here's a reference which describes these structures in terms of energy and evolution.

    DaVinci discovered these relationships - based on Fibonacci relationships and how our human bodies and consciousness are physically and spiritually formed accordingly. see

    the flower of life description links directly to davinci's work and beyond to our individual and collective energy fields in these physical planes, inspired by the relationship of spirit-matter which form the torroidal field(s) of consciousness

    interested in your thoughts

    Fred McEwan

  2. Hi Fred. Good to hear from you.
    Yes, it does resemble the flower of life. Reading up on the subject at Wikipedia and a website or two, it looks like the flower of life usually involves six circles plus the center circle making a pattern of seven circles. This seems to be important to the symbolic meanings people ascribe to the shape. My drawing is only five circles--four and one center. My model doesn't need the other two circles, as I seem to jump straight to the torus shape without incorporating that X axis the other two circles contribute.
    Having said that, sure, I'm claiming pretty much the same things of the torus shape that people associate with the "fruit of life" drawing--from wikipedia: "The Fruit of Life is said to be the blueprint of the universe, containing the basis for the design of every atom, molecular structure, life form, and everything in existence". So we probably are all talking about the same things, each from our limited POV.

  3. Hi, Cyd
    thanks for the reflections
    we are talking about the same things - My sense is that your torroidal models bring the reality of movement to the more static 2 dimensional flower of life model. It's vital to recognize the flow through as we move from model to the reality of the flows of spirit into consciousness into matter in recognizing our identities within these model structures. It is the stream of will/spirit into relationship with more dense, slower movement patterns we call matter that creates these torroidal fields of consciousness - the formulae of diverse forms/formulas of will-substance which then are reflection into dimensions of physical appearance.

    So I value these torroidals flow models as they bring further understanding to the reality behind appearances.

    I have further questions and ideas to explore along these lines of evolving consciousness from human to soul based that I'd explore with you from your insight into the torroidal models - if you're interested my direct email is Blessings, Fred

  4. Keep spreading truth! We the people are waking up :)

  5. Hi & thanks for your post. Very interesting & much appreciated. I am a graphic designer & would be delighted to draw the torus fractal picture for you - it's really fun to play with sacred geometry in illustrator I must say. Let me know your email & I will give it a crack for u

    1. Hi Marie. Thanks for volunteering to make some fractal toroids! Can't wait to see what you have. Can you upload them to a site where I can then repost to here?