Euler > Riemann > Homology { Betti Numbers }

The list of cosmic, sub-catagorical three-ness Ive posted before, I have no document to copy and past for reference.

Arthur Young cosmic version { The Reflexive Universe }  was based on seven-ness as associated with an derived from torus four lines of convergence { 0-3 } and divergence { 4-7 } as derived from his four lines { 0v-1v--v2--3v - ^4-^5-^6-^7 }

Bucky Fullers cosmic version was base on 3-fold { tetra structural integrity }, 4-fold { cubo-octa-hexa operational integrity } and 5-fold { penta alternate systemic integrity } and that is a cosmic three-ness.

 However, those three do no appear to be considerate of a torus topology, at least in Synergetics { for the most part }. My approach resulted in regarding 2-dimensional great circle planes as 3-D great tori.  It was sort of similar way that Jim Lehman took the 1D radii and chords of 0-1 frequency VE and expanded them as 3D PODs { Euclidean and Curved }.

..." But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable. If you want a mathematical justification that a T-shirt and a pair of pants are different, you should turn to a topologist, not a geometer. The explanation: They have different numbers of holes. "....

..."For instance, in 1813 the Swiss mathematician Simon Lhuilier recognized that if we punch a hole in a polyhedron to make it more donut-shaped, changing its topology, then V – E + F = 0. "...

..." Bernhard Riemann was studying surfaces that arose in his study of complex numbers. He observed that one way of counting holes was by seeing how many times the object could be cut without producing two pieces.......A straw can be cut once without disconnecting it, and a hollow torus can be cut twice."...

..." Notable among these was the concept of homology, which Poincaré introduced to generalize Riemann’s ideas to higher dimensions. Through homology, Poincaré aimed to capture everything from Riemann’s one-dimensional circle-like holes in a straw or binder paper, to the two-dimensional cavity-like holes inside Swiss cheese, and beyond to higher dimensions. The number of these holes — one for each dimension — became known as the Betti numbers of the object in honor of Enrico Betti, a friend of Riemann’s who had attempted similar work."...

....." The torus shows us how to visualize Betti numbers. We can produce infinitely many nontrivial loops on one, and they can wind, double back and wrap around multiple times before ending at their starting point. But rather than producing a chaotic mess, these loops possess an elegant mathematical structure. "...

..." Which of the following groups of animals is having tube in tube body plan?

Best you begin with one line of text at a time.  Ex the first line, as i'm here for you when your sincerely interested in understanding.

Also see post #3 ...or not.... for additional elaboration on the topics.

You and I are like most people who fear stepping outside of the classical box of understanding. It is strange foreign/new world to them and much safer to stay within the boundaries, or at least near the boundaries of accepted thought processes.

Exploration always comes with potential for mental { Meta-space }  peril ---ex ego death--  or,

  actual { physical reality } peril--- ex "One of the most well-known explorers to have died in Antarctica was Captain Robert Falcon Scott. Scott's expedition of five men died in 1912 "---

I became fixated on Bucky Fullers { 1895-1982 } thought processes beginning with his book Critical Path around 1984.