Euler's Law

How it may influence Tverse's initial spherical shell collapse, and how it may predict that the Universe should have distantly detectable icosahedral or dodecahedral symmetry

Euler's Law states that the number of EDGES a convex polyhedron has is equal to the number of FACES it has plus the number of VERTICES it has, minus the number "two." To my knowledge, this Law has no exceptions

E = F + V - 2


It is also a fact (I refer you to enormous mathematical resources on the WWWeb for all of these things) that any convex surface can be "triangulated" —broken up into triangles, be they regular or irregular.

Example: take the non-nucleated icosahedron (wire-frame, in the graphic below). It has twelve vertices, twenty faces, and 30 edges. In this case all the faces are triangles, and all the triangles are regular.
Take also while we're at it, my old friend the nucleated icosahedron. (Squeezed tight all-on-one-side, this would be my "Icosahedro(not)" ("Our Lady in Jade"). In this one (below), the triangles are also regular.

(See these two in other ways: non-nucleated in STEREO3D and/or in .TVG, nucleated in STEREO3D and/or in .TVG : )

Both of these obey Euler's Law ( 30 = 12 + 20 - 2 ), as, most likely, does EVERY convex polyhedron.
A sphere is then, in this surface-triangulated sense, a polyhedron whose triangles are so tiny that the surface appears to be smooth, and in the case of Tverse's spherical shell of pionts, is a gigantic triangulated sphere.

If we place balls (think: pionts, plus their "repulsion zones") as tightly as we can on a table, we find the best fit is an hexagonal pattern with six balls evenly spaced around one ball, everywhere. Even if we make the table an infinite plane, six balls surround one everywhere.

Now suppose we want to pick up this plane of balls and curve it into a sphere....

Two things happen: 1) it resists (the tight packing of 6-around-1 is only good in a plane), and 2 it eventually slips a ball out of itself here and there in order to overcome this resistance.

What we end up with, if we succeed in curving the plane into a closed sphere, is a surface-triangulated sphere which has twelve locations on it, more or less evenly spaced, where only five balls surround one ball (or, more likely, five surround none —surround a hole), while everywhere else six still surround one with some slight gaps involved.

(This is, in fact, what enables students of Buckminster Fuller to make "geodesic domes." Note in any picture of a geodesic dome except the smallest ones, the surface of the dome is covered by panels (or spaces enclosed by struts) of six sides (our 6-around-1 balls) interspersed with panels of five sides (our 5-around-1 or 5-around-none balls).)

Now, the meat:

When Tverse's expanding spherical shell reaches the enormous size to which "ambiguous flatness" applies, there are 12 places on it where things are "very uncomfortable" compared to the rest of the sphere-shell's content. These are the 12 locations at which it is most likely that "something will happen" —that balls will "find out" that they can make use of the interior of the now-huge sphere, to continue to get away from each other.

These are also places where, once this destabilizing situation has begun, balls may flood into the interior in a more or less five-around-none manner. This is very important to see; since in these places we may find it easier for Tverse to spew Tverse 'protons' into the interior, while all around them the destabilized shell also spews pionts headed for a more or less random, but once filled, 12-order-vertexed, patterning.

It has long been my desire to model this process, to prove with the math and the geometry that this is, indeed, what ought to happen at this stage of Tverse's "Origin Sequence," but friends, this sort of modelling cannot be accomplished with my program and on my machines. It would I believe require a supercomputer, running a version of my program written for it, to model this in anything like a reasonable amount of time.

Thus I issue a call: Hey! If you've gotten this far in my website, you've probably an interest in this stuff, and if you happen to be a GRADUATE STUDENT looking for a THESIS TOPIC, then please get in touch with me and let's talk. I'm getting old, and this all may die when I do unless someone else with a serious curiosity takes it all over from me.

Email me: John Brawley (When the address here HTML-ized shows up in your email client, DELETE the 'X' in front of it. I keep that there so spiders and bots can't harvest my email address.)

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