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

It is also a fact (I refer you to

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

Take also while we're at it, my old friend the

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

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

(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

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-

It has long been my desire to model this process, to

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

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.)

Back to the Tverse main page, or backbutton yourself to where you were, and thanks for reading!