THE SPHERE

A sphere is a polyhedron 
of an invisible plurality of trussed (triangle) facets.

As vertices & trussed faces multiply 
the condition is rapidly approached 
where both the chord & arc lengths 
& spherical surface angles & their chordal facet angles
become practically indistinguishable 
& the polyhedrons surface becomes indistinguishable from the sphere.

If a further approach 
to the congruence 
of all-trussed chordal polyhedra with arc-structured spheres 
can be accomplished, 
the vertices & trussed facets (penetration points) multiply, 
providing increased advantage in more directions 
against concentrated loads & more directions of penetration.

As the number of trussed facets increase, 
the convex vertexial interactions 
approach a zero altitude condition,
which, though ideal for tension or internal pressure, 
tends to allow 
concentrated external loads 
to push the convex chordal vertices inside-out 
(ie. to a dimpled or concave condition).

In the dimpled or concave condition, 
continuing concentrated external pressure 
will be resisted
by a tension increase 
in omni-surface direction 
(eg. as a rubber ball draws on its skin as it resists punching in
 & gains reaction & springs back causing bounce).

The sphere encloses (defines) 
the most volume with the least surface.

Every time the diameter of a sphere 
is doubled
inclosed volume increases 8  fold  & surface area increases 4 fold.

Concavity within tends to regenerative reconcentrations 
of the withinness 
by angular convergence of vectors 
impinging or echoing from its surface.

The convex outer surface 
separates, diffuses & thins 
the energies impinging upon it or echoing from it
by angular divergence. 
The convex-concave surfaces of systems 
are the prime energy pattern differentializers of universe.
They occasion energy patternings 
either as local regenerative atomic masses 
or as universally divergent radiation.

The inseparably occurring positive (convex) & negative (concave) surfaces
represent the most effective angular valving or shunting 
of the operative energy events of the environment,
purposefully including this , excluding that.

As the no. of trussed faces (facets)
of symmetric polyhedra are increased 
from the chordal to the arc structural tetrahedra, 
through the hierarchy of great circle arc & chord trussed solids
(i.e. the octahedron then the icosahedron thru to the sphere)
the no. of vertices & edges increase 
providing more & dispersed structural interactions.

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