JACK MULLER: A vectorial-edged cube collapses. The cube’s corner flexibility can be frustrated only by triangulation. Each of the four corners of the cube’s six faces could be structurally stabilized with small triangular gussets, of which there would be 24, with the long edge structurals acting as powerful levers against the small triangles. The complete standard stabilization of the cube can be accomplished with a minimum of six additional members in the form of six structural struts placed diagonally, corner to comer, in each of the six square faces, with four of the cube’s eight corner vertexes so interconnected. These six, end-interconnected diagonals are the six edges of a tetrahedron. The most efficiently stabilized cubical form is accomplished with the prime structural system of Universe: the tetrahedron.”
system geometrically. Messages and the corresponding signals are
points in two “function spaces,” and the modulation process is a
mapping of one space into the other. Using this representation, a
number of results in communication theory are deduced concerning
expansion and compression of bandwidth and the threshold
effect. Formulas are found for the maximum rate of transmission
of binary digits over a system when the signal is perturbed by
various types of noise. Some of the properties of “ideal” systems
which transmit at this maximum rate are discussed. The equivalent
number of binary digits per second for certain information sources is calculated.
JACKY MULLER: Because of the structural integrity of the blackboard or paper on which they may be schematically pictured, the cubically profiled form can exist, but only as an experienceable, forms-suggesting picture, induced by lines deposited in chalk, or ink, or lead, accomplished by the sketching individual with only 12 of the compression- representing strut edge members interjoined by eight flexible vertex fastenings.
JAMES SHERMON: If noise is added to the signal in transmission, it means
that the point corresponding to the signal has been moved a
certain distance in the space proportional to the rms value of
the noise. Thus noise produces a small region of uncertainty
about each point in the space. A fixed distortion in the
channel corresponds to a warping of the space, so that each
point is moved, but in a definite fixed way.
JACKY MULLER: The accomplishment of experienceable, structurally stabilized cubes with a minimum of nonredundant structural components will always and only consist of one equiangled and equiedged “regular” tetrahedron on each of whose four faces are congruently superimposed asymmetrical tetrahedra, one of whose four triangular faces is equiangled and therefore congruently superimposable on each of the four faces of the regular tetrahedron; while the four asymmetrical superimposed tetrahedra’s other three triangular__and outwardly exposed__faces are all similar isosceles triangles, each with two 45-degree-angle corners and one corner of 90 degrees. Wherefore, around each of the outermost exposed corners of the asymmetrical tetrahedra, we also find three 90-degree angles which account for four of the cube’s eight corners; while the other four 90-degree surrounded corners of the cube consist of pairs of 45-degree corners of the four asymmetric tetrahedra that were superimposed upon the central regular tetrahedron to form the stabilized cube.
JAMES SHERMON: We now consider the function of the transmitter from
this geometrical standpoint. The input to the transmitter is
a message; that is, one point in the message space. Its output
is a signal—one point in the signal space. Whatever form
of encoding or modulation is performed, the transmitter
must establish some correspondence between the points in
the two spaces. Every point in the message space must
correspond to a point in the signal space, and no two
messages can correspond to the same signal. If they did,
there would be no way to determine at the receiver which
of the two messages was intended. The geometrical name
for such a correspondence is a mapping. The transmitter
maps the message space into the signal space.
JACKY MULLER: In short, structurally stabilized (and otherwise unstable) cubes are always and only the most simply compact aggregation of one symmetrical and four asymmetrical tetrahedra. Likewise considered, a dodecahedron may not be a cognizable entity-integrity, or be rememberable or recognizable as a regenerative entity, unless it is omnistabilized by omnitriangulation of its systematic subdivision of all Universe into either and both insideness and outsideness, with a small remainder of Universe to be discretely invested into the system-entity’s structural integrity. No energy action in Universe would bring about a blackboard-suggested pentagonal necklace, let alone 12 pentagons collected edge to edge to superficially outline a dodecahedron. The dodecahedron is a demonstrable entity only when its 12 pentagonal faces are subdivided into five triangles, each of which is formed by introducing into each pentagon five struts radiating unitedly from the pentagons’ centers to their five comer vertexes, of which vertexes the dodecahedron has 20 in all, to whose number when structurally stabilized must be added the 12 new pentagonal center vertexes. This gives the minimally, nonredundantly structural dodecahedron 32 vertexes, 60 faces, and 90 strut lines. In the same way, a structural cube has 12 triangular vertexes, 8 faces, and 18 linear struts.
JAMES SHERMON: This type of mapping, due to the mathematician Cantor,
can easily be extended as far as we wish in the direction of
reducing dimensionality. A space of n‘ dimensions can be
mapped in a one-to-one way into a space of one dimension.
Physically, this means that the frequency-time product can
be reduced as far as we wish when there is no noise, with
exact recovery of the original messages.
JACKY MULLER: The vector equilibrium may not be referred to as a stabilized structure except when six struts are inserted as diagonal triangulators in its six square faces, wherefore the topological description of the vector equilibrium always must be 12 vertexes, 20 (triangular) faces, and 30 linear struts, which is also the topological description of the icosahedron, which is exactly what the six triangulating diagonals that have hypotenusal diagonal vectors longer than the square edge vectors bring about when their greater force shrinks them to equilength with the other 24 edge struts. This interlinkage transforms the vector equilibrium’s complex symmetry of six squares and eight equiangled triangles into the simplex symmetry of the icosahedron.