986.060 Characteristics of Tetrahedra 
Fig. 986.061 
986.061
The tetrahedron is at once both the simplest system
and the simplest
structural system in Universe (see Secs.
402
and
620).
All systems have a minimum set of
topological characteristics of vertexes, faces, and
edges (see Secs.
1007.22
and
1041.10).
Alteration of the minimum structural system, the tetrahedron,
or any of its structural
system companions in the primitive hierarchy (Sec.
982.61),
may be accomplished by
either external or internal contact with
other systems^{__}which
other systems may cleave,
smash, break, or erode the simplest primitive systems.
Other such polyhedral systems may
be transformingly developed by winddriven sandstorms
or wavedriven pebble beach
actions. Those other contacting systems can alter the
simplest
primitive systems in only two topologicalsystem ways:

Fig. 1086.062 
986.062 As we have learned regarding the "Platonic solids" carvable from cheese (Sec. 623.10), slicing a polyhedron parallel to one of its faces only replaces the original face with a new face parallel to the replaced face. Whereas truncating a vertex or an edge eliminates those vertexes and edges and replaces them with faces^{__}which become additional faces effecting a different topological abundance inventory of the numbers of vertexes and edges as well. For every edge eliminated by truncation we gain two new edges and one new face. For every corner vertex eliminated by truncation our truncated polyhedron gains three new vertexes, three new edges, and one new face. 
986.063 The cheese tetrahedron (Sec. 623.13) is the only one of the primitive hierarchy of symmetrical polyhedral systems that, when sliced parallel to only one of its four faces, maintains its symmetrical integrity. It also maintains both its primitive topological and structural component inventories when asymmetrically sliced off parallel to only one of its four disparately oriented faces. When the tetrahedron has one of its vertexes truncated or one of its edges truncated, however, then it loses its overall system symmetry as well as both its topological and structural identification as the structurally and topologically simplest of cosmic systems. 
986.064 We may now make a generalized statement that the simplest system in Universe, the tetrahedron, can be designaltered and lose its symmetry only by truncation of one or more of its corners or edges. If all the tetrahedron's four vertexes and six edges were to be similarly truncated (as in Fig. 1041.11) there would result a symmetrical polyhedron consisting of the original four faces with an addition of 10 more, producing a 14faceted symmetrical polyhedron known as the tetrakaidecahedron, or Kelvin's "solid," which (as shown in Sec. 950.12 and Table 954.10) is an allspace filler^{__}as are also the cube, the rhombic dodecahedron, and the tetrahedral Mites, Sytes, and Couplers. All that further external alteration can do is produce more vertex and edge truncations which make the individual system consist of a greater number of smallerdimension topological aspects of the system. With enough truncations^{__}or knocking off of corners or edges^{__}the system tends to become less angular and smoother (smoother in that its facets are multiplying in number and becoming progressively smaller and thus approaching subvisible identification). Further erosion can only "polish off" more of the onlymicroscopically visible edges and vertexes. A polished beach pebble, like a shiny glass marble or like a highfrequency geodesic polyhedral "spheric" structure, is just an enormously high frequency topological inventoryevent system. 
986.065 Joints, Windows, and Struts: As we have partially noted elsewhere (Secs. 536 and 604), Euler's three primitive topological characteristics^{__}texes, faces, and lines^{__}are structurally identifiable as joints, windows, and pushpull struts, respectively. When you cannot see through the windows (faces), it is because the window consists of vast numbers of subvisible windows, each subvisiblemagnitude window being strut mullionframed by a complex of substructural systems, each with its own primitive topological and structural components. 
986.066 Further clarifying those structural and topological primitive componentation characteristics, we identify the structural congruences of two or more joinedtogether systems' components as two congruent single vertexes (or joints) producing one single, univalent, universaljoint intersystem bonding. (See Secs. 704, 931.20, and Fig. 640.41B.) Between two congruent pairs of interconnected vertexes (or joints) there apparently runs only one apparent (because congruent) line, or interrelationship, or pushpull strut, or hinge. 
986.067 Returning to our earlyGreek geometry initiative and to the asyetpersistent academic misconditioning by the Greeks' oversights and misinterpretations of their visual experiences, we recall how another nonIonian Greek, Pythagoras, demonstrated and "proved" that the number of square areas of the unitmoduleedged squares and the number of cubical module volumes of the unitmoduleedged cubes correspond exactly with arithmetic's secondpowerings and thirdpowerings. The Greeks, and all mathematicians and all scientists, have ever since misassumed these square and cube results to be the only possible products of such successive intermultiplying of geometry's unitedgelength modular components. One of my early mathematical discoveries was the fact that all triangles^{__}regular, isosceles, or scalene^{__}may be modularly subdivided to express secondpowering. Any triangle whose three edges are each evenly divided into the same number of intervals, and whose edgeinterval marks are crossconnected with lines that are inherently parallel to the triangle's respective three outer edges^{__}any triangle so treated will be subdivided by little triangles all exactly similar to the big triangle thus subdivided, and the number of small similar triangles subdividing the large master triangle will always be the second power of the number of edge modules of the big triangle. In other words, we can say "triangling" instead of "squaring," and since all squares are subdivisible into two triangles, and since each of those triangles can demonstrate areal secondpowering, and since nature is always most economical, and since nature requires structural integrity of her forms of reference, she must be using "triangling" instead of "squaring" when any integer is multiplied by itself. (See Sec. 990.) 
986.068 This seemed to be doubly confirmed when I discovered that any nonequiedged quadrangle, with each of its four edges uniformly subdivided into the same number of intervals and with those interval marks interconnected, produced a pattern of dissimilar quadrangles. (See Fig. 990.01.) In the same manner I soon discovered experimentally that all tetrahedra, octahedra, cubes, and rhombic dodecahedra^{__}regular or skew^{__}could be unitarily subdivided into tetrahedra with the cube consisting of three tetra, the octahedron of four tetra, and the rhombic dodecahedron of six similar tetra; and that when any of these regular or skew polyhedras' similar or dissimilar edges and faces were uniformly subdivided and interconnected, their volumes would always be uniformly subdivided into regular or skew tetrahedra, and that N^{3} could and should be written and spoken of as N^{tetrahedroned} and not as N^{cubed}. 
986.070 Buildings on Earth's Surface 
Fig. 986.076 
986.076
If two exactlyverticalwalled city skyscrapers are
built side by side, not until
they are two and onehalf miles high (the height of
Mount Fuji) will there be a space of
one foot between the tops of their two adjacent walls.
(See Fig.
986.076.) Of course, the
farther apart the centers of their adjacent bases, the
more rapidly will the tops of such high
towers veer away from one another:
(Footnote 2: The Engineer (New York: TimeLife Books, 1967.) If the towers are 12,000 miles apartthat is, halfway around the world from one anothertheir tops will be built in exactly opposite directions ergo, at a rate of two feet farther apart for each foot of their respective heights.) 
986.080 Naive Perception of Childhood 
986.084 In my poorsighted, feelingmywayalong manner I found that the triangle^{__}I did not know its namewas the only polygon^{__}I did not know that word eitherthat would hold its shape strongly and rigidly. So I naturally made structural systems having interiors and exteriors that consisted entirely of triangles. Feeling my way along I made a continuous assembly of octahedra and tetrahedra, a structured complex to which I was much later to give the contracted name "octet truss." (See Sec. 410.06). The teacher was startled and called the other teachers to look at my strange contriving. I did not see Miss Parker again after leaving kindergarten, but threequarters of a century later, just before she died, she sent word to me by one of her granddaughters that she as yet remembered this event quite vividly. 
986.085 Threequarters of a century later, in 1977, the National Aeronautics and Space Administration (NASA), which eight years earlier had put the first humans on the Moon and returned them safely to our planet Earth, put out bids for a major spaceisland platform, a controlledenvironment structure. NASA's structural specifications called for an "octet truss" ^{__}my invented and patented structural name had become common language, although sometimes engineers refer to it as "space framing." NASA's scientific search for the structure that had to provide the most structural advantages with the least pounds of material^{__}ergo, least energy and seconds of invested timein order to be compatible and light enough to be economically rocketlifted and selferected in space^{__}had resolved itself into selection of my 1899 octet truss. (See Sec. 422.) 
Next Section: 986.090 