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Shape Comparisons: Qualities of Space
Once again, we take advantage of the ease of working with planar configurations, before tackling space. We thus start by comparing the characteristics of triangles and quadrilaterals, and then we shall attempt to apply our conclusions to tetrahedra and cubes. We begin by drawing an irregular version of each polygon, and observe the following. If we bisect the edges of the two figures and interconnect these points as shown in Figure 10-la, both shapes are divided into four regions. However, the triangle and quadrilateral exhibit a strikingly different result. A triangle, no matter how irregular, automatically subdivides into four identical trianglesall geometrically similar to the original, that is, the same shape but a different size. Observe in Figure 10-la that this is not true for the quadrilateral. Excluding the special case of a parallelogram, the four small quadrilaterals will not be similar to their framing shape.
Now, on to space! A tetrahedron (of any shape or size) carved out of firm cheese can be sliced parallel to one of its faces, removing a slab of any thickness, to produce a new smaller tetrahedron with precisely the same shape as the original (Fig. 10-lb). This does not work with the cube, or for that matter, with any other polyhedron, regular or not. The ability to "accommodate asymmetrical aberration" without altering shape, observes Fuller, is unique to the minimum system of Universe. We add this observation to a growing list of special properties of the tetrahedron. (Appendix D.) Similarly, as will be demonstrated below, an irregular tetrahedron can be subdivided to create smaller identical tetrahedral shapes, whereas an irregular hexahedron will yield dissimilar hexahedra, in the same manner as its planar counterpart, the quadrilateral. Evidence thus gradually accumulates to support using the tetrahedron (instead of the cube) as the basic unit of structureor mathematical starting point.
Volume: Direct Comparison
Before looking more closely at volume ratios, we review the following mathematical generalization. No matter what unit of measurement is employed, the volume of any container is mathematically proportional to a typical linear dimension raised to the third power. This means that if we have two geometrically similar polyhedra, one with twice the edge length of the other, the larger will contain exactly eight times the volume of the smaller. (Having the same shape, the two systems will share a common "constant.")
To bring this mathematical law into experiential grasp, we consider two familiar shapes. It is easy to visualize that a cube of edge length 2 consists of eight unit cubes (Fig. 10-2). Now imagine a tetrahedron of edge length 2. Observe in Figure 10-3 that, just like its cubic counterpart, the altitude of a two-frequency tetrahedron is two times that of a unit tetrahedron, and similarly that each face subdivides into four unit triangles. The latter observation indicates that the area of the large tetrahedron's base is four times that of the small tetrahedron's unit-triangle base. Emulating the approach employed by Loeb in his "Contribution to Synergetics," (1) we can deduce the following, simply by utilizing traditional geometric formulae.
Let VolT and Volt represent the volumes of the large (two-frequency) and small (unit-length) tetrahedra; AT and At, the areas of their bases: and HT and Ht, their altitudes.
According to the formula
volume of pyramid = constant X (area of base) X height.
and since we observed in Figure 10-2b that the base of the large tetrahedron is divided into four triangles, each of which is equal to the base of the small one, it is clear that AT = 4At. Similarly, the altitude of the larger pyramid is twice that of the smaller, or HT = 2Ht. Substituting, we have
it follows that
or, in words, that the volume of the big tetrahedron is eight times that of the little tetrahedron. The constant K cancels out of the expression when the two equations are compared. This conclusion will be useful in deriving the volume ratios displayed in Table V.
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