Relative Units
In summary thus far, square numbers are every bit as much triangular in the new way in which we are seeing them, that is, from a scale structure point of view.
Numbers of the form pq are composite, occupy area, and are analogous to planar shapes having two unequal sides. By the same token, if p = q, then pq = p2 and this is the algebraic equivalent of (symmetric) polygons in the geometry of the plane. The shapes best suited to serve as general units of area in Euclidean geometry will be those regular polygons able to be joined together without gaps. These are the equilateral triangle, the square, and the hexagon.
Scale structures & units of area
- Areas can be expressed in different ways depending on the units we choose (square, triangular, hexagonal, etc). An area of nine square-units is not the same absolute area as nine triangular-units (i.e. equilateral triangles) even though the numbers are the same. In other words, we note that a 3 x 3 square has a greater absolute area than a 3 x 3 equilateral triangle. The traditional perspective has been to emphasise the differences between them (in terms of absolute areas).
- But this difference is balanced by its very opposite: an equivalence principle of great beauty. The equivalence becomes clear when you ignore the artifice of arbitrary units (inches, square centimetres, etc) and focus instead on the pure numbers and on the relative units of shape enclosed by each (Figure 6.9) . For a given side length p (expressed in pure numbers) the area in each of these systems (expressed in terms of the number of enclosed self-similar shape-units) is the same in each, p2. This relation we have short-sightedly described as the square of the side. Stop for a minute and let this point sink in. The number 4, for example, can just as comfortably represent any of several different shapes (a square in a square system; a triangle in a triangle system, etc). And these shapes occupy different amounts of space (area) on the plane. But in each, the number 4 has the same relative meaning: a regular polygon of side-length 2 (square, equilateral triangle, or hexagon), with an area of 2 x 2 (or 22) relative units (figure 6.9).
- So, these square numbers (1, 4, 9, 16, ...) need not necessarily be associated with squares, and only squares. The same numbers are equally at home in the shape of equilateral triangles or hexagons, and in fact can be associated with other less regular shapes which can tile the plane, as we shall see.
Figure 6.9
We have traditionally expressed areas in terms of square-units. It is now apparent that areas may just as easily be expressed in triangular units (fig. 6.9), in hexagonal units, or in any units which can gaplessly tile the plane. The units we choose depend on the situation. Thus we have introduced the idea of relative units. Why all the fuss when square units have done such a fine job for so long? The answer will become apparent as we consider the nature of numbers and geometries. The application of appropriate units to a problem simplifies the mathematics and reveals a more beautiful connectedness. The answer has to do with wholeness, simplicity, and connection. The connection of numbers to the shape of space; of quantity to quality. In a fundamental way, it has to do with our very being and our place in the Universe.
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