# Scale structure theory applied to the square numbers

This web page is quoted from the author’s book Principles of nature; towards a new visual language, WA Roberts P/L Canberra. 2003. [Used with permission of the author] – minor editing for the web

In the regular division of the plane, there exists a scale structure of only three distinct regular polygons which can tile the plane to infinity in every direction without gaps. They are the square, the equilateral triangle and the hexagon.

These gapless tiling patterns (known as tessellations of the plane) are demonstrated in the next figure (Figure 6.7)

Figure 6.7

There are only three regular polygons whose sides and angles are equal and which can tile the plane without gaps: the square, the equilateral triangle, and the hexagon

That only these three regular polygons can tile the plane without gaps and in self-similar manner is a fact that has been known by geometers for centuries and is easily proven (Dunham, 1994, pp. 108-111). What is perhaps not recognized or undervalued is the potential significance of such a fact. [This appears to be symptomatic of a problem in current cultural attitudes to knowledge and research, and will be briefly addressed later.]

We should therefore expect that these three shapes contain within themselves and within this triadic scale, important clues to the structure and form of the infinity which they divide. Just as the time it takes for an echo to return reflects the distance of a rock face on the other side of a gorge, so these shapes ought to reflect something about the nature of the infinity they divide. And because they divide this infinity in a self-similar way, that is, because they divide this infinity into integral units, we expect these shapes will reflect something important about the whole-numbers themselves. More particularly, they should reflect something important about special sorts of whole number, because, of all the regular polygons (i.e. having equal side-lengths and internal angles), there are just three which can tile the plane without gaps. Since polygons are shapes which occupy area, and since areas can be expressed numerically in a multiplicative manner (length x breadth), we should expect numbers of a symmetrical composition in the form p x p (that is, p2) may be connected to these particular regular polygons in certain special ways. That is exactly what we find. These special sorts of whole number we currently know as the squares.

Since polygons are shapes which occupy area, and since areas can be expressed numerically in a multiplicative manner (length x breadth), we should expect numbers of a symmetrical composition in the form p x p (that is, p2) may be connected to these particular regular polygons in certain special ways. That is exactly what we find. These special sorts of whole number we currently know as the squares.

The line of reasoning may be schematically summarized as in figure 6.7 below.

Figure 6.7

Line of reasoning leading to the scale structure of the plane

The square property of numbers of the form p2 has been widely known for more than two thousand years. But on the preceding page (click button b ) we have seen a less familiar triangular representation of the ‘square number’ 4. We need to show if this triangular characteristic is a property of just some of the square numbers, or all of them.

### Do all square numbers have a corresponding triangular form?

It has long been known that the squares can be generated by the successive addition of the members of the set of odd numbers (1+3=4, 1+3+5=9, 1+3+5+7=16, …). Geometrically, these odd numbers may represent integral areas which are added to a given shape, and such added shapes were called gnomons by the Greeks if they could regenerate the original figure (see figure 6.8). Clearly, it can be seen that for each unitary increase in the side-length of either squares or equilateral triangles, the gnomon consists of successive odd numbers of component self-similar shapes. Each successive gnomon contains two more units than its predecessor. And therefore each successive gnomon includes successive members of the set of odd numbers and can therefore form successive numbers of the form p2. Thus every square number can just as easily be represented in equilateral triangular form.

Figure 6.8 Illustrating gnomons which when added to a given shape reproduce the original