 Principles of Nature: towards a new visual language

## Scale-structure theory applied to units of area

This web page source: Roberts,W. Principles of nature: towards a new visual language, WA Roberts P/L Canberra 2003, p.97)  The problem of finding the area of a closed planar shape is known as quadrature, or squaring. The word refers to the very nature of the problem: to express the area in terms of units of area, that is, squares. To the Greeks this meant that the given shape had to be transformed into an equivalent one whose area could be found from fundamental principles. E. Maor, 1994   ### Finding suitable units Crucial to the notion of quantifying an area is the unitthe unit of area. How is such a unit to be chosen? Firstly, units must be congruent (or regular in at least some way) so that we can directly relate these to the integers for example, and so that we can make economical mathematical statements and comparisons like 'an area of four units is quantitatively twice that of an area of two units'. Also it helps enormously if the units can be joined without gaps because the gaps are part of the area we are trying to measure. If the units all have the same shape and size, we can simply count the number of these units which fit the space to quantify the size of the area in terms of our defined units.

#### History has favoured the square unit

Historically, the defining unit of area now widely established is the square unit. And for many good reasons. Firstly, squares fit the criteria we have already mentioned, and have served us well for centuries, even millennia. But as we shall see there are other units we might have chosen and which work equally well as units of area. What are they, why have they been neglected as units, and what are the implications of their inclusion into the realm of units? ....We shall proceed by examining the issue from a scale-structure point of view.

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