Principles of Nature: towards a new visual language
© copyright 20032015 Wayne Roberts. All rights reserved.
The area of an equilateral triangle in etuThis web page is quoted from the author's book Roberts, W, Principles of nature; towards a new visual language, Canberra. 2003. We have established in our discussion of socalled 'square numbers' that p^{2} numbers are profoundly connected to equilateral triangles and not just squares. We can put our finding this way: equilateral triangles can be dissected into p^{2} numbersofsmallerequilateraltriangularunits. And we can therefore express an equilateral triangle's area as being equal to p^{2} etu, in other words, that the number of etu's which fills its area without gaps is equal to p^{2}, where p is the number of etu sidelengths which fit along a side of the equilateral triangle in question. In equation form, The area of an equilateral triangle of sidelength p (expressed in etu) =p^{2}
The equilateral triangle has a sidelength ptimes as large as the smaller equilateral triangular unit of area (or ‘etu’), and its area is p^{2} as large... Notice the simplicity and symmetry of such a relationship. The beautifully simple relationship given above is muddied if we try to understand equilateral triangular areas and sidelengths in terms of squares (which is what we have done for more than two thousand years). To remind us of this, let us compare and contrast our traditional 'square' handling of equilateral triangular areas [see diagram below]. What is the area of an equilateral triangle of unitary sidelength expressed in square units? [This is easy for any highschool student but is not as easy as finding the area in etu which is simply p^{2} as given above!]
The area of any triangle in square units is 2 bh where b is the baselength and h the perpendicular height of the apex from the base. We can find h by Pythagoras' theorem ... which equals 3/2. The area (½bh) is therefore equal to 3/4 square units. This irrational number results from applying a square peg to a triangular hole (see figure on this page). The fit is awkward, complex, and dysfunctional. [See diagram below]

