Is the dominance of right triangles and squares justified from a scale structure perspective?
This web page is reproduced from my book (W. Roberts, 2003, pp.119—121) with some minor editing and reformatting for the web.
The importance of the right triangle and Pythagoras' Theorem to geometry and mathematics cannot be overstated. We have already mentioned their anchor role in trigonometry, analytic geometry, vector analysis, etc.
But earlier... we found that certain numbers of the form p2 have been somewhat naïvely named ‘the squares’, since they are in fact, equally at home in other fundamental geometric shapes (such as the equilateral triangle). This simple fact has enormous ramifications, which is becoming startlingly apparent at every turn: first, in our discussion of the very fundamental concept of the unit in geometry and mathematics, (in particular, units-of-area), and now, as we are about to see, in identifying certain fundamental and important new classes of triangle.
School students are familiar with the following classifications of triangle: scalene, obtuse, acute, equilateral (or equiangular), isosceles, and of course, right triangles*. The importance of the right triangle is at once connected to the importance of the Pythagorean Theorem, but is also linked to the importance of the square and right-angle to mensuration. Add to this the orthogonal axes of analytic and coordinate geometry, and we find the square and right triangle ensconced at the head of a 'rectangular table'. But perhaps our table, being of mathematical bent, is given to topological tricks, and can metamorphose into a round table. Let us see which knights are now sitting around it.
Reassessing the rationale of triangle classification
Scale-structure theory forever changes the way we see numbers of the form p2. It is no longer enough to call them the ‘squares’. Similarly, scale-structure theory also has implications for triangle classification.
To that end, let us firstly consider the ‘spirit’ of the right triangle. In a manner of speaking, it is a hybrid of both the square and the triangle: it has three sides and angles, yet it has one right angle (which reflects its 'square ancestry'). It is as if it has gained one chromosome from one parent (the triangle) and a complementary chromosome from the other (the square). It is powerful since it has inherited 'the best of both worlds'. It is like a queen in chess which can move in 'square' as well as 'diagonal' manner.
New important triangle-type points to new Pythagoras-like theorem
Scale-structure theory implies that the right triangle should not be the only interesting or powerful ‘hybrid’ geometric shape. We expect others since our understanding of the scale-structure of the Euclidean plane ...indicates that squares do not have precedence over and above equilateral triangles, or hexagons. In other words, right angles, whilst of utmost importance, do not have a monopoly of importance in the scheme of things. The 60- degree angle (of the equilateral triangle) and the 120-degree angle (of the hexagon) must have some significance beyond what is commonly known today since our geometry to date has been dominated by a square view of numbers of the form p2 and also a square method of mensuration. Scale-structure theory implies that there ought to be some other important Pythagoras-like theorems for other 'hybrid' triangles containing one of these other 'resonant' angles. Let us find them.
If we look more closely at the pedigree of the right triangle, we find that it is really closest to a hybrid of a scalene triangle (reflected in the fact that all sides may be unequal) and a square (since one angle must equal a right angle).
Let us first replace the 'square heritage' with an equilateral one. This means that our fixed angle must now be 60-degrees instead of 90-degrees. I earlier defined just such a triangle, and named it the eutrigon... To reiterate, a eutrigon is a triangle with one angle equal to 60-degrees. We have also previously determined the area of such a triangle (in etu) to be ab, where a and b are the lengths of its legs...
We shall begin our search for new Pythagoras-like theorems by considering the eutrigon, but from a scale-structure point of view: we shall be constructing 'geometric scale structures' (wholes) which are resonantly divided (consisting of a 'grammar' of parts) and resonantly 'measured' (i.e. using relative units, in this case, equi-triangular units or ‘etu’).
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