Why Pythagoras *
seemed obsessed with triangles *
This is a question that I often asked myself, provided this basic premise: undoubtedly, Pythagoras had an intelligence that couldn't be lesser than ours; so it couldn't be that he devoted such great part of his life to the study of triangles only because he would have been engrossed by futile subjects that wouldn't have interested more mature minds.
Whatever we may think of men like him when the distance between we laymen and genius induces us into believeing we would have been smarter than them because we feel we would have been allegedly more "practical" than genius is, it would be preposterous presuming that what may appear obvious to our intelligence, couldn't be considered obvious by a man like Pythagoras as well, and that what may seem uninteresting to us, seemed interesting to him because of a limitation of his, limitations from which we would have been free. Genius is not such because it sees less than us, but because s/he sees more.
Besides, that Pythagoras was right can be inferred also considering Cartesian Coordinates *
, where the whole of the analytical geometry *
we inscribe within such coordinates is concerned with triangles as well, by drawing the projections of every point in its space onto the orthogonal axes and thus deriving rectangles, each of which on its own turn can be (and normally is) divided into two triangles in order to afford further inspections and considerations.
Why this happens, why we so often fall back to triangles?
Thinking this issue over I have found a possible answer that, though it may not coincide with the reasons Pythagoras might bring forth, is none the less both satisfactory to my own intelligence and appears at least generally plausible too.
defined the point as what has no parts.
This is an ideogram *
, of course: because whatever point we may draw, has both a width and a breadth, no matter how small. So when a point is defined as what has no parts, we are proposing a definition that is the highest abstraction of a point we can make, regardless of any specific and real point we may consider; Democritus *
was the first to speak of "atom" linking its concept to that of "idea": the smallest unit we can conceive (the atom), theoretically coincides also with the highest idealization we can conceive.
Analogously, Euclid defined the line as width without breadth: this is, again, the highest abstraction we can achieve, defining the line as another ideogram. The definition of a line comes after that of the point, because a sequence of points describes a line.
Therefore, points are the minimal constituting unit of every line: we could say they are the atoms of all geometry. Lines, on their own, are then the minimal constituting brick of every polygon *
: they are the atoms of all shapes.
Now we may wonder what is the minimal constituting polygon we can think of?
The first polygon we may meet is exactly the triangle, because the first simplest area we can circumscribe is the one circumscribed by three
crossing lines: which his what we call a tri
If a triangle is the simplest polygon we can think of, it means that every other and more complex polygon (a square, a rectangle, a pentagon, and whatever other irregular shape we may conceive) is composed by a sum of triangles as if triangles were their building bricks or, in other words, every polygon may be decomposed in a set of inner triangles, by drawing lines within it, that build it up completely.
This is just another evidence, derived by geometrical inspection, that every shape is composed by a sum of triangles.
Of course, within a polygon we could derive any other polygon too by drawing lines in it: what is meant here is that when we derive triangles within a polygon, we are deriving the set of the simplest
shapes we can divide it into, and not just shapes whatever.
If so, namely if the triangle is the simplest polygon we can think of and if every other polygon may be divided perfectly into triangles as the simplest shapes we can make out of it, it also means that whatever theorem we may understand about triangles, will be valid about every other polygon
too: so that, understanding rules about triangles, we can by construction attain rules also about more complex areas because all areas can be decomposed and translated into sets of triangles.
Now, this conclusion may not be very impressive, although it is probably rational and right both. Yet it still leaves out a significant element: Pythagoras was not interested in triangles whatever, but in a specific
type of triangle: the right
That is, Pythagoras did not elect as the most important triangle to study an equilateral, isosceles, scalene, or obtuse or acute triangle, but a right
triangle namely a triangle that included an angle of 90 degrees in it.
If there is an angle of 90 degrees in a shape, it means that the two lines that inscribe the 90 degrees angle are perpendicular *
Perpendicularity is defined as follows: the shorter route between a point and a line.
That is, given a point in space and a line, the shorter route we may travel in order to conjoin them, is the one that makes a right angle at the intersection: all the other routes would be longer, or might be curves, and therefore would not be the shortest route.
So, if a triangle is the first simplest shape we may meet, a triangle with a 90 degrees angle is the more essential
among triangles, the simplest
one we may envision, the less convoluted one, the sharpest one, and, so to say, it is "more" triangle than any other triangles: it is the atom of triangles.
So, whatever theorem we may devise about a right
triangle must be true for all the other polygons and triangles too, because we are dealing with the simplest and more concise version
of what is already the simplest shape we may envision.
Therefore a right triangle can be envisioned as the atom of all polygons, and therefore all rules that prove valid for it, rule everywhere: consequently, by recursive application, or by construction or deconstruction namely by logical inference about the rules that apply to a right triangle, valid rules for all the other polygons may be deduced or/and induced, and generalizations may ensue.
Besides, since each right triangle can be divided again into a set of two other right triangles by simply drawing the perpendicular to the hypotenuse from the right angle, all polygons can go on being divided into right triangles that are smaller and smaller, as a fractal *
At this point one may wonder: yes, but triangles, right or whatever they may be, cannot provide us with any means of inspecting curves
This is wrong: curves, in analytical geometry, are also called Conic sections *
That is, if we get a cone and we cross it with a plane parallel to the base, and we see its plant, what we see is a circle
If we get a cone and we cross it with a slanted plane that falls outside its base, and we see its plant, what we see is an ellipse
If we get a cone and we cross it with a plane perpendicular to its base, and we see it from the side, what we see is a parabola
, and if the plane is slanted and ends on the base of the cone, we may find an hyperbola.
These: circle, ellipse, parabola, and hyperbola are curves indeed.
One may wonder: yes they are, but what has this to do with right triangles?
The fact that a cone is
a right triangle: in fact, a right triangle revolving
around itself describes a cone.
This may be why right triangles matter so much, that Pythagoras devoted such great part of his life to their study. When we are at geometry, everything seems to end in triangles, and among them the right one is the more synthetical.