Pythagorean Tetraktys and Masonic Delta by Arturo Reghini: The numbers starting from three, in addition to a linear representation, concede even a surface representation ie in the plane.
Three is the first number that admits a linear representation over the plane representation, through the three vertices of a triangle (equilateral). Three is a triangle or triangular number, it is the result of the mutual coupling of the monad and the dyad, two is the unit analysis, and three is the synthesis of unity and dyad.
Thus we have with the trinity the manifestation or epiphany ( Theophania, Embodiment Day, Θεοφάνια) of the monad in the superficial world. Arithmetically 1 + 2 = 3.
Proclus (The Theoretic Arithmetic of Pythagoreans, 2 ed., Los Angeles 1924, pag. 176) observed that two had a character in a way intermediate between unity and three. Not only because it is the arithmetic mean, but also because it is the only number which, summed with itself or multiplied by itself, you get the same results, while, as for the unity, the product gives less than the sum and the product of the three gives more, ie, it has
1 + 1 = 2 > 1.1 ; 2 + 2 = 4 = 2.2 ; 3 + 3 = 6 < 3.3
In Modern times it was instead noted that 1, 2, 3 are the only positive integers whose sum is equal to the product. You can also easily recognize that 1, 2, 3 is the only triple of consecutive integers for which it happens that the sum of the first two is equal to one-third because the equation x + (x + l) = x + 2 admits unique solution x = 1. It is also recognized immediately by the geometric representation that the sum of several consecutive integers always exceeds the number following the last of its parts, except where the summands are two that we have: 1 + 2 = 3. Concluding the triad, the holy trinity, you can only get it through the addition of the monad and the dyad, and so providing continuing basis under the four points to get ten.
This geometrical development of the first triangle to one of three vertices taken as the center of homothety gives us so later the triangular numbers, and the triangular gnomon is called the base which is added to switch from a triangular to a triangular row. Arithmetically, written as a first line the sequence of integers, we can deduce the sequence of triangles, writing the units under the unit, then by the sum of one and two, and then taking for elements of the second row the numbers which are obtained by the sum of the first integers, or doing, to obtain an element of the second row, the sum of the element preceding it in the same row with what precedes it in the same column:
Therefore, by definition, the triangular No. t is the sum of the first n integers and is therefore equal to ‘(n-1) ° triangular increased by n. If triangle three does an equilateral triangle, proceeding with the development of homothetic, the other triangular numbers too have a regular shape, namely, the similarity is preserved in the development of form. Moreover, since around a point you can have six angles of 60 ° (as was known to the Pythagoreans), ie there are six congruent equilateral triangles around a point, building upon all six against this common vertex, taken as the center of homothety, we obtain the total coverage and isotropic plane through regular triangles.
The number four too, in addition to its linear representation allows a single flat representation: It is, therefore, a square, it is the second square because the unit is a square of one. The gnomon of the square, ie the difference between 4 which is the second square, and the previous square is above 3, the third square, or as we say the square of base 3, is obtained in the geometric representation by adding to the right and below a gnomon shaped like a team and composed of 5 points, and so you move from one square to the next adding the odd numbers. Thus we see that squares grow while preserving the similarity of the form and since around a point we can have four angles congruent and each of them a square, it follows that, by developing homothetic four squares at the common top as a center of dilation, you get the total and isotropic coverage of the plane by means of squares. Arithmetically you can just write odd numbers in the first line, and in the second act as we did for the triangular numbers to get the squares:
It follows the important property: The sum of the first n odd numbers is equal to N. square, a property that allowed Galileo to find the formula of naturally accelerated motion.
A square is a number in a rectangle shape whose sides contain the same number of points. A number having a rectangular shape was called eteròmeco if containing only one point more than the consecutive and was named promech if the difference between points on one side and consecutive exceeded one. For example, 15 was a promech and 20 was an heteromech.
Leading a straight line to the side and parallel to a diagonal it divides a number heteromech into two parts, that are two triangles equal; and as the number of points of n ° heteromech, consisting of n columns and n rows is n (n + 1) it follows for the No. triangular number the following formula
Recalling the definition of triangular we have
If one leads the parallel to a diagonal to a square number, the square is divided into two consecutive triangular, ie the sum of two consecutive triangular squares is equal to one, and this allows us to deduce from the sequence of triangular that of squares. Written in the first row is the triangular series, in the second row the sequence of squares
writing below every element of the first line the sum with the previous one…”
To be continued at Arturo Reghini Sacred Pythagorean Numbers 3 .
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