Pythagorean Tetraktys and Masonic Delta by Arturo Reghini: Unlike number three, number four also admits a geometric representation of space. Specifically, conducting the perpendicular to the plane of an equilateral triangle to its center, there is a point on it that has the three vertices of the triangle equal distance to the side.

Four points are the vertices of a tetrahedron, called a pyramid by the Greeks (1) that’s to say a regular pyramid with a triangular base, which is the space representation of the number four.

We are continuing our translation from Arturo Reghini’s “ Sacred numbers in masonic Pythagorean tradition” 1947, Chapter 1.

Again it is possible the homothetic develops to one of the summits, which may be disposed below the base the consecutive triangular number and tetrahedral numbers are thus obtained. The gnomon of the tetrahedron is triangular which is added to the previous tetrahedron. The first tetrahedral (2) number is the unit: the second is 4 because 1 + 3 = 4; the third is 10 because 4 + 6 = 10. Starting from an initial line composed entirely of units, and writing in the second line the sequence of natural numbers, in the third that of the triangular and fourth tetrahedral, we obtain the following framework:

The law of formation of this framework is as follows: each element of the framework is equal to the sum of all elements of the previous line beginning with the first up to the above considered item, for example 5 = 1 + 1 + 1 + 1 + 1, 15 = 1 + 2 + 3 + 4 + 5, 35 = 1 + 3 + 6 + 10 + 15, or even every element is equal to the sum of what precedes it in the same line and that above it in the same column, eg 20 + 35 = 15.

There is only one linear growth of numbers. However, there are an infinite number of surface developments and infinite solid developments. For example, the number 5 can be represented in the plane by means of the 5 vertices of a pentagon and in space as the five vertices of a pyramid with a square base. Development for the pentagon is made by taking as a center of dilation one of the vertices of a pentagon, and for the tetrahedron with a square base as a center of homothety taking the top of the pyramid. Arithmetically to obtain the pentagonal it is enough to start from the succession of the terms of the arithmetic series of three that’s to say for the numbers: 1, 4, 7, 10, 13, 16 … and make the sum. The sum of the first n. is equal to N ° pentagonal, and therefore pentagonal are 1, 5, 12, 22, 35, 51 … The pyramid with a square base is instead obtained by the sum of the first n square consecutive: 1, 4, 9, 16, 25 … and are the numbers: 1, 5, 14, 30, 55 … In such a way we held hexagonal numbers starting from the series of arithmetic reason 4, or series of hexagonal gnomons: 1, 5, 9, 13, 17 …, And hexagonal are: 1, 6, 15, 28, 45 … He readily acknowledges that the No. the hexagonal number is just the (2n – 1) ° triangular number. It could also demonstrate that the development of pentagonal and hexagonal keeps the similitude of shape, but not the isotropy; therefore, although the plane satisfied a breakdown in regular hexagons, one can not obtain total and isotropic coverage through the homothetic development of three hexagonal matching around a common vertex. Similarly one can show that a space satisfies equipartition only by cubes whose vertices fill it in the whole and isotropically, but it does not agree to equipartition although tetrahedrons and also the octahedron are homothetically developed and fill completely and isotropically the angle-void within which they develop. We make this observation because Aristotle, after stating (3) correctly that the plane may be in equipartition only through regular triangles, squares, and regular hexagons, added that space can be in equipartition by using cubes and pyramids. It is a fallacy in which Aristotle is committed, and, as the three polyhedral regular numbers tetrahedral, octahedral, and cubic, homothetically developed within one of the angle-voids, fill this angle-void totally and isotropically.

Aristotle is in error for he has confused the space with angle-void space, but if the error comes from such confusion that has indirect evidence that the Pythagoreans of that time were already dealing with cubic, tetrahedral and octahedral numbers, besides with the equipartition of the plane by regular polygons and of space by a regular polyhedron, and in particular of the space within an angle-void. In addition to such plane numbers said polygonal numbers, and to the pyramid numbers represented in space by the polygonal base pyramids, Pythagoreans admitted plane and solid numbers as rectangle-shaped and shaped like a regular polyhedron. The formula which gives the polygon number N. r-sided has been known by Diophantus and it is

for n = 4 and r = 6 this formula gives for the fourth hexagonal number P (6. 4) = 28, with the points that represent having the following provision:

The formula that gives the number N pyramidal with base r-gonal is

which appears in another form in the Codex Arcerianus, a Roman code of 450 A.D. (4). For example for r = 4 and n = 5 we find that the fifth pyramid with a square base is F ( 4.5) = 55.

As to mark a straight line takes two points, the minimum number of lines that delimits a portion of this plan is three, between all the plane numbers three is the minimum; so the minimum number of planes needed to define a portioned space is four; between all solid numbers the four, namely the tetrahedron, is the minimum. According to Plato (see Timaeus) this tetrahedron, or pyramid, as he calls it, is the latest particle constituent body, that’s to say the atom or molecule of matter (4). Of course, we now know that atoms or molecules do not have this form and that are not indivisible, but it is worth noting that the body has the greatest molecular firmness, that’s to say diamond, has a molecule composed of four atoms arranged in a regular tetrahedron shape. The mistaken etymology from πῦρ [pyr] = fire explains Plato’s error in giving tetrahedron the symbol of fire.

Adding the unit to the unit we have gone from point to line, identified by two points, by adding to these two points another point you can move to the plane through the triangle, and by adding the unit you may still move to space through the tetrahedron. But remaining within the limits of human intuition of three-dimensional space, you can not add a unit to the four vertices of a tetrahedron taking a point out of the three-dimensional space and representing the five as a pyramid of hyperspace as a basis for the tetrahedron. In other words, the unit switches to two and you have the line, the two passes to three and you have the plane, you go from three to four and have the space, and then you have to stop because proceedings have ended. Now, according to the meaning that Aristotle and simply Greeks gave to the word perfection, things are perfect when they are finished, and completed: the end is perfection. In our case, since the four is the last number that you get going from point to line, from line to floor, and from the floor to space, because you can not figure out a fifth number of the space defined by the four vertices of the tetrahedron, the four is, in the generic greek and Pythagorean sense of perfection, a perfect number. The assembly of the monad, the dyad, the triad, and the tetrad includes everything: the point, line, surface, and the concrete world of solid material, and you can not go further. Therefore the sum

1 + 2 + 3 + 4 = 10

That’s to say the sum or the quadruplet or quadruplet of unity, duality, trinity or the tetrad, in other terms the decade, it is perfect and contains the whole…”

To be continued at Arturo Reghini Sacred Pythagorean Numbers 4 .

Previous article Arturo Reghini Sacred Pythagorean Numbers 2 .

- The Greek word pyramid is a slight corruption of the Egyptian pirem-us designating the height of the pyramid (see REVILLOUT E., Revue Egypt., 2e année, 305-309);
- ARISTOT., De coelo, III, 8;
- See CANTOR M., Die Römischen Agrimensoren, Leipzig, 1875, pagg. 93, 127;
- See WILLIAM BRAGG, L’architettura delle cose, 2a ed. ital., Milano, 1935, pag. 157;