The fourth chapter of Arturo Reghini’s Sacred Numbers in Masonic Pythagorean Tradition. Pentagons, pentalphas, and dodecahedrons with twelve regular pentagons
Title: “The Pythagorean pentalpha and the blazing star”.
Do not enter my school which ignores the geometry, Inscription on the entrance of Plato School.
We arrived at number five starting from the tetrachord of Philolaus or by consideration of the Egyptian triangle. Another way, similar to the first of two, who led the Pythagoreans to the evaluation of the number five, is that starting from the consideration of the golden divine section of a line segment, and leads to the study of pentalpha or pentagram, a characteristic symbol of the Pythagorean brotherhood, that’s to say the blazing star symbol characteristic of the Masonic brotherhood.
The rigorous study in terms of geometry and arithmetic of this topic would require a long development that we have done in previous work (1). So we will generally omit the proofs referring the reader to our work, where we come to the results and the property, which we will use, in a Pythagorean way, that’s to say without the postulate of Euclid.
One of the most important discoveries of the Pythagoreans is that of incommensurable magnitudes and consequently of irrational numbers. The simplest case is that of the incommensurability of the diagonal and side of a square, and Aristotle gives the proof the Pythagoreans gave. It is a consequence of the theorem of Pythagoras. In fact, if for absurd the diagonal and the side of the square admit a common measure, that is, if the diagonal contained m times a segment and the side contained it n times, the square on the side could be divided into n² squares all equal and having as a side this common segment, and the square of the diagonal could be divided into m² squares equal to them: and being, according to the Pythagorean theorem, the sum of the squares on the sides equivalent to the square on the hypotenuse, the number of squares 2 n² contained within the squares of the sides should be equal to the number of the squares m² of the hypotenuse, that’s to say it should be 2 n² = m². Now, being n and m two integers, the two numbers of the previous equality should contain the same prime factors because a number can be decomposed in a unique way into a product of prime factors, and this is not possible because m should contain two and then m² would contain the two an even number of times and then also n should contain the two, n² would contain it an even number of times n² and 2 would contain it an odd number of times.
In particular, if the side of the square is one, the square of the diagonal is two and the diagonal is equal to the irrational number √ 2. Then, by dividing the circle into four equal parts and neatly combining the four points of division, you get the square inscribed, one can also say that the side of the square inscribed in the circle of the unit radius has as a measure the irrational number √ 2. This segment is incommensurable with the unit segment being determined geometrically with the utmost simplicity. Similarly, considering the right triangle in which the hypotenuse is double the minor cathetus, one would find that the larger cathetus has as measure the irrational number √ 3, and considering the triangle that has a cathetus double than the other, one would find that the hypotenuse has to measure √ 5. And, since it is easy to demonstrate that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle, it follows that the side of the equilateral triangle inscribed is equal to the segment that has as measure √ 3 . The two irrational numbers √ 2 and √ 3 are respectively the measure of the side of the regular square and triangle inscribed in circumference, and they are two segments incommensurable with the unit segment which can be easily determined by geometry.
The number √ 5 is connected instead, though less simple, with the division of the circumference into ten and five equal parts, and with the measure of the side of the inscribed pentagon and the side of the regular inscribed decagon. It is called the golden part of a segment, or also the divine section, that part of the segment such that the square that has this side is equivalent to the rectangle whose sides are the entire segment and the remaining part. The determination by geometric of the golden part of a segment can be achieved by two constructions, and with the theory of proportions, the golden part of a segment can also be defined as the geometric median or proportional between the entire segment and the remaining part. It can also be shown that in the isosceles triangle that has the vertex angle equal to half the angle at the base, the base is the golden portion of the side; and, since this angle to the vertex has the width of 36 °, it follows that divide the circumference into ten equal parts, the side of the regular decagon inscribed is the golden ratio of the radius; vice versa the arc that has as a rope the golden part of the radius has the width of 36° degrees and is the tenth part of the entire circumference. It follows the usual determination of the golden part of the ray OA of a circumference and the division of the circle into ten equal parts.
It is led through the center O the radius OB perpendicular to the radius OA and took the midpoint C of this radius OB, describes the circle with center C and radius CO: the diameter AC meets this circumference in two points D, E, and it happens that the radius OA is a median proportional between the entire secant AE and its outer A D. Dividing this ratio we can deduce that the outer part AD = AM is the golden part of a radius AO. For the uniqueness of the golden part, the triangle isosceles side OA and the base AD = AM has the vertex at the angle of 36° degrees and so AM is the side of the regular decagon inscribed, therefore bringing the AM segment as rope ten times as from point A, it divides the circle into ten equal parts, and then also into five taking in an alternating way the points of division (2). If the radius OA is equal to one, the radius OC is 1: 2, and the hypotenuse of the right triangle AO AOCE is
and the golden part A D has to measure
Then the side of the regular decagon inscribed in the circumference of radius one is the golden ratio of the radius and has to measure
If instead of uniting the division point A of the circumference into five equal parts with the next point C, one meets point A with the third division point E, and this with the fifth, and so on the starry pentagram is obtained, so-called because it consists of five lines, also called pentalpha because it contains five times the letter A formed for example by two ropes AE and AG and by the segment of rope MR C I. The term pentalpha is located in the arithmetic of the father Kircher (1665), but the term decalca, evidently formed similarity to the first, is already in Plutarch. However, this is not what we want.
Since IC is clearly parallel to the quadrilateral C E, the C E G R is a parallelogram, more it’s a diamond because EC and EG are the same as the sides of the inscribed regular pentagon, and it is easy to recognize that the isosceles triangle A E G is the apex angle of 36 °, and then that E G = E C = E M is the golden part of the side A E pentalpha. We will call l5 the side E G of the regular pentagon and s5 the side A E of pentalpha, and we can say that: 1 – the side l5 of the pentagon is the golden side of pentalpha s5, 2 ° that the side s5 = A E of the pentalpha is divided in two points M, N by the other two sides of the pentalpha so that the part R C V = E M is the golden part around the side s5.
Since then the isosceles triangle C E M has a vertex angle of 36 °, the C M is based on the golden part of the side E C, and as the five points of the starry pentagram are obviously all the same it follows that AM = E N is the golden part of E M = A N. Then determined over a segment its golden part, the remainder is the golden part of the golden, that’s to say A E: A N = AN: E N = E N: N P. ..
The sides of the pentalpha determine a regular pentagon M N P Q R of side M N = l”5 whose vertices are also vertices of another pentalpha whose side s’5 is equal to A M, and it has the proportion
s5 : l5 = s’5 : l’5
in which each term is the golden part of the previous. It has namely:
s5 : l5 = l5 : s’5 = s’5 : l’5
The second pentalpha determines in turn a third inscribed pentagon’ of side l”5 and a third inscribed pentalpha of side s “5 etc… And you have the equal chain of ratio
s5 : l5 = s’5 : l’5 = s”5 : 1″5
in which each term is the golden part of the previous. We observe in passing by that if we consider the successive arcs equal respectively to a tenth, two-tenths, three-tenths, and four-tenths of the circumference and whose sum is equal to the entire circumference, their chords A B, U D, D G, G A form a quadrilateral whose sides are respectively the side l10 of the inscribed decagon, the side l5 of the inscribed pentagon, the side s10 of the inscribed decalpha, and the side s5 of inscribed pentalpha and whose diagonal B G is diameter and divides the quadrilateral into two right triangles, and then:
l2 5 + l2 10 + s2 5 + s2 10 = 8 r2
these four sides form a Tetractys whose sum is equal to twice the square of the diameter. We observe now that if with a, b, c, d we denote four segments such that each is the golden part the previous one has:
a = b + c and b = c + d
a + d = b + c + b – c = 2 b
Therefore, the second term in the sequence of the four segments is the arithmetic median of the extremes. It has then for the definition of golden part
b² = a c c² = b d
and then b ² c ² = a b c d and finally b c = a d, and the four segments form a proportion. On the other hand, indicating with M the harmonic median of the ends a, d it is such that
and therefore also
b c = b M
and then c = M, that’s to say the third term of the sequence is the harmonic median of extremes. We can therefore state the property: If four segments are segments following a sequence such that each segment is the golden part of the former they form a proportion and the second segment is the arithmetic median of the extremes and the third is the harmonic median of externals.