It has been the great secret of mathematics and ancient art and was used in the form of mathematical and mystical knowledge in all the great architectural works throughout history.

The Baghdad Wall Gate responded in its design to the divine proportion, and we can find this relationship in the pentacles engraved on Sumerian clay tablets from Mesopotamia in 7000 BC. It appears in the Bible in different forms, in the construction of Noah’s Ark and in the dimensions of the Ark of the Covenant; and we can find it in the design and construction of the great pyramid of Giza.

In imperial Egypt, every religious and artistic manifestation dedicated to the gods was not conceived without the presence of PHI and thus appears in all architectural, sculptural and pictorial creations: as in the plan of the temple of Seti I in Abydos, in the Temple of Dendera and in the Temple of Edfu. In the Great Pyramid of Cheops, it is manifested in the quotient of the height of the triangles and its side, which is 2 PHI; in the ratio between the height of a face and half the side of the base; in the ratio between the total area and the lateral area of the Pyramid, and also in the ratio between the lateral area and the area of the base, which is also PHI. We even find it in the King’s Chamber of the Great Pyramid, not to mention that it is found in the faces of all the great sculptures of the ancient period.

For the Pythagoreans, an initiatory community of scientific, philosophical, artistic, and religious character, the pentalpha or starry pentagon represented the triumph of the spirit over the elements of nature. Its mystical relations with PHI and the Spiral turned the Pentagram into the symbolic emblem of Health and Life of its initiatory school, and symbol of perfection and beauty. The Pentagram (symbol of the physical and mental man), and especially the Dodecahedron (the representation of the macrocosm) of the Platonic Solids were the most representative elements of his school. This knowledge came from the Egyptian mystery schools, in the sense of revealed sacred wisdom. For Plato, the Golden Ratio was the key mathematical relationship to understand the cosmos.

The golden number, also called the golden ratio, golden ratio or divine proportion, is represented by the Greek letter Phi Φ in honor of the sculptor Phidias, the most famous sculptor of Ancient Greece. Phidias studied this proportion and applied it in the design of the sculptures of the Parthenon, but it is exhibited in all its spaces: in the front facade, in the plant and in the side facades, in the temple of Apollo and in the statues of Zeus. We also find it in the Doric temples, as well as in its sculpture and pottery.

Euclid defines it for the first time in his book Elements of Geometry: «A straight line is said to be divided by half a ratio and extreme ratio when the length of the whole line is to the greater part, as that of the greater part is to that of the lesser part». The ratio of the whole line to the larger section is equal to the ratio of the larger section to the smaller section: C=A+B.
Most of the theorems and mathematical proofs about the geometries that compose Euclid’s elements are due to the Pythagoreans, although the Greeks never specified the way to find this golden number.

The legacy of ancient Greek mathematics passed centuries later to the West through the Islamic world of Al-Andalus. This growth of knowledge was mostly due to the schools of translators, mainly from Seville and Toledo, from where this knowledge spread throughout Europe in the 12th century.

Islam combined the mathematics of Greek, Syrian, Persian and Hindu culture and algebra, geometry and arithmetic would have a contemplative, spiritual and intellectual aspect, as well as a practical and rational one. The Alhambra in Granada with its prodigious geometries shares the secrets of the Gothic cathedrals and is the best example of PHI in Arab architecture. In Islam, Paradise is directly related to geometric perfection, and PHI appears in arches, facades and patios, and in the design of the tower of Comares, from where all the radial symmetries that provoke the incredible play of perspective start.There were many compendiums and mathematical and astronomical treatises on the symbolic and metaphysical aspect of arithmetic and geometry. Hindu ideas on infinity and its algebra were synthesized with Greek methods and together with the use of Indian numbers, known as Arabic numerals, were the main contributions, a fundamental contribution to the way mathematics is today.

On the other hand, a great number of works had music as a protagonist, where its effects on the soul were described. Based mainly on the theories of Pythagoras and Aristotle, the Islamic world dealt with music as a branch of mathematics, which was considered a philosophical discipline and was the first to give a scientific explanation of sound and to establish the rules for the construction of musical instruments.

The studies of Leonardo Fibonacci and Luca Pacioli were based on these Arabic sources. Fibonacci described in «Liber Abaci» the Arabic numerals and rediscovering the mathematical properties of this proportion connects it with the number PHI. The Fibonacci sequence or series begins with the numbers 0 and 1, and from these «each term is the sum of the two previous ones», this is the recurrence relation that defines and shapes this infinite sequence of natural numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597….

Luca Pacioli’s essay «De Divina Proportione» which contains the drawings of the 5 Platonic solids made by Leonardo da Vinci, is a compendium of the mathematical knowledge that Pythagoras had obtained from Egypt and had an enormous importance in the history of art since this knowledge would be used throughout the Renaissance by the great artists. Luca Pacioli called it Divine Proportion because of the religious and philosophical correspondences he found between this relationship and divinity.

Although no Renaissance writings are preserved that demonstrate the use of PHI in the construction of cathedrals, all medieval Gothic architecture maintains the golden ratio as a mathematical principle for its constructions. The secrets of Platonic mysticism wereexpressed again throughout Europe and stand out in the Cathedral of Notre Dame de Paris, in Santa Maria del Fiore in Florence and in the Cathedral of Cologne. The Pythagorean pentagram also appears in many constructions in the form of stonemasons’ marks or directly in the form of rose windows, as in the Cathedral of Seville, the Cathedral of Amians or in the Hermitage of San Bartolomé de Rio Lobos.

PHI is a perfect ratio and can be calculated in several ways, but the most interesting are the deep philosophical aspects that emerge from the results of the operations of the unit relations with PHI.

In addition to the pentagram, a symbol of health and power, and also associated with magic, the spiral is one of the most universal symbols in existence and is directly linked to the creation of the world and eternal life. The golden spiral or logarithmic spiral can be built from a golden rectangle, so it grows in geometric progression, and is one of the most symbolic figures that exist: it represents growth, evolution and expansion or vice versa.

PHI is also closely related to PI, as it is related to the fact that 4 divided by the square root of PHI is practically PI. On the other hand, PHI can be expressed as a function of PI and the numbers 1, 2, 3 and 5 of the Fibonacci series. Also PHI, PI and 5 can be related through trigonometry, since by dividing a circle, 360° in 5 equal sections, 72° each, we obtain the five points of a pentagon, whose dimensions are all based on PHI. We can also discover these relations reduced in the golden circle (diameter PHI) and in the golden ellipse (diameter PHI and 1).

The relationships between the Golden Number and PI have been transcendental in the history of the human being, since PI is symbolically linked to the creative power that originates and dynamizes the cosmos, while PHI expresses the power that configures it in harmonic parts; the manifestation of creation.