Pythagoras and the Mystery of Numbers


Kate Hobgood



         Pythagoras was the first of the great teachers of ancient Greece. Born in 580 B.C., Pythagoras became one of the most well known philosopher and mathematician in history. Creating the Pythagorean Brotherhood, his teachings greatly influenced Socrates, Plato, and Aristotle. Pythagoras is considered as the first pure mathematician; he also founded a community/society based on religion and science. He is most widely known as the author of the Pythagorean Theorem. Some even venture to believe that the word “philosophy” was invented by Pythagoras. He claimed to be a lover of wisdom, for which the Greek word philo means love and Sophia means wisdom.

         The Teachings of Pythagoras revolved around the idea that when considering the deepest level, reality is essentially mathematical in nature. Pytahgoreans believed there was a system of principles existed behind numbers. The principles form a foundation for many concepts of traditional Western thought. One of his most basic notions revolves around the symbolism and beauty associated with the Divine Proportion.

         Pythagoras expressed his thoughts and ideas by using words and symbols rather than numbers as we know them today. Instead, his ideas were based on many thoughts that we associate with numbers today. For instance, a point is the essence of a circle. A series of symbolic forms were constructed to mirror the concepts of the universe. Using a compass, straightedge, and writing utensil, mathematical philosophers created these symbolic representations.

The Mystery of Numbers

         Of all shapes, the circle is the parent of all following shapes. When a circle is mirrored, two mirrors are created. These two circles side by side build a foundation for all numbers. The overlap of the circles allows each one to share the center of the other. This shaped created is called the vesica piscis (Latin for “fish’s bladder”). From this shape, a triangle, square, and pentagon can be produced. And the relationship between these figures justifies the existences of further number principles.

Vesica Piscis

         Believing that nothing exists without a center, mathematical philosophers started with a point and drew a circle around it. This symbol is called the monad and represents the number one. This figure is the most stable, and the mathematical philosophers also called it The First, The Essence, The Foundation, and Unity. Pythagoras believed the monad to be god and the good. The monad is origin of the One. The monad is the seed of a tree for which the numbers are to the monad as what the branches of a tree are to the see of a tree. The monad in relation to other numbers preserves the identity of every other number or anything it encounters. Any number multiplied by one is itself, and any number divided by one is itself.

         “The Pytahgoreans believed that nothing exists without a venter around which it revolves. The center is the source and it is beyond understanding, it is unknowable, but like a seed, the center will expand and will fulfill itself as a circle” (Hemenway 51).


         In order for ‘one to become many’, the circle (or one) must be transformed by a reflection. By contemplating itself, the circle is able to become many. It is replicated with each circle sharing the center of the other. The geometry of creating a line that connects the two center of the circles furthers the principles of all following numbers.

         The dyad involves the principles of “twoness” or “otherness”. Greek philosophers referred to the dyad as “audacity” because of the boldness of separation from the one, and “anguish” because there is still a sense of tension of a desire to return to oneness. They believed that the dyad divides and unites, repels and attracts, separates and returns. “Pythagoras held that one of the first principles, the monad, is god and the good, which is the origin of the One, and is itself intelligence; but the undefined dyad is a deity and the evil, surrounding which is the mass of matter” (Aet. 1. 7; Dox. 302). The dyad is the door between the One and the Many. Recall the symbolic figure of the vesica piscis. The vesica piscis is a passageway to the journey of spiritual self discovery. The notion of fertility is associated with its vulva shape, and is thus related to the passage of birth.


         The triad represents the number three. It is the first born and the eldest number. The equilateral triangle serves as its geometric representation and is the first shape to emerge from the vesica piscis. The triangle contains the smallest area within the greater perimeter. The number three is the only number equal to the sum of the previous numbers. For instance, one plus two equals three. And three is also the only number whose sum also equals their product. Or, one plus two plus three equals one times two times three. The triad signifies prudence, wisdom, piety, friendship, peace, and harmony. The triangle represents balance and is a polygon of stability and strength.


         The triangle can be extended beyond the vesica piscis by extending the lines through the corners to the opposite sides of the circle. Connecting these new lines with horizontal lines creates a larger triangle. Extending lines to further fill the vesica piscis results in a ‘profound harmony’.

The next shape emerging from the vesica piscis is the tetrad. Using logic to construct the tetrad, philosophers drew a horizontal and vertical line connecting the centers and the intersecting points of the two circles. Then, when a circle is drawn along a line that connects the two centers, a perfect shape of a square exists within the circle. Four is associated with justice, wholeness and completion. There are four seasons, four ages of man, four directionsFour is the first number formed by addition and multiplication of equals. It is the first “female” number because it is the first even number.


The shape of the pentad follows as the symbol of life itself. The pentad consists of all previous numerical symbols: monad’s point, the dyad’s line, the triad’s surface, and the tetrad’s three dimensional volume. The pentad also refers to the well known five point star. The pentad arises in the existence of our five fingers, five toes, the symbol that wards off evil, and a symbol for power and immunity. Because of the importance of the pentad, the construction of the symbol was first kept secret from society. The pentad was used as a secret sign among the Pythagoreans for them to distinguish themselves and recognize other members.


The pentad’s symbolism can be directly related to the Divine Proportion. And the regeneration of the pentad is related to the value of phi (Ф). The image of the pentad is found in nature in leaves and flowers. The Greeks believed each point of the pentad to represent an element: water, earth, air, fire, and idea. Early Christians used the pentad to represent the five wounds of Christ. The symbol, when the point is directed downwards, was later used as a sign for Satan and the Devil.

Regenerations of the Pentad

The decad represents the number ten. Instead of simple numeric and geometric interpretations, the decad further extends to the idea of a new beginning of limitlessness. Symbolizing both world and heaven, the decad helps us to understand the creation of the universe. The monad times the dyad times the pentad (one times two time five) results in the decad. Since any number times ten is similar to any number times one, it is similar to the monad; however, the number is brought to a higher level.


“Ten is the very nature of number. All Greeks and all barbarians alike count up to ten, and having reached ten revert again to the unity. And again, Pythagoras maintains, the power of the number 10 lies in the number 4, the tetrad. This is the reason: If one starts at the unit (1) and adds the successive number up to 4, one will make up the number 10 (1 + 2 + 3 + 4 = 10). And if one exceeds the tetrad, one will exceed 10 too…. So that the number by the unit resides in the number 10, but potentially in the number 4.” (Aetius 1.3.8)


Early philosophers found harmony in numbers. The symbolism and beauty behind each number can be further extended to the essence of all following numbers. The mysteriousness behind the theories founded by Pythagoras and his followers is certainly deeply inspiring and symbolic.


For further reading:

Hemenway, P. (2005). “Divine Proportion: Ф (Phi) In Art, Nature, and Science.” New York:  Sterling Publishing Co.