Quaternities
Why 4 is the most important number
There are four cardinal directions, four seasons, four forces of nature. There are four nucleotides in the DNA, four covalent bonds of carbon, four chambers of the heart. There are four Christian gospels, four Islamic holy books, four Buddhist noble truths, four Hindu yogas.
Enter the Quaternity: a set of four.
Wherever we look, we discover that the magic number four reigns supreme. In philosophy, religion, psychology, music, and many other fields, we find that the central concepts with which we organize our world, our understanding of the world, and our place in it have their roots in a quaternity. In physics, chemistry, and biology we discover that the substrata of matter, energy, and life emanate from some quaternity. It seems that our mind and modes of understanding, as well as the physical world, gravitate towards the formation of fourfold structures. Since the dawn of civilization, be it Ancient Greece or India, and throughout history up to modern times, quaternities have made their appearance in almost all fields of life, whether practical or theoretical.
Ancient roots
The Ancient Greeks, starting with the pre-Socratic philosopher Empedocles, saw the world and everything in it as consisting of four elements: earth, air, water, and fire. These were the equivalent of what we would today call atoms, out of which everything is made. Aristotle would subsequently relate these elements to another quaternity: that of the sensible qualities (hot, cold, dry, and wet), and later to his four causes (material, formal, efficient, and final) that purported, if not to explain, then at least to classify all possible answers to the question “why?”
The Pythagorean school actually assigned the number four a mystical quality via its sacred symbol, the tetractys — an equilateral triangle consisting of ten points arranged in four rows, each row representing one of the first four numbers symmetrically. The Pythagoreans were the first to realize that by adding the first four numbers, we get the number ten, which corresponds to the number of our fingers and toes. They saw the creation of the world as proceeding along the tetractys from the monad (one), to the dyad, the triad, and the tetrad. They even had a prayer to the tetractys that elaborated on the creation of the important number ten out of four:
Bless us, divine number, thou who generated gods and men! O holy, holy tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all.
What a hymn to the number 10, centuries before it became the foundational basis of our decimal numeral system!
The tetractys was related to the Pythagoreans’ religion, metaphysics, and even music, thereby transcending the world of mathematics that engendered it. They saw musical harmony as being derived from the tetractys through the ratios of each consecutive line of the triangle: pure fourths (the 4:3 relation), pure fifths (3:2), the ratio of the unity (1:1), and the octave (2:1) — yet another quaternity of musical intervals. Last, each level of the tetractys gave rise to the dimensions of reality that also formed a quaternity: the point defines a singular dimension, the two points a line, three points a surface area, and four points a solid in three-dimensional space.
Quaternities were and still remain central in the Indian civilization too, where they appear in all fields of life. First, we have the infamous four castes on which Indian society has been divided and structured throughout history (Brahmins, Kshatriyas, Vaishyas, and Shudras). Then the four goals of a human life, the Purusharthas. These refer to the main preoccupations that every person must have in their life: dharma is right conduct, artha refers to financial prosperity, kama is the enjoyment of worldly pleasures, and moksha is the ultimate aim of life — the freedom from the cycle of death and rebirth (samsara) and unification with the Divine. To achieve the latter, the Hindus again have four paths, or yogas: Bhakti yoga, the path of Love; Raja yoga, the path of meditation and self-discipline; Karma yoga, the path of disinterested works; and Jnana Yoga, the path of knowledge, understanding, and wisdom. Even consciousness is divided into four states: jagrat is our state of wakefulness, svapna the state of dreaming, sushupti the state of deep dreamless sleep, and turiya is the state of pure awareness during which ignorance disappears. At any moment, we exist in only one of these states.
When encountering all these Indian quaternities for the first time, one may question the need or even the practical usefulness of such categorizations and divisions. But upon closer examination, all of the above fourfold divisions make sense and are recognized as the product of deep preoccupation with the subjects at hand. A good test of whether such divisions are arbitrary is always to ask oneself: is there a fifth element missing? For example, can you come up with any other state of consciousness apart from the four mentioned?
Moving on to Buddhism, we have the four noble truths: dukkha, the affirmation of the unsatisfactoriness of life; samudaya, craving or attachment to worldly things, which is the cause of dukkha; nirodha, the realization of the possibility of liberation; and magga, the Noble Eightfold Path that leads to liberation — these “eight folds” obviously being derived from a double quaternity. There are also the four Buddhist virtues, the Brahmaviharas for which each must strive: metta, loving-kindness; karuna, compassion; mudita, empathetic joy; and upekkha, equanimity. Furthermore, and by now not surprisingly, Tibetan Buddhism, which has conquered the West over the past several decades, is also divided into four major schools, namely the Nyingma, Kagyu, Sakya, and Gelug. The number four is also encountered in much of Buddhist mythology and ritual. The (usually giant) mythological four heavenly kings, for example, can be found throughout Asia adorning Buddhist temples.
In Islam we have the four holy books: the Quran itself, considered the supreme revelation of God; the Tawrat or Jewish Torah; the Zabur, which is the Book of Psalms from the Old Testament; and the Injil, which is the New Testament. There are also the four Sunni schools of thought: Hanafi, Maliki, Shafii, and Hanbali. And, in a parallel to the Buddhist heavenly kings, Islam has four archangels — who also reside in heaven.
In Christianity, we have the symbol of the cross, which itself is a pure quaternity dividing space into four regions while also pointing in four directions; the four gospels and four gospel writers; and the four-character name of God, brought in from Judaism, the tetragrammaton, YHWH — pronounced Yahweh. The number four is also encountered very often in the Bible via its brother number 40. For example, we have the flood of Noah lasting 40 days, the Hebrews wandering 40 years in the desert, Jesus fasting 40 days in the desert, 40 days of Lent before Easter, 40 days separating the Ascension of Jesus from his Resurrection, and more.
Quaternities appear in many tribal societies as well. We may take one truly astonishing example from the Native Americans to make the point. This is how Ed McGaa, a member of the Sioux tribe, explains the significance of four in his book Mother Earth Spirituality:
There are four faces, or four ages: the face of the child, the face of the adolescent, the face of the adult, the face of the aged. There are four directions or four winds … There are four things that breathe: those that crawl, those that fly, those that are two-legged, those that are four-legged. There are four things above the earth: sun, moon, stars, planets. There are four parts to the green things: roots, stem, leaves, fruit …
At the foundations of the universe
But quaternities do not only appear in abstract mathematics, philosophy, and religion. Nor are they the naive outcome of the human compulsion to divide and classify, nor the product of an imagination gone rampant! They are actually to be found at the very foundations of the sciences and the natural world.
Let us begin with physics: we have the four fundamental forces of nature — gravity, the weak nuclear force, the strong nuclear force, and the electromagnetic. Although physicists have been trying for decades to unify them, they have yet to achieve this, so we are … stuck with a quaternity of forces that we use to explain everything that happens in the universe.
Yet what physicists have failed to achieve with the forces of nature, they managed to do in another field: that of electromagnetism. In 1865, the Scottish physicist James Maxwell created the greatest quaternity up to that moment in science: he conceived and developed a unified theory of electricity, magnetism, and light — that became known as the Theory of Electromagnetism. This theory is considered a model of elegance and beauty not least because it consists of four simple equations that describe the fundamental behavior and interaction of electric and magnetic fields. Maxwell’s equations introduced for the first time the idea of invisible force fields that govern the movements of objects, as well as the concept of self-propagating electromagnetic waves — that underpins all our modern communications. In effect, he created a new mathematical language for describing the physical world. The following incident is characteristic of how important these four equations are in the description of the physical world: When Einstein visited the University of Cambridge in 1922, he was told by his host that he had done great things because he stood on Newton’s shoulders. But Einstein replied: “No I don’t. I stand on the shoulders of Maxwell!”
Standing on these shoulders, Einstein would in turn create another revolution in physics, in 1905 (exactly … 40 years after Maxwell!), with his Special Theory of Relativity. In it, he showed that the three dimensions of Space are connected with the dimension of Time, and that they in fact form a four-dimensional entity (manifold) that he termed Spacetime. Rather than standing aside and “observing” the happenings of the cosmos, Time now became one of the basic building blocks of the universe, permeating space and matter. What is even more astonishing (and unfortunately this is rarely explained) is that this new entity of Spacetime that describes the universe has its own “measuring rod” called “invariance.” And what this invariance tells us is that our universe is not only a four-dimensional manifold involving the three dimensions of Space and the one of Time but that it is the same for all observers at every corner of the universe. Einstein’s Relativity theory, contrary to what its name suggests, is actually about the Absoluteness of the fourfold nature of Spacetime. The structural foundation of our universe is not only a quaternity, but an absolute one! Einstein regretted the name Relativity (which was coined by Max Planck two years after Einstein’s paper was published) and said that he would have preferred to call his theory the exact opposite (Invariantentheorie), highlighting the fact that Spacetime is invariant throughout the universe.
Moving on to chemistry, we have the four states of matter: solid, liquid, gas, and plasma (unrelated to blood plasma). There are also the four major classes of biological macromolecules that form the basis of all life: carbohydrates, lipids, proteins, and nucleic acids. These molecules are organic, which means they contain carbon. As it turns out, carbon itself contains four free electrons in its outer shell and can form four covalent bonds with other atoms or molecules. Carbon, the most important element for life on earth, is basically a chemical quaternity! Being one of very few elements that can form long chains of its own atoms, carbon gives rise to a large number of organic molecules that have become the focus of the field of study known as organic chemistry.
In biology, we yet again encounter what is probably the most important fourfold structure in nature: the DNA, which carries the genetic material of living cells. The famous double helix of the DNA macromolecule is made up of nucleotides that are constructed of four types of nitrogen bases — adenine (A), thymine (T), guanine (G), and cytosine (C). It is these four bases that determine the DNA’s genetic code. Of course, there are many other fourfold structures to be found in the living world, be it the four limbs of most land animals, the four types of human blood, the four wings of most insects as well as the four stages of the development of insects — egg, larva, pupa, and adult.
In physics, chemistry, and biology, wherever we go deep into the most fundamental elements of reality and into the most fundamental structures of matter, energy, and life, we discover that at the foundations of the universe and nature lie fourfold structures. Quaternities lie at the very heart of the makeup of the cosmos.
In the world we create
So, it is not surprising when we humans, who function in such a universe, end up reproducing or imitating, in a sense, the way the universe is. We have already seen how a big part of our abstract and concrete thought, our philosophies and religions, enclose important quaternities. But it is not just our thought. Most of the material objects we construct end up relating to some form of fourfold structure: Just consider our square houses with their rectangular doors, windows, and sofas, our four-legged tables, chairs and beds, our four-wheeled cars, our rectangular mobile phones and TV sets, our books and bookshelves — it seems that the majority of what we create has four sides!
Even our arts, and especially our music, have gradually evolved many fourfold structures: A symphony orchestra has four types of string instruments … each with four strings (violins, violas, cellos, and double basses), four main woodwind instruments (flutes, oboes, clarinets, and bassoons), and four main brass instruments (tubas, trombones, trumpets, and French horns). In chamber music, the most important ensemble is the string quartet (two violins, a viola, and a cello). Even the piano trio is actually a quartet, because the piano is played with two hands and has two melodic lines! Similarly, our choruses have four main voices (soprano, alto, tenor, and bass). As for the musical compositions themselves, most symphonies and quartets have four parts or movements.
3 + 1
But it is time that we examine the structure and features of the quaternity qua quaternity. An important question is the following: Are all the members of a quaternity equal?
Although all members of some quaternities are equal, as it turns out, most quaternities take the form of 3 + 1, where one member of the four is primary. In the four cardinal directions, we orient ourselves by locating the north star, and we always place the north at the top of maps — so the north is primary. In the tetractys that incorporates the four first numbers, the number four is the most important. In Aristotle’s four causes, the Final Cause is primary because it defines or explains the movement towards which each living organism, including humans, moves. In Christianity, we have the three synoptic gospels and the Gospel of John — by many considered superior to the others. In the Indian four goals of life, moksha is the primary, and of the four yogas, Jnana yoga is traditionally considered the superior.
In the examples from physics we examined earlier, of Maxwell’s four equations the last one is the most important and transformative, because it basically brings the other three together into a unified framework and predicts the existence of electromagnetic waves. In Einstein’s theory we have again the 3 +1: the three dimensions of Space and one dimension of Time. Moving on to chemistry, of the four states of matter, plasma is distinct from the other three, because it is not composed of molecules and atoms, but of free electrons and ions. And of the four DNA bases, Thymine (the only one to be found exclusively in the DNA) is considered somehow more important than the other three, because of its unique role in maintaining the integrity of genetic information.
The same applies in music: the violins are the primary instrument in an orchestra (not least because they constitute the majority of instruments); the first violin is the primary instrument in a quartet, while the first violinist is traditionally the head of the ensemble; the soprano is usually the primary voice in an opera; the first movement of a symphony defines its whole character.
Carl Jung — the great Swiss psychologist who studied the psychological significance of quaternities deeper than any other — knew of the 3 + 1 scheme and was the first to point out that when we encounter a trinity, we should immediately start searching for a fourth element that will complete it. He believed that the Christian Trinity was lacking in two ways: it included neither evil nor the feminine. When Pope Pius XII promulgated the dogma of the corporeal Assumption of the Virgin Mary in 1950, Jung considered this inclusion of the feminine as the most significant Christian religious event since the Reformation. He saw it as the final completion of the Christian Trinity.
We could actually use the same Jungian approach to transform the Indian trinitarian gods into a quaternity. In Hinduism, Brahman is divided into three deities: Brahma, Vishnu, and Shiva, respectively representing the acts of creation, preservation, and destruction. This is another masculine Trinity like the Christian one, even though its form and symbolism is entirely different. But each of these three deities has a wife: Saraswati, Lakshmi, and Parvati. These three female deities are often grouped together under an overarching feminine goddess, Shakti. With the inclusion of the feminine, the fundamental male trinity is thus transformed into a quaternity that includes all aspects of the Divine.
The archetype of Wholeness
But why is the quaternity so important? And why does it appear so often throughout history and around the world?
There are a few reasons for this. Let us start from what is probably the most important one, which was discovered and addressed by Carl Jung: He came to the conclusion that the quaternity is a symbol of, or rather, to use his own terminology, the archetype of Wholeness. He often related it to the mandala, which is usually a circular symbol, divided into four parts. For Jung, the quest for wholeness becomes one’s vocation that slowly but inevitably directs one’s life. Wholeness was related to Jung’s other important concept: Individuation. This is usually defined as “the process by which every living organism becomes what it was destined to become from the very beginning.” In order to “individuate,” one must follow one’s vocation, which Jung thought of as being “an irrational factor that destines a man to emancipate himself from the herd and its well-worn paths.” The quaternity symbolizes the final attainment of this supreme goal of Man, and it is for this reason that it often appears in our dreams and in so many symbols and fourfold divisions around the world. There is a sense in which four becomes a symbol of the completeness of a human life.
floats above a Swiss village — from Carl Jung’s “The Red Book.”
There seems to be in nature, and also in our mode of understanding nature, a tendency whereby completeness is achieved once the number four is reached. But why? I have meditated a lot on this subject and I have come to the conclusion that there is another concept that may explain it: stability. Wholeness and completeness are achieved when movement and activity cease. Stability is the quality of something to remain what it is without seeking any further movement. The square or rectangular structures seem to command a stability by virtue of their strong symmetrical structure. The same happens with all solids. A cube or a pyramid “sitting” on a square base will be more stable than anything standing on a triangle — i.e., there’s a reason that our tables and beds have four legs, our cars four wheels, our houses four walls. One may claim that these objects end up taking this structural form because we humans construct them by incorporating (consciously or unconsciously) in them our own form that has four limbs. This may be true to some small degree. But still, can you imagine a three-walled house or a triangular swimming pool? Tables with three legs tend to be wobbly, and few would venture on a long road trip by hopping … on a tricycle. Even the apparently stable two-wheeled motorbike, upon closer examination, is found to obtain its stable grip on the asphalt by two other factors: the handlebar managed by the rider, which allows for stable turning, and its speed — the less the speed, the less its stability. This is actually a great example of moving from a dyad to a quaternity: even the motorbike’s structure and mobility are part of a covert quaternity!
Harmony and beauty
But wholeness and stability are not all. Quaternities have an inherent harmony and beauty. From the very first analyzed quaternity, the tetractys, to the symmetrical proportions of the rectangular Parthenon and the Egyptian Pyramids, to the mental classification of material objects and ideas from Aristotle to Jung, we discover that there is a harmony and beauty in fourfold structures.
To elucidate this aspect of quaternities, let us take Maxwell’s equations again as a first example. What is astonishing is that Maxwell did not do any experiments in electromagnetism himself, nor did he truly discover any new phenomena. He collected a number of already known laws discovered by others (Gauss, Faraday, Ampere) and fiddled with, rearranged, and bound them together to create a unified framework. The equations ended up bearing his name in order to honor his transformative contribution in the whole field. What Maxwell did is a work of the creative imagination. It resembles a piece of art. The equations themselves, when written in their compact vector calculus form, have an elegance and symmetry. That so few terms are used to represent the relationships between electricity and magnetism is aesthetically pleasing. That such simple relationships exist in the first place, and that physicists can understand them at a glance, is not only awesome but points to an inherent harmony in nature that Man has simply unveiled. It was in search of a similar beauty that Einstein would later discover his own elegant equations describing Spacetime.
Another example, this time concerning the proportions of our human body, is the beautiful symmetry and harmony of the relationship between our four limbs. Everybody is acquainted with Leonardo da Vinci’s famous sketch of the Vitruvian man (named after the Roman architect Vitruvius, on whose discoveries da Vinci based his drawing). The figure shows that our body can be enclosed in both a square and a circle — just as Jung had likewise highlighted the relationship between quaternities and the mandala (a circle). The beauty of da Vinci’s drawing is due to the inherent harmony and beauty of the quaternity — he did not invent it, he simply brought it out in the open.
But perhaps the best example of the harmony and beauty of quaternal structures is none other than the human heart! Our heart has four chambers — neither two nor six — and four valves, because four is the most economical number for it to perform its function. The heart is basically a mechanical pump: the “dirty” blood enters the right atrium, moves into the right ventricle, and then it is pumped to the lungs to be cleaned; the “clean” (oxygenated) blood then returns to the left atrium and moves to the left ventricle in order to be pumped to all the organs of the body. But why does it need to enter an atrium first before moving on to a ventricle? The answer to this question explains both the necessity of four but also highlights the wisdom of nature: A pump must first become full of liquid in order to pump. It also has to be filled from a tank separate from the pump. This other tank are the two atria, in which the blood enters and then waits for the pumping of the ventricles to be finished so that they may refill the ventricles. But during this “filling stage,” the heart muscle works very little, so in effect the heart rests — nature incorporated the heart’s rest in its mode of work. The heart muscle, which is the strongest single muscle of the human body, is also the only muscle in nature that never stops working, exactly because it works and rests simultaneously!
The heart works with electricity — it transforms the chemical energy of the body into electric — and uses Bernoulli’s principle of liquid dynamics to perform its function. Shockingly, the most important equation describing the blood’s flow between the chambers of the heart, which is used on a daily basis by cardiologists, includes the number four: ΔP = 4V2. Furthermore, the four chambers of the heart are not equal — they again follow the pattern 3 + 1. The two large ventricles are unequal; the left ventricle (which is the right one in the video below) is larger, stronger, and has thicker walls than the right because it pumps the blood to the whole body. So, it is the primary of the four chambers.
When observing the human heart at work and the coordination of the opening and closing valves with the pumping and refilling of the chambers, one can become hypnotized by the immense beauty, harmony, and economy of its function. Last but not least, I would dare claim that the cardiologist is the most complete of doctors, since his range of knowledge must perforce span the whole gamut of sciences — mathematics, physics, chemistry, biology, and more. Both the heart and the cardiologist are symbols of wholeness and completeness.
In our everyday life
But at this point, you may ask: If quaternities are important in so many fields and in so many ways, how do they enter our everyday life?
Well, whether we are aware of them or not, they already do enter our life since they are ubiquitous — as described earlier with the examples of the many objects we create. But they can become even more important in our life if we start paying more attention to them or, still better, if we consciously seek them. I actually think we ought to consciously seek them for three important reasons. First, constructing any form of quaternity helps us to organize our thoughts and what we already know. Second, the simple recognition or discovery of a new quaternity makes us view the world in a new way and from different angles. We discover new interrelationships and patterns among our familiar objects and ideas. When, for example, I was first acquainted with Jung’s interpretation of the Ascension of Mary completing the Trinity, I realized that yes, it is true that the feminine element was missing, yet Mary has been ever-present, silent, and unacknowledged all along. Another example, elucidated in the essay “Legs and Arms” in my book Destination Earth, is the deeper symbolic relationship between our legs and arms (four limbs). In an insight I had while traveling around the world, I saw my legs as being the symbol of my solitude, my individual path, and my uniqueness, while my arms are the symbol of togetherness, my connection to others, and my belonging to the human race. My legs make me who I am; they create my solitary path. My arms make me who I belong to; they connect me to the world. Although the source of my inspiration had come from traveling, I feel that the idea holds true for life in general. This personal discovery, when shared, may, of course, lead others to view the same meaningful relationship between their limbs in their own lives.
Third, and most important, quaternities seem to enclose hidden meanings that are revealed the moment the fourfold structure makes its appearance. Let me give another personal example from the period of my midlife crisis over two decades ago: I was trying to understand the forces that shape my life by examining the character, drives, and lives of my parents. But I felt that I was not getting anywhere with the parent-dyad. Since I was reading Jung at the time, I playfully decided to examine the lives of my four grandparents. Suddenly, all fell into place! The main driving forces in each of my grandparents’ lives perfectly described the four main forces that had been guiding my life. I immediately created and drew “a grandparental quaternity” and delved deeper into these forces, discovering a 3 + 1 form where the one was my main highway in life (corresponding to my namesake paternal grandfather’s main drive and path in life) and the other three created the environment in which this drive may find its own fulfillment. The moment I found this new fourfold structure a hidden reality was revealed that infused my life with new meaning. I would dare say that there are not just hidden meanings enclosed in newly discovered quaternities but sometimes even wisdom.
Seeking out quaternities, finding them, and meditating upon them, but also constructing one’s own, are creative endeavors of the active imagination that may enrich one’s life immensely. Such seeking will reveal new beautiful things and meaningful interconnections in both the cosmos and our inner self.
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Check out my latest books, both published in 2024, Conversing with Cavafy: Reflections on the Journey of Our Life and The Eternal Ragpicker: Essays on the Human Condition. My first book is the award-winning Destination Earth: A New Philosophy of Travel by a World-Traveler. All three books are available on Amazon and at your local bookstore.
