In our quest to define a metric to quantify intuition, and reduce the reliance on flawed information, we might ask what would such a unit of intuition look like, and where might we start looking for clues? First we must define intuition. Intuition is our ability to understand something instinctively without the need for conscious reasoning. It is, therefore, a psychological process, and consequently a biological one as well. At the same time it is instinctive, but a higher form of instinct that applies to vast stores of symbolic memory. As neuroscience research reveals, the neural activity of intuition involves large columnar clusters of neurons on the neocortex. There are billions of neurons involved in this process. Indeed they process the sensory signals of the external world and relay regular patterns as well.
The inner world of consciousness is built upon the symmetries of the brain, an organ found in the outer world, along with all the other symmetrical structures surrounding it. Indeed not only is the world full of symmetry but human bodies, being part of that world, have embedded all of that symmetry. As our brain is built on that foundation, our conscious mental operations can be seen as arising from it as well.
Symmetries seem all-pervasive and are found across all species, making it, in a sense, independent of all the many possible Umwelts in the world. It is a rule that is independent of all of the human race. We see radial symmetry manifesting across the universe, at all scales, from the tiniest to the largest, and throughout many flora and fauna species. Indeed, symmetry can be considered a glue that binds us all together. Since symmetry is so universal that its expression is invariant across all forms, it would seem natural to seek the fundamental laws of physics through symmetry. Indeed, our culturally based knowledge is a repository of observed patterns, and characteristic patterns reflecting the symmetry encoded throughout nature at every level.
One reason for pursuing these ideas is that our psychology is fundamentally tied to symmetry. Psychologists have demonstrated that visual compositions with symmetries, such as bilateral or vertical symmetry, are more readily detected. In 1897, the scientist Ernst Mach conducted an experiment using irregular shapes that demonstrated people could easily perceive bilateral and radial symmetries. He proved that we could detect symmetry before recognizing the pattern. This led Mach to conclude that symmetry is computed at a shallow level of image representation. Magnetic resonance imaging research has shown that when we look at an object, our brain detects visual symmetries in less than 50 milliseconds (Tyler). This is one of the quickest reaction times the brain has.
Pythagoras established the famous Pythagorean school in Croton, Southern Italy around 530 B.C.E. One of the unique results of this school was establishing the beauty of mathematical symmetries.
His Greek counterpart, Euclid was equally influential in the exploration of symmetry in space. Like Pythagoras before him, Euclid took logic as an organizing principle to another level, systematizing 465 known but disparate theorems and tying them all together in a work revered for its beauty as much as its power. It found the commonalities, or symmetry, in all these ideas and combined them into a beautiful and symmetric operation. Euclid’s work established Euclidean geometry as well as the axiomatic method and logical deduction, all fundamental elements of modern mathematics.
Across the ages, scientists have explored similarities and commonalities, and in the process have proven that symmetry is not only socially and psychologically necessary to humans but are also present in great degrees in structure the universe itself. Thus, it makes sense that we are fascinated by it as a species, consciously and unconsciously, which in turn has led to more investigation.
One of the recent attempts to try and quantify the universal nature of symmetry was in the field of aesthetics. The modern study of aesthetics found one of its greatest proponents in the late Harold Osborne, founder of the British Journal of Aesthetics. Osborne devoted his life to studying aesthetics and placed a great deal of emphasis on the perceptual mechanisms of aesthetics. To Osborne, that beautiful works of art took on an aesthetic quality was a direct result of our heightened state of awareness and arousal surrounding them, inducing within us alertness, vitality, and wakefulness. The layperson often cannot appreciate art for art’s sake, but Osborne thought that as reason is cultivated for its purpose in fields such as logic, pure mathematics, philosophy, and pure science, perception must also be developed for its own sake as well. He thought that somewhere in there, there must be some seed that could lead to a new understanding of symmetry.
As Osborne thought, If order is the key to understanding beauty, then perhaps there are fundamental laws of attraction that apply to all beautiful objects. Thus, Alexander Baumgarten (1714-1762) was the first to propose a new science based on laws of beauty, aesthetics. In his book Aesthetica (1750) Baumgarten argued that the appreciation of beauty is the ultimate goal of the aesthetic experience. Unfortunately, this branch didn’t manage to come out with much that could be of use to us. However, that wasn’t the end of the exploration of beauty.
Other attempts to explore symmetry as a phenomenon have occurred in art. Art historians have generally observed and commented upon the symmetries found in great works of art, but it wasn’t until 1963 that Charles Boleau’s classic, The Painter’s Secret Geometry, revealed the hidden mechanics of art appreciation. Boleau takes us behind the magician’s tricks: secret symmetries such as patterns, ratios, and vanishing points that great painters throughout history employed in their artwork. Art appreciators who have read Boleau’s book are astonished at the geometric overlays within the patterns that subconsciously attracted them to the piece. So insightful was his book that geometric practice still pervades today’s art world.
His book has led us to ask: is the division between art and science a false one? For one, Einstein said, “The greatest scientists are always artists as well.” Foremost among those who embodied this dexterity was the interdisciplinary genius, Leonardo Da Vinci. In spite of his prodigious scientific output, famed art historian E.H. Gombrich argues that Da Vinci took up his diverse scientific and engineering interests in anatomy, biology, civil engineering, and astronomy to elevate his artistry as a painter. This is one proof that intuition and intelligence have to work together.
Pythagoras, Euclid, Epicharmus, Da Vinci: these luminaries of Western thought explored the mysterious intersection of symmetry, mathematics, and aesthetics. Through these thoughts, they’ve managed to impact thought across millennia. Their ideas still reverberate throughout the world today, in science, math, art, literature, medicine, and other fields.
The symmetry found in the lessons of Pythagoras was only the beginning of the influence of symmetry on modern thinking. Our modern world lies upon the intersection of these ideas. Newton and Leibniz formulated the laws of nature as differential equations, but this changed utterly with the shift to symmetry.
Symmetry, as we have shown, is a very complex and universal relationship between many things in the universe. Molecules are always looking for balance, as are plants, as are the stars. Humans find the most symmetrical things the most beautiful. Planets are always looking to be symmetrically round, and snowflakes to be snowflakes. Since this tendency permeates the universe, we can easily find other phenomena that mirror these relationships.
What do all new-born mammals have in common? Offspring are genetically coded to find security with their parent(s). As a baby begins its new life in the world, their relationship with their mother is critical for protection, and sustenance. Subsequently, offspring become intimately familiar with the shape of their parents, and since those parents are symmetrical, symmetry is entrenched as one of the main elements of safety. Furthermore, this is an essential element of survival, as the child begins to see this symmetry in other living organisms in its environment as well. These easily recognized symmetrical patterns create a code for safety, food, and danger.
We can find movement in symmetry, and food through unique symmetrical shapes. We can discover ideas by comparing symmetrical shapes and discuss abstract ideas more clearly through the same concepts. Symmetry, because it is so universal, is a powerful all-purpose discovery tool.
Then, when we go out and look for symmetry, we can find it embedded in reality at every scale, from spiral galaxies to the spiralling paths of subatomic particles. The same applies to living systems. We can observe bilateral symmetry of most animal species, insects, leaves; rotational symmetry in eyes, jellyfish, worms, and flowers; and helical symmetry in scale patterns in pinecones or the double helix of DNA, to name a few. Fractal symmetry, a self-similarity between a part of the whole and the whole itself, appears in numerous plant and animal species at every scale.
As living organisms grow, certain structures repeat. Our symmetrical bodies are built from symmetrical genetic transcription laws that, in turn, reflect symmetrical molecular structures such as DNA. As we have said before, there is much evidence for the significant impact symmetry has on the human race every day.
Related to this is the idea of invariants. In recent years, conventional science has discovered the power of invariants, properties that remain unchanged under applied transformations. Invariants are always ratios and are always symmetrical. In other words, symmetry is the essence of invariants. Symmetrical objects can be transformed by applying specific operations to one part of an object to create another part of it. Hence in mirror symmetry, reflecting an object across a line can recreate the same shape on the other side. In radial symmetry, patterns are duplicated at fixed angles. Thus invariance principles provide a structure and coherence to the laws of nature just as the laws of nature provide a structure and coherence to the set of events. Indeed, it is hard to imagine that much progress could have been made in deducing the laws of nature without the existence of certain symmetries.
However, the unchanging nature of symmetry was not clearly understood until recently. Historically, symmetry and invariance were perceived as different concepts: Until the 20th-century principles of symmetry played little conscious role in theoretical physics. The Greeks and others were fascinated by the symmetries of objects and believed that these would be mirrored in the structure of nature. Even Kepler attempted to impose his notions of symmetry on the motion of the planets. Newton’s laws of mechanics embodied symmetry principles, notably the principle of equivalence of inertial frames, or Galilean invariance. These symmetries implied conservation laws. Although these conservation laws, especially those of momentum and energy, were regarded to be of fundamental importance, these were considered as consequences of the dynamical laws of nature rather than as consequences of the symmetries that underlay these laws. Maxwell’s equations, formulated in 1865, embodied both Lorentz invariance and gauge invariance. However, these symmetries of electrodynamics were not fully appreciated for over 40 years or more.
The connection between symmetry and invariance was only made in the 20th century when Albert Einstein proposed his theories of relativity. Einstein’s great advance in 1905 was to regard the symmetry principle as the primary feature of nature that constrains the allowable dynamical laws. Thus the transformational properties of the electromagnetic field were not to be derived from Maxwell’s equations, as Lorentz did, but instead were consequences of relativistic invariance, and indeed largely dictated the form of Maxwell’s equations. This is a profound change of attitude. Lorentz must have felt that Einstein cheated. Einstein recognized the symmetry implicit in Maxwell’s equations and elevated it to a symmetry of space-time itself. This was the first instance of the geometrization of symmetry. Ten years later this point of view scored a spectacular success with Einstein’s construction of general relativity. The principle of equivalence, symmetry, and the invariance of the laws of nature under local changes of the space-time coordinates dictated the dynamics of gravity, and of space-time itself.
With the development of quantum mechanics in the 1920s, symmetry principles came to play an even more fundamental role. In the latter half of the 20th century, symmetry has been the most dominant concept in the exploration and formulation of the fundamental laws of physics. Today it serves as a guiding principle in the search for further unification and progress.
This invariance/symmetry relationship is notable in modern gauge theory. Gauge theories have assumed a central position in the fundamental doctrines of nature. They provide the basis for the enormously successful standard model, a theory of the fundamental, non-gravitational forces of nature, the electromagnetic, weak, and strong interactions. To be sure gauge invariance is a symmetry of our description of nature, yet it underlies dynamics.
Symmetries such as Einstein’s relativistic invariance or the symmetrical gauge invariant have predictive power and provides us with an essential tool for the exploration of the fundamental laws of nature. Moreover, there is no answer for why nature should be symmetrical yet. The currently shaky knowledge of science forces us to once again examine our most basic assumptions. Out of that, we find a deeper epistemology of the universe that contains even more fundamental and symmetrical concepts that describe the universe.
Invariance and symmetry were discovered separately, but over time they were found to have significant properties that supported each other. These shared properties led us to believe that the universe is founded upon these two concepts. Invariant symmetry is truly a fundamental part of the universe and is found everywhere, from the quantum foam to the orbits of galaxies. This insight helped us discover a new type of predictive math. So, this math has powerful applications.