While not known for a finely coiffed hair and flawless visage, Einstein was motivated by the pursuit of scientific beauty. He looked at the universe with awe, always convinced that a great range of physical phenomena could be elegantly explained by a few simple principles. He obsessively contemplated the deep patterns of the universe through creative thought experiments, trying to see through the eyes and feel through the bodies of imaginary observers in fantastical situations. Chasing a beam of light or being in a windowless box accelerating through space were among his creative “thought experiments.” These could have led him to produce science fiction stories, for he was a very good writer, and early drafts of his theories resembled premises like HG Wells’ stories of time travel. His vignettes, like a traveler on a sub-light-speed train passing two bolts of lightning, remind me of brief art videos or an Escher print, challenging us to go beyond usual conceptions. Einstein’s brother and many others noted that Albert had more the temperament of an artist than a scientist. However, for all his creative fantasizing and meditation, punctuated by frequent violin playing breaks, Einstein was after scientific beauty, symmetry, simplicity and deep coherence in the laws of the Universe.

Mathematician Ian Stewart says of Einstein, “He drew radical theories from the simplest of principles and was guided by a sense of elegance rather than a wide knowledge of experimental facts. The important observations, he believed, could always be distilled into a few key principles. The gateway to truth was beauty.” The beauty of which Stewart speaks is a particular kind, not the beauty of hair salons or fashion runways. When scientists and mathematicians like Stewart speaks of Beauty, he means pattern and symmetry.

Symmetry in Objects

In mathematics (the science of patterns), symmetry is an immunity to change. Move an object, or do an operation on it, and if the object does not look different, it is a symmetry. Turn an equilateral triangle one third of a full turn (120 degrees), and it looks the same. Turn it two thirds of a turn and, likewise, it still looks the same. The triangle looks the same because it is essentially the same triangle that is was before the rotations. This is rotational symmetry. By contrast, turn a triangle 90 degrees, a right angle, and the triangle tilts to one side, so we say that move is not a symmetry of the triangle.

Rotating a square by 90 degrees is a symmetry of the square, because it will look the same as it did before the move. In this case, the “move” is a rotation, but there are other moves that can be a symmetry. For example, take a grid pattern of squares and move it exactly sideways one square to the right or left and the pattern will look the same. Similarly a grid of hexagons, like a honeycomb, or a grid of triangles, has this kind of symmetry. This is called translational symmetry.  It is easy and fun to come up with patterns that have interesting combinations of symmetries. (you can find several translational symmetries and rotational symmetries in this image).

In our Beautiful Discovery boxes, we explore such symmetries with Escher tessellations, natural wonders like crystals, honeycombs, and origami. The simplest kinds of symmetries are translational symmetries, which we show with grids, tilings, games with tiles, and artful origami of the type that uses repeated patterns across paper. Physical objects such as cardboard pieces on a game board show simple visual and tangible kinds of symmetries that apply to objects.

Symmetries of rules and laws

Not only can objects have symmetries, but so can rules, like rules of checkers on a board, or the board games that come with our Beautiful Discovery boxes. The rules are the same anywhere on the board. In the same way, symmetries apply to physical laws, like the laws of motion. Throw a ball and it will behave according to the same laws of motion, regardless of which direction we face. We can rotate any direction before we test the laws of physics with our ball. The laws of physics have rotational symmetry.

Similarly, move 1000 feet to the right and the laws of physics are the same. For example, we can stand on a pitcher’s mound or over there 1000 feet away. It doesn’t matter to physics where we are in space when we throw the ball. The laws hold even when “translated” in space. This is a “translational symmetry” of the laws of physics. Likewise, the laws of physics are the same at any time. We can throw the ball today or next week and the laws of physics governing the ball stay the same. This is another translational symmetry of the laws of physics. It is a translation in time that doesn’t change the laws.

Finding simple symmetries and patterns is essential to learning at the most basic levels. A developing infant is constantly detecting patterns and regularities to predict and cope with the world. For the same reasons, if you suddenly found yourself in another world, you would naturally seek out patterns and similarities to familiar things in your own world. We made pattern seeking the central theme of our Beautiful Discovery boxes because it is a visual and intuitive way to grasp our world and it leads to the most advanced scientific discoveries. While this search for patterns is as fundamental as the first sights, sounds, and textures of newborn babies, searching for beautiful patterns also has been essential at the most advanced frontiers of scientific research.

Searching for symmetries and patterns has been the key to many great discoveries in the sciences. This includes mathematics, which is the science of patterns, and physics, which uses mathematical models to describe the real world. Amazingly, many real-world phenomena are well described by abstract mathematical models of great beauty. In fact, the search for beauty often has been a key ingredient in the discovery of the nature of our world.

Symmetries of Electricity, Magnetism and Light

A beautiful discovery in the 19th century was that electricity and magnetism were two sides of the same phenomenon. Building on Faraday’s empirical findings, James Clerk Maxwell was able to show in his equations that changes in electric fields affect magnetic fields in a way that is symmetrical to how changes in electric fields affect magnetic fields. Electric and magnetic fields are simply two symmetrical cases of a more fundamental electromagnetic field (E is electric field, B is magnetic field in the image, a lightwave). Moreover, wave-like disturbances in both fields produce light. Magnetism, electricity and light were not separate and unrelated as was previously assumed. This was a great unification. “It was beauty and symmetry that guided Maxwell and his followers–that is, all modern physicists–closer to truth.” (Wilczek, 2015)

Scientists describe Maxwell’s equations as elegant, beautiful, and symmetrical. Yet, it took much thought to understand the symmetry because it was not like the symmetries of the laws of physics known at the time, in which laws of motion treated time and space as separate. Instead, the equations of electromagnetism led to new ways of describing the relationship of space and time. In fact, deeply contemplating these equations had much to do with Einstein’s revolutionary re-conceptualization of time and space. It was the elegant symmetry of Maxwell’s equations that led Einstein to this dramatic discovery. The symmetry suggested to Einstein that the speed of light was fixed, invariant. This invariance of the speed of light posed a puzzle.

Symmetries of Motion

Normally if we throw or shoot an object from a moving vehicle, we add up the speeds. So, if we throw a ball from a moving car the ball goes that much faster to the stationary observer at the side of the road. Say, we have the ability to throw a ball 80 miles per hour from a stationary position. We can get a ball to travel faster by throwing while running. If we can run 20 miles per hour then we can get the ball to reach 100 miles per hour. The ball still is moving 80 mph to the runner, but the ball goes 100 mph to a stationary observer. We add up 80 and 20 mph to get the total speed relative to a fixed observer, say a sitting spectator in a stadium. Relative to a non-moving position, running makes the projectile travel faster. Based on this principle, a javelin thrower can get the javelin to go faster by running while throwing.

If it’s hard to imagine someone able to throw with competency while running, then think of a passenger shooting a bullet straight ahead from a moving car. A bullet goes 1,700 mph relative to the gun itself. If the gun is held by a passenger going 100 mph in a car, the bullet goes 1,800 mph relative to a stationary observer. Similarly, if you throw a ball at 20 mph inside a train traveling at 50 mph, your ball goes 70 mph to a stationary observer at the train station. 

Bizarre Consequences of Light Symmetry

So, do the same laws apply to the speed of the beams from headlights of a moving car? To a stationary observer standing at the end of the road and facing the headlights. The speed of light is known to be 670 million miles per hour, so that’s the speed of light emanating from the headlights. Since the car is going 100 miles per hour, do we add 100 to 670 milllion miles per hour to get the speed of the light coming from headlights of a moving car?

This question posed a puzzle when physicists first accepted that the speed of light should always travel at the same speed, 670 million mph. Einstein’s reaction to this puzzle was to accept the absoluteness of the constancy of the speed of light. He realized that the absolute constancy of the speed of light must hold also for different observers, for example, an observer at rest or one or an observer moving at a constant speed relative to the observer at rest. This is astonishing because it means both the driver of a car will see the light moving this speed, and so will a pedestrian standing still down the road watching the car approach. We will always observe light to travel at the same fixed speed.

Imagine you are traveling in a car nearly the speed of light, say 80% the speed of light, and you turn on your headlights. How fast do you, the driver, see the light traveling? How fast does a stationary observer see the light traveling?

The answer is that both observers see the light traveling at the same speed, 670 million mph. And no matter how fast we attempt to chase after a beam of light, even if we ouselves are flying at 500 million mph or only 10 mph, the light will be moving away from us at 670 million mph. In the language of symmetry, the speed of light is invariant for all observers.

The way to make this work, Einstein reasoned, was to revise the very model of time and space. We should think of motion not through just space but rather motion through spacetime.

It is easy to notice that we are always moving through time. Just look at a clock or listen to a heartbeat. But would you believe that we also are constantly moving through spacetime? Even if we are standing still on the ground, we still move in spacetime, just with all our motion directed through time. So, we can’t stand still in spacetime. It’s like being a flying drone that can’t stop moving. It can’t even change its speed of 80 miles per hour. The drone can fly straight forward, north, parallel to the ground at 80 mph so long as it doesn’t veer upwards or downwards. It would register as 80 mph to someone watching the drone from the ground, like the police do when searching for drivers’ speed violations on the highway. If the drone veers upwards a bit, then it is moving north at less than 80 mph, despite that its total speed registers as 80 mph on its speedometer. If the drone goes straight upwards at 80 mph, then it makes no progress north, because all its constant motion is diverted upwards.

Likewise, our motion through spacetime is all diverted into time, when we stop moving in space. If we had the amazing ability to divert most of all our motion into space, then we move less through time relative to a stationary observer. If we move through space near the speed of light, then time for us is significantly slower than time is for a person who is just standing on the ground. The faster we move through space, the slower we move in time (a trade-off that is only noticeable for an object moving at extremely high speeds). The wonderful conclusion is that the person moving near the speed of light ages more slowly compared to those who keep relatively still. 

Chasing Moonbeam to Stay Young

An Einstein chasing a moonbeam is an Einstein who stays relatively young. Is that Einstein’s beauty secret? The beauty Einstein revealed is a deep symmetry in the laws of the Universe. The person standing on the ground is directing all their motion through the time dimension of spacetime. But both persons are moving through spacetime at the same speed, which is the speed of light. This is the absoluteness of the speed of light. For all observers, despite different motion relative to each other, the speed of light is the same, a great symmetry.

The speed of light is absolute, invariant, symmetrical, for all observers, even those moving relative to each other. Einstein added this symmetry to the list of other symmetries in the laws of physics. This was a great triumph for physics and for symmetry.

Many scientists describe Einstein’s special relativity, and Einstein’s reasoning about this topic, in terms of elegant symmetry: “Einstein was led to special relativity by thinking in terms of the symmetry principles that defined light, as it was understood in the late nineteenth century. In fact, this is, in a sense, the greater import of Einstein’s vision. It was Einstein who radically changed the way people thought about nature, moving away from the mechanical viewpoint of the nineteenth century toward the elegant contemplation of the underlying symmetry principles of the laws of physics in the twentieth century.” (Lederman and Hill, 2011)

The symmetry of light is a counter-intuitive concept at first. Most people never learn much about it, and may seem there is little need to, although it is a great joy to think about such symmetries. There are much simpler wonders of patterns and symmetries in our world that also provide ways to understand math, science, art and nature itself, as we explore in our Beautiful Discovery boxes. Even where nature seems very diverse, we often find underlying unities, repeating patterns. What seems to be a dizzying diversity in our world often has underlying symmetry.

As Brian Greene said, “Much in the same manner that they affect art and music, such symmetries are deeply satisfying; they highlight an order and a coherence in the workings of nature. The elegance of rich, complex, and diverse phenomena emerging from a simple set of universal laws is at least part of what physicists mean when they invoke the term beautiful.” (Greene 2003).


More reading

Brian Greene. The Elegant Universe. 2003.

Brian Greene. The Fabric of the Cosmos: Space, Time, and the Texture of Reality. Alfred A. Knopf. 2004.

Ian Stewart. Why Beauty is Truth: a History of Symmetry. Basic Books. 2007.

Frank Wilczek. A Beautiful Question: Finding Nature’s Deep Design. 2016. Penguin Books

Leon M. Lederman, Christopher T. Hill. Symmetry and the Beautiful Universe. 2011