If you have ever noticed a straw in a cup of water appear “broken,” you have probably witnessed a universal principle at work. We encounter the same principle when driving home from work on a Friday evening: we seek the quickest route, not necessarily the shortest one. From tiny ants on the ground to distant galaxies, systems tend to follow the most efficient, namely, the fastest, path available.

Imagine you are running on a beach toward a friend in the water. Running on sand is faster than swimming, so the quickest route is not a straight diagonal line. Instead, you instinctively run along the beach for a distance before cutting into the water at an angle. Without knowing any physics, you naturally follow this principle: Fermat’s principle of least time.

The shortest path

In the 1600s, the French mathematician Pierre de Fermat proposed something radical: Light doesn’t always take the shortest path, but it does take the fastest one. At first, this seems odd. Why wouldn’t the straight-line path – the shortest possible route – always be the fastest? But the medium matters. Light moves at different speeds through different materials. It slows down in water, speeds up in air, and changes direction when moving between the two. Fermat’s insight was that light chooses the route that allows it to arrive in the least amount of time – even if this means curving or bending (refraction) along the way. Light bending through a glass prism or the formation of a rainbow are all results of this phenomenon.

Snell’s law

About thirty years before Fermat, the Dutch mathematician Willebrord Snell had already discovered a precise rule describing how light bends when it passes from one medium into another. This rule, now called Snell’s Law, tells us exactly at what angle the light will bend. But Snell’s Law is basically the mathematical expression of Fermat’s idea.

Amazingly, Fermat’s principle does not only apply to light. It is astonishing to see that the same logic that guides a ray of light also guides living creatures and physical systems. Fermat’s principle which initially described the behavior of light has now evolved to a broader idea in physics: systems tend to follow efficient paths, the ones that minimize time or energy. This theme appears throughout the natural world, from animal behavior to planetary motion.

The quickest path

It may be surprising to learn that ants – tiny, busy, seemingly chaotic – can mimic the path of a bending light ray. In certain experiments, scientists placed food on one side of a barrier and ant colonies on the other. The ants could walk across a smooth surface or move into a rougher, slower terrain. Over time, the ants collectively chose a route resembling the path light takes when moving between two media with different “speeds.” Their path looked like a refracted (bent) light ray obeying Fermat’s principle.

Much like light, the ant colony “searches” through many paths at first. Eventually, they figure the quickest route. The result is a collective behavior that finds almost the same solution a physicist would compute to predict how light bends. Ants obviously don’t do calculate equations, but they do choose the quickest path.

The most efficient path

Rivers carve their meandering routes over centuries, forming bends and curves. Flowing water tends to follow the easiest route downhill, not the straightest. It avoids obstacles and seeks out weaker soil, which means the river’s path is determined by minimizing energy loss. Though not exactly Fermat’s principle, the idea is remarkably similar: a physical system is finding the most “efficient” path.

The least amount of time

You might have experienced the illusion of water on the road on a hot day, or a pool of water in the middle of a desert (mirage). These are other examples of Fermat’s principle at work. On a hot day, sunlight bends as it travels through air layers of varying temperatures. Warm air is less dense, so light moves faster through it while cool air slows light down. These variations in speed cause light rays to bend as they travel through the layers, allowing the path from sky to eye to take the least amount of time and producing shimmering distortions, such as the illusion of water on a road. What looks mysterious is really optimization playing out in real time.

Minimizing travel time

If ants give us a small-scale analogy, galaxies give us a cosmic one. According to Einstein’s theory of general relativity, massive objects like galaxies bend the fabric of spacetime. Light traveling through this distorted space follows the “straightest possible” path—which looks curved to us. This effect, called gravitational lensing, can create rings, arcs, or stretched images of distant galaxies. Even here, at astronomical scales, the path of light still reflects an extremal principle: in curved spacetime, it follows a route that locally minimizes travel time (this specific line on a curved surface is called a “geodesic”).

If you dip wire frames into soapy water, the film that forms always stretches into a shape of minimal surface area, because that configuration requires the least energy. This is a different efficiency principle, but the connection remains: the resulting configurations minimize something – time or energy. Soap bubbles don’t consciously “choose” their shapes any more than light chooses its path. They simply obey this universal law of “efficiency.”

Dirt paths in the park

Even people unknowingly follow Fermat-style logic. Imagine walking across a city park: although paved walkways exist, people often cut diagonally across the grass if it is faster. Over time, these shortcuts become visible dirt paths. The same principle appears in GPS algorithms, which calculate routes that minimize travel time based on traffic conditions.

Economy over extravagance

Efficiency is baked into our decision-making, just as it is into light’s behavior. In this sense, Fermat’s idea feels almost philosophical: In the universe, "economy" is preferred over extravagance. What begins as a simple observation about light bending in water becomes a window into a deeper unity in nature. Fermat’s principle isn’t just a rule of optics – it’s a guiding thread woven throughout the natural world. Light bending at a glass surface, ants finding efficient routes, rivers snaking across landscapes, galaxies warping the paths of light, soap films forming perfect curves.

These are not random coincidences. They highlight a universal pattern. Fermat’s principle teaches us that even something as simple as a beam of light carries within it a profound law: the pursuit of the fastest path. What’s remarkable is that this principle doesn’t stop at optics but is in action at every scale. When we observe these systems side by side, we are reminded that the universe is not a collection of isolated phenomena but an interconnected framework of elegant principles. They have been in action since the beginning of the universe whether or not we have discovered them.

The next time you notice light bending in a glass of water or watch ants tracing a trail, you might see something deeper. You might be witnessing an invisible yet universal principle connecting the smallest creatures to the largest structures in the cosmos. Our universe is a book full of patterns. Fermat’s principle is one of its clearest, most beautiful expressions.

References

Oettler, Jan, Michael Kreuzer, and Jürgen Heinze. “Ants Follow Curved Routes to Minimize

Travel Time — A Potential Analogy to Fermat’s Principle.” PLOS ONE, vol. 8, no. 4, 2013,

e59772. https://doi.org/10.1371/journal.pone.0059772.