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.

My most recent visits were to Pakistan and Finland, two countries that on the surface have very few in common, if any, in terms of continent, climate, people, culture, population, economy, politics, stability… you name it. And my conviction was once again confirmed that it is absolutely a necessity to see a country with your own eyes and get firsthand knowledge to be able to see beyond what you have through other means.

The Fountain presentation on media and values was conducted in four universities with great interest and participation. Students and faculties actively joined the discussion with invigorating questions on the universal themes and values The Fountain cherishes. Despite what the Western mind would expect of Pakistan, interestingly, the female ratio in the academia is visibly very high and their voice is no less strong than their counterparts in the West.

Finland has its own texture and character by all means. Although it was mid-April, there were still pockets of muddy snow especially in the northern-facing shades of the landscape and buildings. Hardly anyone in the streets was around, even at the airport terminal, with the exception of those enjoying the spring sun during their lunch break.

However different any given country might be from another, culture of coexistence is inevitably a common theme in both regardless of continent or culture. The Minister of Culture and Sports, Paavo Arhinmäki, who opened the conference organized by Finland Dialog Association and The Fountain in Helsinki on April 24, said “Encountering ethnic and cultural diversity is no longer the experience of just a few. It has become an everyday reality for many more people worldwide. Interactive skills and conflict resolution must become part of everyone’s civil preparedness. The desire to participate and act only becomes greater through personal involvement.”

This quote summarizes the universality of our need for greater awareness of peaceful coexistence and proves that our differences are a richness of diversity. Hence, this is one of the many reasons why The Fountain is actively engaged in different continents and societies and what we try to reflect in the contents. In the same vein, this issue covers themes that are universally relevant, like What It Means to Be Human, food safety, environment, bad memory, and senior citizens. This issue also offers thought-provoking articles on natural sciences, appreciating the compatibility of the type of food animals consume with their digestion system, the perfect physical make-up of birds, and the very interesting balance in the world that is bound with its rotation. Challenges in the gifted education are also separately discussed.

For those of you in the northern hemisphere, we wish you a good summer, and a reasonable winter for those of you in the south.

The Second Fruit of the Fifth Branch

soul! Worship is not the introduction to additional rewards, but the result of previous bounties. Yes, we have received our wages, and are accordingly charged with the duties of service and worship. Because, O soul!, since the All-Glorious Creator, Who clothed you in existence which is pure good, has given you a stomach and appetite, through His Name of Provider, He has placed before you all foods on a table of bounties. Then, since He has given you a life decked out with senses, life too requires sustenance like a stomach; all your senses, like eyes and ears are like hands before which He has placed a table of bounties as broad as the earth. Then, because He has given you humanity, which requires many immaterial foods and bounties, He has laid out before that stomach of humanity, in so far as the hand of the mind can reach, an extensive table of bounties as broad as the worlds of both the inner and outer dimensions of things. Then, since He has given you Islam and belief, which require infinite bounties and are nourished through countless fruits of mercy and are supreme humanity, He has opened up before you a table of bounties, pleasure, and happiness which includes the sphere of contingency together with the sphere of His sacred Names and attributes. Then, through giving you love, which is a light of belief, He has bestowed on you an endless table of bounties, happiness, and pleasure. That is to say, with regard to your corporeality you are an insignificant, weak, impotent, lowly, restricted, limited particular, but through His favour, it is as if you have risen from being an insignificant particular to being a universal, luminous whole. For by giving you life, He has raised you from particularity to a sort of universality; and by giving you humanity, to true universality; and by bestowing Islam on you, to an exalted, luminous universality; and by giving you knowledge and love of Him, He has elevated you to an all-encompassing light.

O soul! You have received this wage, and you are charged with the pleasurable, bountiful, easy, and light duty of worship. But you are lazy in this too. If you perform it half-heartedly, it is as though the former wages are insufficient for you and you are overbearingly wanting greater things. Also, you are complaining: Why was my prayer not accepted? But your right is not complaint, it is supplication. Through His pure grace and munificence, Almighty God bestows Paradise and eternal happiness. So seek refuge in His mercy and munificence constantly. Trust in Him and heed this decree:

Say: In the bounty of God, and His mercy -in that let them rejoice; that is better than the [wealth] they hoard.¹

If you say: How can I respond to these countless, universal bounties with my limited and partial thanks?

The Answer: With a universal intention and boundless belief… For example, a man enters a king’s presence with a gift worth little, and he sees that other gifts worth millions have arrived from acceptable people, and have been lined up there. It occurs to him: My present is nothing. What shall I do? Then he says suddenly: My Lord! I offer you all these valuable gifts in my name. For you are worthy of them. If I had the power, I would have given you gifts equal to them. Thus, the king, who has need of nothing and accepts his subjects’ gifts as a sign of their loyalty and respect, accepts that wretched man’s universal intention and wish, and the worthiness of his elevated belief as though it was the greatest gift.

In exactly the same way, while performing the five daily prayers an impotent servant of Almighty God declares: Salutations be to God! That is, I offer You on my own account all the gifts of worship all creatures offer you through their lives. If I had been able, I would have offered You as many salutations as them, for You are worthy of them, and worthy of more besides. Such an intention and belief comprise extensive universal thanks. The seeds and grains of plants are their intentions.

And for example, the melon utters a thousand intentions in its heart in the form of the nuclei of its seeds: O my Creator! I want to proclaim the embroideries of Your Most Beautiful Names in many places on the earth. Since Almighty God knows how future things will come about, He accepts their intention as actual worship. The rule, A believer’s intention is better than his actions² alludes to this mystery. The wisdom in offering glorifications in also infinite numbers is understood from this mystery. For instance:

Glory and praise be unto You to the number of Your creatures, that may be as pleasing to You as the extent of Your Throne and the ink of Your words, and we glorify You with all the glorifications of Your prophets and saints and angels.³

Just as an officer presents all the duties of his soldiers to the king in his own name, so man, who acts as officer to other creatures, commands the animals and plants, has the capacity to be God’s vicegerent over the beings of the earth, and in his own world considers himself to represent everyone, declares:

You alone do we worship, and from You alone do we seek help.⁴

He offers the worship and seeking of help of all creation to the All-Glorious True Object of Worship in his own name. He also says:

O God! Grant blessings to Muhammad to the number of the particles in existence and all their compounds!

He offers benedictions for the Prophet (pbuh) in the name of everything. Because everything is connected with the Muhammadan Light. Thus, you may understand the wisdom in the countless numbers mentioned in the glorifications and benedictions for the Prophet (pbuh).

Footnotes

1. The Qur’an (10:58)
2. al-Manawi, al-Fayd al-Qadir, vi, 291, No: 9295.
3. Muslim, Dhikr, 79; Tirmidhi, Da’wat, 103; Nasa’i, Sahw, 94; Musnad, i, 258, 353.
4. The Qur’an, 1:4.