Fumfer Physics 28: Why Pi and Fibonacci Appear in Nature

Pi recurs because circular and spherical geometry minimize surface area and energy: surface tension rounds droplets; for fixed area a circle has the shortest boundary; in 3D a sphere resists stress and encloses volume efficiently. Fibonacci patterns arise from local growth rules near the golden angle (~137.5°), packing leaves and seeds without overlap. Those rules produce spiral counts that match consecutive Fibonacci numbers. Iterative branching and logarithmic spirals extend the effect across pinecones, sunflowers, shells, and more. Beneath both patterns is information shaped by constraints: simple optimization rules yield stable forms nature reuses, from eyeballs to orbits to seed heads.

Scott Douglas Jacobsen: Here’s a question: why does nature produce things like pi and Fibonacci? We see them in living organisms and ordered structures. Why do those patterns keep showing up?

Rick Rosner: Pi shows up in nature because many physical and geometric relationships involve circles and periodicity. For example, liquid droplets tend to become spherical because, for a given volume, a sphere has the minimum surface area; surface tension then favors that shape. In two dimensions, a circle has the maximum area for a given perimeter (equivalently, the minimum perimeter for a given area), which is why circular boundaries often arise. Fibonacci patterns often appear in plants (like spirals in sunflowers or pinecones) because new growth placed near the “golden angle” of about 137.5° distributes seeds or leaves efficiently, leading to spiral counts that are typically consecutive Fibonacci numbers.

Of course there are forces. Gravity and air resistance turn what would be a sphere in zero gravity into a drop shape. Once that drop contacts a surface, it forms a spherical cap set by surface tension, gravity, and the surface’s wettability (contact angle), rather than pulling itself into a free sphere.

It happens because when everything is pulling on everything else, the surface area is where you have less material pulling, since it's the outer boundary. The particles on the surface don’t have neighbors on all sides, so everything “wants” to be pulled by as much surrounding material as possible, which minimizes the surface area. The minimum surface area configuration in two dimensions is a circle; in three dimensions—like a planet or a star—it’s a sphere.

A circle has a fixed ratio of its circumference to its diameter, which is where π (pi) arises. For spheres, π appears in surface area (4πr²) and volume ((4/3)πr³). There are many other ways that pi appears in nature.

Fibonacci patterns, on the other hand, appear through iterative branching. The golden ratio—basically the Fibonacci constant—can be derived from the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, where each number is the sum of the two preceding ones. Continuing, you get 34, 55, 89, and so on. As the sequence progresses, the ratio between consecutive numbers approaches the golden ratio, approximately 1.618. The formula for the golden ratio (φ) is (1 + √5)/2, which equals about 1.618.

Fibonacci patterns show up in nature through iterations—start with one branch on a tree, then that branch produces another, and each new branch produces more. It’s not exact doubling but rather a growth pattern governed by addition of previous states, like in the sequence itself.

This pattern appears in many natural forms. For example, sunflowers have seed arrangements that follow Fibonacci spirals, distributing seeds efficiently without overlap. The same kind of phyllotaxis—the arrangement of leaves, seeds, or petals—appears in pinecones, pineapples, and many other plants.

Jacobsen: So why do those patterns appear in nature?

Rosner: The simplest processes are also the most common—they’re the most likely to happen. You see circular structures in nature because they represent a minimum surface area. They’re also strong. For instance, eggs aren’t perfectly spherical—they’re mostly spherical but pointy enough to pass through a chicken’s cloaca. In cross-section, they’re circular, and for having just a thin shell of calcium, they take a lot of force to break.

A sphere is the most efficient shape for protection. Your skull, for example, isn’t perfectly spherical, but it’s close enough to minimize the amount of bone needed to protect your brain. You don’t want a bunch of pointy protrusions unless you’re an animal that uses them for fighting. Humans don’t fight that way—though some people might headbutt others—but we don’t have antlers. The skull’s rounded shape provides the least investment in bone while maximizing protection.

A sphere is also resistant to crushing. You can knock a corner off a cube or a tetrahedron, but a sphere has no corners to break. If you look up why spheres are so common in nature, you’ll find many more reasons.

Think of a pit bull chained in a yard. If it runs in circles at maximum distance, it’ll wear a circular path in the dirt. Wheels are circular because you want a smooth ride, not constant jolting. Your eyeballs are roughly spherical because that’s the most efficient shape to contain the vitreous humor, and they move easily in their bony sockets.

The same principle applies to your shoulders and pelvis—if you want a full range of motion, you need a spherical joint in a spherical socket. There are countless reasons spheres show up in nature—they’re simply the most efficient, stable, and versatile shape for both structure and function.

We’re less well-versed in what information actually is and what it requires. I’ve been saying for a few months now that information requires a context. Usually, the context is so obvious that we overlook it—we’ve failed to develop a theory of information in context, of how it takes an entire universe to have information.

There’s information that exists for the universe itself. There’s information tied to how things work at the quantum level, down to the smallest scales. But many of those quantum interactions don’t rise to the level of “information” for the universe—they’re information-like because they follow the basic principles of existence.

Then there’s information as it exists for us—conscious beings living on a planet in this universe. We haven’t yet developed, or even felt the need to develop, a comprehensive theory of how information functions within existence. We’re happy to use it moment to moment. We like sports scores. We like knowing when it’s safe to cross the street. We like having some idea of whether the person we just hit on thinks we’re creepy.

But when it comes to a theoretical understanding of how information becomes information, we’re not great. Yet it’s essential—it’s as crucial to existence as tires are to a car.

Scott Douglas Jacobsen is the publisher of In-Sight Publishing (ISBN: 978-1-0692343) and Editor-in-Chief of In-Sight: Interviews (ISSN: 2369-6885). He writes for The Good Men Project, International Policy Digest (ISSN: 2332–9416), The Humanist (Print: ISSN 0018-7399; Online: ISSN 2163-3576), Basic Income Earth Network (UK Registered Charity 1177066), A Further Inquiry, and other media. He is a member in good standing of numerous media organizations.