Tag: symmetry

The Mathematical Beauty of Autumn Leaves

Here I reflect on the mathematical beauty of falling leaves from two trees, revealing order amidst perceived chaos in nature’s patterns.

MichaelStephenWills17 Comments

Standing before these two trees on a unseasonably warm autumn day, I am struck by the intricate patterns of their fallen leaves. Against the vibrant green of the grass, the golden leaves form halos around the trunks, as if nature herself were sketching mandalas. There’s something profoundly mathematical about these arrangements—a quiet order amidst what might initially seem like chaos.

The first tree, its barren branches reaching skyward, stands on a carpet of yellow that radiates outward in near-perfect symmetry. The leaves have fallen in such a way that their density decreases as the distance from the trunk increases. It reminds me of the inverse square law—a principle in physics that governs how light, gravity, and sound diminish with distance. Here, instead of energy dispersing, it’s the leaves thinning out, their graceful scatter dictated by the wind’s whims and gravity’s pull. There’s an undeniable harmony in this seemingly random process, a convergence of natural forces creating an elegant gradient.

Stewart Park, Ithaca, New York

The second tree presents a different story, yet one equally mesmerizing. Its leaves, still clinging in part to the branches, form a looser ring at the base. The distribution is uneven, hinting at prevailing winds or the sheltering influence of nearby buildings. But even in this asymmetry, I see fractals—the self-similar patterns that repeat at different scales throughout nature. Look closely, and you’ll notice clusters of leaves mimicking the broader structure of the canopy above. It’s as if the tree’s essence is echoed in the ground below, a reminder of how deeply interconnected every part of a system can be.

Robinia pseudoacacia, commonly known as the black locust

These patterns invite reflection on the mathematical principles governing our world. Fibonacci sequences, golden ratios, and fractals—abstract concepts are not confined to textbooks. They are etched into the fabric of existence, visible in the spiral of a sunflower’s seeds, the curve of a nautilus shell, and the fall of leaves beneath a tree. Even the chaos of autumn is underpinned by order, a dance choreographed by countless variables: the angle of the branches, the strength of the wind, the moisture in the air.

I find myself wondering about the unseen forces at play. How many leaves fell straight down, obeying only gravity? How many were carried aloft by a breeze before settling farther afield? Could we model these patterns with algorithms, tracing the arc of each leaf’s descent? Would the data reveal a perfect equation, or would it remind us that some mysteries resist full comprehension?

As I stand here, I feel a deep gratitude for these natural equations. They ground me in the present moment while also connecting me to the infinite. The pattern of leaf fall is a reminder of life’s balance: chaos and order, randomness and structure, fleeting moments and timeless principles. The trees, now shedding their golden crowns, invite me to pause, observe, and marvel at the beautiful mathematics of autumn.

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Growth and Beauty

a exploration of logarithmic spirals and symmetry

MichaelStephenWills25 Comments

Growth

An early thought of mine, as a child, was to wonder, “How large does a person grow?” If growth was perpetual, there was no end to how large I will become; yet, tested against observed reality, “Why was it the case this was unlikely?” Years later, when recalling this, I understood my intuition touched upon the logarithmic spiral and mollusk shell.

Three Scallops and One Tallin

Sea Oat stalk, photographed above, after it dries slowly in the sun and wind, curls into a logarithmic spiral. One two dimensional spiral may be compared to another by measuring the rate and direction of opening, the increase in distance between the part closer to the source and the outer swirl. The growth of all shells follow a logarithmic spiral in three dimensions where the progression from a staring plane, as well as the direction, up or down from the plane, is an element.

Sea shells give evidence to my question of “how large can one grow.” The size of each of the millions encountered on a beach is an example of a life ended. Each of record of the length and character of the organism. For example, a close inspection of the bottom shell of the above photograph, a tellin of the family Tellinidae, reveals the spiral is growing toward the surface of the sand. Imagine wrapping your hand around the outer edge of the tellin with your thumb pointed down.

Each of the four shells of the above photograph had a mate, were one of a pair. Types of shells share characteristic pair symmetries. For example, a pair of tellins display a type of asymmetry called chirality, also called “handed-ness” after the same property of your right and left hands. One shell half (from the same individual) is the mirror image of the other, each unbalanced as the growth spirals toward opposite directions.

Asymmetry, halves from different individuals

When I started beachcombing, examining collected shells I did not have a pair from the same individual and incorrectly concluded direction of growth was unique to an individual. The ribbing of the above two shells illustrate three concepts: the logarithmic spiral growth pattern, chirality, as well as how I came to that wrong conclusion: that two individuals can grow in different directions. It was a logical hop to understand how, to make two shells hinged at the source of the growth spiral, each individual requires two halves, each a mirror image of the other. That every member of the species demonstrated the same asymmetry, each half grows in the opposite direction.

Asymmetry, attached matching halves

The above photograph shows attached matching halves. The attachment point was a surprise: the apparent source point is not attached to the ligament joining the halves? I have yet to understand this. Do you?

Beauty

The association of beauty with scallop shells bridges thousands of years. For example, a fresco of the Roman goddess Venus, born from the ocean riding a shell, was unearthed from Pompeii. The living organism is not part of the story, just the shell. Why the scallop? My answer is, “Each half is completely, in itself, symmetrical.”

The top three shells of the first photograph are scallops. The first and last, broken by the waves, are missing parts. The middle scallop, small and off-white, is complete. Place an imaginary line down the center and each side is identical. Applying the real world (i.e., physics) to myth, a scallop shell allows the goddess to move forward in a straight line. Sailing an asymmetrical shell, she moves in an eternal circle.

An object with symmetry is visually complete unto itself, self-contained; functionality aside, one scallop does not required a partner. The paired shells are interesting in they do not match, one is deeper, it encloses more volume. The deeper side rests under the surface, allowing the top halve to present a lower profile the better to hide from predators.

Calico Scallop Shell

The scallop echoes the beauty of Venus. Symmetry enhances human features (earch “Venus (mythology)” for images of her face through the ages), though it does not define beauty. An overly symmetrical face seems strange. I will close with an extreme example, the other day I came upon this beach crab wandering around in the daylight. Symmetry does NOT enhance the alien eyestalks, menacing claws, a pallid, tough exterior. Safe travels, little one.