Data suggests that claims about the golden ratio underlying the geometry seen in nature, from mollusk shells to human proportions, are largely exaggerated.

The science of symmetry | The Royal Society

Nature is full of symmetries, from spiral galaxies to florets of broccoli. So why do complex creatures break the pattern?This short video discusses why symmetry came to dominate life on Earth, and how many complex life forms, such as humans, transitioned from radial to lateral symmetry.

Why patterns in art are so pleasing

Repeated visual cues like shapes or lines derive from the brain's innate compulsion to identify faces, places, bodies, and more. With repeated shapes, the brain seeks out resolution to the subtle variations between them. See examples in art and music.

In mathematics, a 'perfect number' is equal to the sum of the numbers it is divisible by, excluding itself, such as 28, which can be evenly divided by 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 equals 28.

The most bizarre numbers in the universe

Even for math-y types, some numbers are incomprehensible. This article dives into several kinds of numbers more complicated than even irrational numbers with endless decimals like "pi" or Euler's number: Noncomputable numbers. Read the bizarre...

The Ishango bone suggests mathematics was born about 20,000 years ago

Ancient civilizations developed mathematical systems for various purposes, including the Egyptians for the construction of the pyramids and the Ancient Greeks for logical reasoning in theorems. However, these systems were predated by the Ishango bone—found in the Democratic Republic of the Congo—whose notches suggest a base ten system, prime numbers, and simple mathematical operations.

How the buildings you occupy may affect your brain health

Beautiful spaces filled with light and natural motifs aren't just aesthetic, but likely key to your health. Researchers had study participants view photographs of two distinct buildings, one designed with more natural elements while the other featured more abstract shapes and lines. Using electroencephalograph readings of each participant, they discovered participants had less brain inflammation while viewing the "biophilic," or natural, architecture. Learn more about the interdisciplinary framework of so-called neuroarchitecture here.

The spiralling structure of the Guggenheim Museum in New York is itself a work of art

The Solomon R. Guggenheim Museum is a UNESCO World Heritage Site in Manhattan's Upper East Side. The contemporary art museum, known for its unique spiral structure, was Frank Lloyd Wright’s final work.

The Koch snowflake has an infinite perimeter enclosing a finite area

Counterintuitively, although it would be impossible to build a fence along the edges of the shape because it would require an infinite amount of materials, this perimeter can be contained in a larger shape with a finite perimeter and limited area. This is equivalent to enclosing an infinitely long spiral that never reaches the center within a circle.

Spiral galaxies get their arms, in part, from cosmic traffic jams, in which faster orbiting matter slows down and bunches up before proceeding at higher speed once more.

A fascinating look at why cucumber tendrils coil in opposite directions

The tendrils of cucumber plants coil when they hit a surface, creating a distinct telephone-cord look. Charles Darwin wrote about the mystery in the 1850s, though it wasn't until the 21st century that scientists figured out just how mysterious the coil is: its tendrils coil in opposite directions.

Apophenia, the cognitive bias that leads people to find meaning in randomness, helps explain why humans look for meaning in their dreams and even in their tea leaves.

The world's weirdest flowers

We mostly marvel at the beauty of flowers, but some kinds of flowers are incredibly weird. This video highlights 15 of the world's strangest flowers, from the massive Titan Arum to the rare Lady Slipper Orchid.

Each design detail on the US $1 bill has significance

The eye in the triangle floating above the pyramid, for instance, is “the Eye of Providence watching over and ensuring the endurance of the structure of the American government.” This article details every design choice’s significance on the dollar—and a few conspiracies around these choices as well.

A mathematical and visual analysis distinguishing chaos from randomness

Random systems lack any underlying pattern or causal relationships, while chaos is mathematically deterministic because the behavior follows established laws of nature. In 3D plots, chaotic systems produce outputs that approximate a mathematical function, while random points appear as statistical noise.

How the number 15 solved a longstanding math challenge

The “packing coloring” problem asks how many numbers are needed to fill an infinite grid so that identical numbers never get too close to one another. A new computer-assisted proof finds a surprisingly straightforward answer—only 15. While...

Taxicab geometry defines displacement as the sum of vertical and horizontal distances

In this mathematical system, pi is valued at four, and circles aren't round—they appear as diamonds with corners instead. The system simplifies pixel distances in digital imaging and helps engineers with robot navigation and chip design.

Organic architecture marries man-made buildings with the natural context around them

Frank Lloyd Wright’s nebulous concept of organic architecture is one that even he struggled to define. Its basic premise is to make a building look as though it grew from the land it stands on by emphasizing natural materials and incorporating the surrounding environment into the design.

Why are there as many males as females?

Pretty much every species breaks down with an equal sex-ratio, with roughly the same number of males as females. But on the surface, it appears that most species wouldn't need many males to keep the population growing, with their energy-efficient, abundant sex cell production. So why is it equal? Scientists argue an equal ratio is the most evolutionarily stable. Learn about this quirk of biology here.

Apple's first logo was a portrait of Isaac Newton sitting under a tree

Before the minimalist image of an apple with a bite taken out of it, Apple's logo depicted the fruit dangling above the physicist, likely a nod to the anecdote about his epiphany regarding gravity.

Frank Lloyd Wright's landscape design relied on abstractions

Wright was interested in finding abstract patterns in nature and attempted to match those patterns through the landscaping on his properties. He also often introduced foreign plants into his gardens, a choice that some experts say was a commentary on American identity and aesthetics.

Figuring out how to get a sofa around a corner has been a math problem since 1966

A sofa with the largest possible area that can make a 90-degree turn is known as Gerver's sofa, named after the mathematician who determined the shape in 1992. In 2025, a 119-page paper provided the proof confirming this shape.

Earth's total mass is equivalent to 13 quadrillion Egyptian pyramids of Khafre

Newton's law of universal gravitation was used to calculate the value after Henry Cavendish torsion balance experiments determined the missing value of the universal gravitational constant from Newton's equation.

Our brains trick us into seeing an enlarged Moon when on the horizon

Humans perceive the sky as a flattened bowl and judge the size of an object by accounting for its perceived distance away from us. Combined, studies have found that the Moon on the horizon is perceived to be larger because the terrain makes it seem like it is farther away, similar to the Ponzo illusion.

Calculators convert input values into a form of scientific notation—a fixed set of digits with a decimal multiplied by a base to some exponent—to store and process very large or very small quantities efficiently at the cost of rounding errors.

Explore interactive simulations of strange attractors, the graphs that help visualize chaos

Although simple, linear equations reach an explicit value for a given input, such as *f(x) \= x* + 1 equaling 4 when *x* equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

Mathematics proves turbulence's chaos creates super diffusion

Turbulent flow is responsible for the rapid scattering of dandelion seeds, volcanic ash, and untied balloons. Its swirling behavior accelerates particle scattering faster than other mixing methods, as particles acquire energy from the interactions of eddies.

By identifying patterns in the periodic movements of the sun, moon, and Earth, ancient civilizations realized that identical eclipses repeat about once every 6,585 days, known as the saros cycle.

Color originates from the brain processing light of varying wavelengths

Various geometric models, including Newton's color circle of hue and saturation and the horseshoe-shaped XYZ space for mixing colors via vector mathematics, have been proposed to describe colors accurately. However, these models do not encapsulate the variety of psychological effects that cause perceived colors to differ from mathematical representations.

Vincent Van Gogh's 'The Starry Night' accurately depicts turbulent flow

Analysis of the swirling patterns in the painting found an alignment with the mathematics used to describe turbulence, despite Van Gogh's unfamiliarity with the science. The depiction may indicate he possessed an uncanny ability to capture nature's details.

In the 1950s, Alan Turing developed a mathematical model to explain why animals have stripes, spots, and other patterns based on differences in how quickly two interacting chemicals spread throughout a developing organism.

The science of optical illusions

The human brain has developed cognitive shortcuts to help us navigate in a world where our visual senses are constantly bombarded of information. But these shortcuts can also be exploited, tricking the brain into telling us something is happening when it really isn't. This short video unpacks the neuroscience behind some of the most common (but perplexing) optical illusions.

Real-time tracking of individual hair cells has revealed that hairs don't actually get pushed out from the root—instead, hair cells are tugged upward by surrounding tissues that act like tiny motors.

View the Lorenz attractor, a graphical solution to chaotic equations that resembles a butterfly

The discovery of chaos theory came from Edward Lorenz's computer model of the atmosphere, which used three equations to quantify atmospheric circulation. However, the chaotic evolution of this system creates wandering figure-8-like shapes—some resembling butterfly wings—as it repeatedly attempts to settle on one of two values along unique trajectories.

NASA designed the Golden Record using hydrogen to ensure it's universally readable

The Golden Record holds what is essentially a human "time capsule," featuring music, nature sounds, and greetings from around the world. NASA used the radiation period of hydrogen's simplest transition as a baseline to convey timing and structure. The first coded image is included on the cover as a test so aliens can check if they decoded it correctly.

No map of the Earth is perfect because each introduces inaccuracies in the shape, size, or orientation of landmasses and oceans by trying to flatten the planet's curved surface.

As crystals form, their properties and the shapes they take on, including cubic, tetragonal, and hexagonal patterns, reflect the arrangement of their constituent atoms.

Besides the periodic table, there are over 1,000 alternative ways to organize the known elements, with many arrangements emphasizing specific trends among them.

Infinity can come in different sizes and be ordered by these sizes

Russian mathematician Georg Cantor demonstrated that certain sets of numbers (e.g., the natural numbers) form countable infinities, while others cannot be counted (e.g., the real numbers) and are therefore uncountable, representing larger versions of infinity. Through the axiom of choice, German mathematician Ernst Zermelo proved this concept of ordering infinities.

Pearls form when a saltwater oyster or a freshwater mussel coats a parasite or other irritant that has entered its shell with concentric layers of calcium carbonate and organic protein.

Flowers have been around for 123 million years

Fossilized pollen grains reveal that flowering plants have existed on Earth for 123 million years, and their emergence considerably enriched the planet's biological diversity. They may also have been more widespread than initially thought—prior assumptions were that the eudicots, the first flowering plants, appeared 121 million years ago.

Within mathematics, some infinities are bigger than others

While the concept of infinity refers to a countless set of items, these infinite sets can be compared to one another by matching an item from one set to an item in another, one by one. This analysis shows that certain sets must be larger than others because some infinite sets have leftover, unmatched items.

The life of Edward Lorenz, who originally used a seagull in his 'butterfly effect' metaphor

By accidentally finding that rounding one variable to its third decimal place and re-running a weather simulation produced a drastically different two-month forecast, the American mathematician established the field of chaos theory. His paper—"Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?"—introduced the well-known metaphor for the behavior of chaotic systems. (Some readers may experience a paywall.)

Learn how natural systems exhibit characteristics of chaos theory

In biology, some mathematical models of ecosystem populations and heart irregularities show strong sensitivity to initial conditions, which scientists can simulate to determine quantitative tipping points at which these systems become unstable. Within the social sciences, crowd behavior and the virality of social media posts have shown how human behavior fits within the framework of chaos theory.

Earth's circumference was estimated over 2,000 years ago using geometry and shadows

By measuring the angle of the shadow cast in Syene when no shadow was cast in Alexandria, Eratosthenes used the distance between the two cities and trigonometry to estimate the Earth's circumference to within a few percentage points of the actual value.

In the third century BCE, the Greek astronomer Aristarchus of Samos estimated the distance to the moon using geometry after measuring the length of Earth's shadow during a lunar eclipse.

People who feel connected to nature also report a higher well-being

Based on data collected from over 38,000 participants across 75 countries, spending time appreciating Earth's beauty appears to increase one's satisfaction with life. Whether it's a hike in the woods, a dip in the ocean, or time spent watching the sunset, the findings held true across a wide range of economic, cultural, and environmental contexts. The findings also indicate that connection to nature fosters experiences of being present and attentive, which are strongly linked to mental health.

In 2000, the Clay Mathematics Institute placed million-dollar bounties on seven unsolved problems in mathematics—the Millennium Prize Problems—in the hopes of spurring solutions.

The 'Old Guitarist' challenged art's standards of beauty

For much of Western art history, the definition of “beauty” in art was dictated by academies rather than personal taste. Artists like Picasso challenged these standards with works considered “ugly” for discarding anatomical accuracy or linear perspective in favor of a feeling.

The fast Fourier transform breaks down wave signals into discrete components

The algorithm was initially discovered by German mathematician Carl Friedrich Gauss in 1805, but his notes were not found until after his death. Today, FFTs form the basis for most compression software in media and are essential to technologies such as radar, sonar, and telecommunications.

The origin of chaos theory, colloquially known as the 'butterfly effect'

The theory explains that systems following well-understood physical laws can still produce unpredictable outcomes due to small variations in their initial conditions. Originally identified in weather data, elements of chaos have been observed in stock market fluctuations, cardiac arrhythmias, and the evolution of online social movements.