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

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.)

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.

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.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

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.

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.

Visualize chaos with the orbit of three or more objects bound by gravity

In this simulation, users can simulate randomized and preset orbital configurations for up to 10 celestial bodies to observe the system's evolution in an interactive 3D space. The same configurations can produce different orbital behavior—a feature of chaos—by making small changes to an object's mass under advanced settings.

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.

An analysis of the logistic map, which shows how chaos can emerge from simple equations

When modeling systems such as the predator-prey cycle of an organism's population in an ecosystem, the logistic map function shows the population normally settles into an equilibrium value. However, at high reproduction and starvation rates, multiple such values arise, making it difficult to accurately model long-term population growth in some environments.

The story of two female programmers running the simulations behind chaos theory's discovery

Margaret Hamilton and Ellen Fetter worked on MIT's 800-pound Royal McBee LGP-30 computer, which was fed binary code on paper tape to perform computations. Among the computations performed were those involving equations describing the motion of gas in a circulating fluid for Edward Lorenz, widely considered the father of chaos theory.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

View the transition from laminar to turbulent flow

Using a dye released in the water, streamlines, which are parallel to one another and consistent at low flow velocities, become visible. As the water moves more quickly, viscosity cannot suppress the velocity change, and streamlines become chaotic, creating turbulence.

Researchers identify a hidden instability pattern in simple ocean wave models

Stokes waves—a solution to the mathematics of water waves—were once assumed to be stable. Italian mathematicians have proven disturbances cause these waves to collapse and such collapses follow an infinite repeating pattern. To complete the proof, the team offered a $100 donation to solicit online help.

The Navier-Stokes equations are fundamentally accurate but unsolved in 3D

The equations model fluids and have been shown to have solutions for all systems in two dimensions. Averaging and approximations let the equations be used in climate models, aerodynamics, and medicine, though their use in some systems produces nonsensical answers.

Tracking butterfly migration via the pollen they collect

Trillions of insects migrate around the world each year, but many are too small to track with physical trackers, leaving their migratory routes a mystery. In recent years, scientists have utilized pollen grains collected from butterflies to analyze which plants that pollen originated from—providing substantial data to infer the insect's specific route. Research has shed light on many insects' migrations since, including the Painted Lady butterfly. Read more and see a map here.

View a breakdown of the mathematics behind the Standard Model

The Lagrangian version of the theory—related to the system's energy—presents a vast mathematical equation, with different components representing various force interactions and types of particles, including ghost particles to cancel out mathematical redundancies.

How a competition honoring a king's birthday uncovered history's first chaotic system

A committee established a contest for King Oscar II's 60th birthday, featuring four unsolved problems in mathematics, including one on proving the stability of a system of more than two objects held together by gravity. Focusing on three celestial objects, Henri Poincaré won the prize for his proof, which he later learned was incorrect and, upon correcting it, instead proved these systems to be unstable.

View the quantum mechanical depiction of electrons as waves around the atomic nucleus

Unlike Bohr's fixed paths, the quantum model describes electrons as exhibiting behavior resembling standing waves—waves that have fixed points at the end while vibrating, like a plucked guitar string. Rather than determining their exact positions, only the probabilities of electrons being in a specific region can be determined.

Their 1901 Wind Tunnel system revolutionized the study of aeronautics

Only the second to be built in the US, the wind tunnel allowed the brothers to study how air moves around more than 200 different wing shapes. This data helped them refine a key math equation, long held as the standard, that had been based only on one wing shape.

Albert Einstein's equations predicted black holes decades before their discovery

For nearly 50 years, scientists dismissed black holes as implausible oddities, despite solutions to the equations indicating their existence. But the 1970s discovery of intense X-rays from Cygnus X-1 and its massive hidden companion flipped the consensus.

Within topology, time can be modeled as restarting after reaching its 'edge'

A manifold is any space that appears flat when zoomed in close enough, even if it is globally curved. Some of these structures can be curved to connect back on themselves, much like bending a line into a circle. This property can be mathematically established in a four-dimensional manifold by allowing time to loop back on itself.

Lichtenberg figures capture lightning-like patterns within insulators

When high-energy electrons from particle accelerators are fired into materials like acrylic, they form a charged plane before discharging into scarred patterns within the material. By evenly charging a cylindrical bottle through rotation during the irradiation process, a 3D figure of lightning in a bottle can be created.

Visualize real-time and historical atmospheric conditions from around the world

This tool provides live maps of wind, rain, thunderstorms, and other atmospheric phenomena overlayed on an interactive map of the Earth. Users can enter their location to explore local atmospheric dynamics and review historical weather conditions dating back to 1979.

English mathematician Ada Lovelace wrote the first computer algorithm a century before computers existed and hypothesized that computing devices would one day be able to compose music—foreseeing the generative AI systems of today.

Earth’s tilt and orbit cause the sun to create a figure-eight pattern

The shape—called an analemma—can be obtained by photographing the sun from the same location and time daily for a year. The Earth's eccentricity created the analemma's width, and its tilt shifts the sun's height in the sky over the seasons.

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.

The ideas of Einstein and Freud combined in 'The Persistence of Memory'

The Persistence of Memory was painted on a canvas roughly the size of a sheet of paper, depicting a desolate landscape with melting clocks, ants, and a distorted figure. The piece incorporates Freud and Einstein's ideas, resulting in a haunting composition symbolizing time's fluidity and Dalí's personal anxieties.

Watch how algorithms can be designed to train themselves and self-improve without human input, to the point where the optimized code is too complex for programmers to understand.

Explore our favorite visualization of the immense scale of space, from an astronaut to the size of the observable universe

How Amazon operates a 'flywheel' business model, a self-reinforcing cycle that drives continuous growth

Learn how your body is, scientifically, like a firework with its own slow, controlled burn.

While the sun's gravity keeps the planets in orbit, their gravity also pulls on the sun, causing it and everything else in the solar system to orbit around a mutual balance point called the barycenter—the true center of the solar system.

Planeterrellas create artificial auroras that mimic the atmospheric displays of light by setting up an electric field between two metal balls in a jar filled with a small amount of air.

Clever Hans was a 1900s horse that was believed to be able to solve math equations, tell time, and read and comprehend German, until a psychology student discovered how he got his answers.

The Mona Lisa's celebrity status and repeated restorations have obscured what made the painting so revolutionary for its time

All rainbows are 'raincircles' caused by water droplets separating white sunlight into a spectrum of colors, but Earth's surface blocks the bottom half of the circle from our line of sight.

In recent years a child anxiety expert has created and tested a novel phobia and anxiety treatment approach for children, focused on changing how parents respond to the fears.

El Niño, Spanish for 'the boy,' was named after Jesus Christ because the phenomenon resembled the weather changes observed around Christmastime.

Mid-century modern graphic designers, known for their bright colors and bold lines in advertising, also found success with children's books.

Exploring the Larry Bird punch that almost changed March Madness

This site reaches millions of visitors who are tired of AI chatbots and seeking human connection. Pretend to be an AI chatbot and ask questions, provide answers, and draw for people in real time.

In 2009, the nation was gripped by the story of a missing young boy who was believed to be floating over Colorado in a homemade balloon, until it was revealed that he was at home the whole time.

American short story writer Flannery O'Connor raised 40 peacocks, which became potent symbols for her fiction's elegant violence

Ebola is a filovirus, meaning it resembles a thread under a microscope, though it sometimes twists and can resemble a bowl of spaghetti.

Some Pokémon have been named after real and fictional organisms, while some real-world species have been named after Pokémon, such as Chilicola charizard and Binburrum zapdos.

SpongeBob exemplifies Aristotle's philosophy for living well because he understands that happiness doesn't come from external factors like money, recognition, or power.

The American cockroach is actually from Africa, an apparent hitchhiker aboard ships carrying enslaved people, with evidence suggesting the bug may have arrived in the New World as early as the 1600s.

A 2019 Facebook event called on attendees to "Storm Area 51, They Can't Stop All of Us," resulting in countless memes, the creation of a music festival, and eventually the arrests of those who did try to approach the area.

See a ranking of the best (and worst) Federal Reserve Chairs since 1914 out of the 16 men and women who have held the office since.