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

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.

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.

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

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.

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

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.

An interview with James Yorke, the man who coined the mathematical term 'chaos'

Along with co-author Tien-Yien Li, the former University of Maryland mathematics department chair published the paper, "Period Three Implies Chaos" in 1975. "Chaos" then became the terminological placeholder for systems with sensitive dependence on initial conditions, which had been well known to mathematicians for over a decade.

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.

View 3D renderings of atomic orbitals created with quantum mechanics

Scientists use the Schrödinger equation to determine how a system governed by a wavefunction—an equation that predicts the probability of certain outcomes—evolves with time. Computers can animate atomic structure and behavior by depicting the wavefunction as a fluid that guides a particle, like water moving a speck of dust.

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

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...

AI systems identify patterns involving physics and time through 'world models'

These realistic virtual environments, such as driving simulations seen in video games, allow these systems to "learn" properties about reality beyond what can be captured in samples of text, images, and videos. Training AI in these simulated environments can help develop "physical AI" for automated robotic systems. (Some readers may experience a paywall.)

Explore an interactive map of the unified theories of everything

This interactive concisely describes many highly complex theories ranging from quantum gravity and black holes to dark matter and energy. Supplementary resources, key questions, and possible solutions accompany each description. As of 2026, no developed theory provides an accepted explanation of all observed phenomena in the universe at all scales.

Visualize how data becomes organized through a variety of sorting algorithms

This interactive shows how different algorithms shift elements in a randomized set to order them from smallest to largest. By comparing the different procedures and their corresponding functions, users can explore how runtimes vary across algorithms that complete the same task.

Explore a simulation of ideal gas behavior, which can be explained by kinetic molecular theory

The theory describes particles as moving in random directions, with a distribution of velocities determined by the gas's temperature. The particles collide elastically like billiard balls, are negligibly small, and do not exert forces on one another. The gas laws can be derived from these principles using statistics or observed via this interactive.

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.

Simulate the effects of changing supply and demand on the electric grid

In this application, users can modify the demand of fictional cities and the supply of various generators, including a nuclear power plant and a wind farm, which can be disconnected from the grid. Notice that current may flow into or out of external systems to compensate for available generation. See what it takes to cause a blackout in different cities or shut down the entire grid.

Accurate climate models incorporate calculus to account for complex feedback loops

The simplest models assume uniformity across the Earth, with incoming energy from the sun being offset by the planet's albedo, or reflectiveness, which prevents the radiation from being fully absorbed. Incorporating elements such as atmospheric layers, latitude bands, geographic variability, and emissivity yields coupled differential equations quantifying Earth's interconnected systems.

A breakdown of algorithms, how they are constructed, and their applications

Algorithms are instructions for accomplishing a goal and typically take the form of a step-by-step procedure. Given the conditions, these procedures can take various paths, increasing an algorithm's complexity. While traditional computer algorithms are programmed to account for all possible scenarios within a given process, machine learning systems build their own rules from data-driven patterns.

Mathematicians often solve problems using wordless mental processes

It's a common technique among expert mathematicians, where thinkers rely on intuition, blurry perceptions and subconscious work—no writing, no formulas. The state is distinct from daydreaming, and often relies on sudden illuminations stemming from extended periods of subconscious reflection.

Studying 'sonic' black holes

If a fish screams as it falls down a cascading waterfall accelerating to ridiculous speeds, will his fish friends above hear him scream? This analogy, odd as it is, has become an essential tool in the astrophysicists kit for learning how enigmatic black holes operate. Lab experiments mirroring the so-called sonic black hole have achieved insights for their cosmic counterparts. Read how it works in this brief piece.

Watch a visualization of a fission chain reaction

When a neutron collides with an atom, it is absorbed into the nucleus. If the resulting nucleus is unstable, it can undergo a series of decays that turn it into a less massive nucleus while releasing energy and other particles.

Explore an interactive 3D model of the brain

The brain contains about 86 billion neurons—nervous system cells—and controls vital bodily functions via electrochemical signals. Neurons form circuits through axons, and different brain regions use these circuits to perform specific tasks.

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.

Five charts that changed the world

Data visualization helps us understand the world, providing a powerful way to communicate complicated information in a quick and efficient way. There are many different ways to create graphs, and these five examples—the pie chart, geographic dot...

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.

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

When you complete a CAPTCHA on a website to prove you are human, you may be unknowingly training algorithms for self-driving cars, whose systems require training data to help identify crosswalks, bicycles, and other outdoor objects.

According to the physics of black holes, our 4D reality may be a hologram—a projection of information coded on a 2D surface on the edge of the universe.

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.

Explore an interactive map built from more than 100,000 survey responses on where people find joy in their lives

Browse a detailed visual analysis of over 200,000 similes—comparisons that often use 'like' or 'as'—from popular fiction to see how frequently writers the same idiomatic expressions.

Visualize the path of every solar eclipse from 1900 to 2099 and see what each looks like when observed from different locations on Earth.

The term 'algorithm' originated from the Latinized name of al-Khwārizmī, a 9th-century mathematician whose works introduced Arabic numerals and the decimal-based numeral system to Europe and whose treatise 'Al-Jabr' gave algebra its name.

Visualize how CRISPR-Cas9—a type of immune system used by single-celled organisms to attack invasive viruses—targets and makes precise changes to a gene's DNA.

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

See these 16 pieces of art evoking phobias of snakes, flying, spiders, and more. 

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

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.

How fusion—which releases energy when two lighter atoms combine to form a heavier atom—differs from fission

Play the role of a student in a game where you try to navigate college and remain debt-free, selecting different income levels, high school experience, race, ethnicity, and more.

Hedge funds, the private investment firms that trade a wide swath of financial assets, make money by imposing steep fees on investors and from the fund's investment performance.

Children are now three times more likely to say they want to be a YouTuber when they grow up than an astronaut, with some even attending 'YouTube camps' that teach kids how to create content for the platform.

Walt Disney's early film 'Alice's Wonderland' was a groundbreaking innovation in the silent film era

Discover the catastrophic consequences of a failure of the US electrical grid—something that is more likely than many Americans think.

Hear 50 years of iconic hip-hop samples, the genre's foundational looped tracks

In 2019, a laptop loaded with six of the worst malware in history and described as 'an exhibit of historical weaponry' sold for $1.3M.