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

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 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 Foucault pendulum is direct evidence that the Earth rotates, as the apparent change in the plane of its swing results from the ground beneath the pendulum rotating over several hours.

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 the effects of tornadoes of different scales

In this simulation, users can adjust the tornado's diameter and rotation speed to see its impact as it passes through a series of trees and a one-story home.

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

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.

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.

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.

Experimenting with space-time

Einstein's theory of general relativity—that mass literally bends space and time—can be difficult for the human brain to imagine. This interactive resource lets you create celestial objects, imagine their effect on space-time, and see how they interact with each other (best viewed on desktop).

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.

Build atomic nuclei to explore their instability and resulting radioactive decay

In this simulation, users can add protons and neutrons to create an atomic nucleus, view its half-life and decay options, and identify patterns in how specific combinations of particles create extremely stable nuclei, from which the nuclear shell model of stability emerged.

The extreme precision of atomic clocks highlights gravitational time dilation

Scientists at the National Institute of Standards and Technology have constructed clocks made out of stacked strontium atoms, which show differences in the passage of time across just a millimeter of height consistent with predictions from general relativity, where atoms closer to Earth's center tick off time more slowly.

How chaos helps explain why small market corrections can produce major economic instability

Oxford University complex systems professor Doyne Farmer discusses how predictions from many economic models are overly weighted toward simple relationships, such as supply and demand, and external shocks, such as the COVID-19 pandemic. However, real-world macroeconomic systems involve public and private actors interacting, producing feedback loops and cascading effects that resemble the oscillatory behavior seen in chaotic systems.

The 'butterfly effect' is the informal name for chaos theory

In mathematics, chaos theory involves complex systems that are highly sensitive to their initial conditions. Originally discovered when analyzing weather data, slight variations in the data input into such systems create drastically different results, despite following known physical laws.

Play an incremental game that teaches the history of particle physics

Developed during the CERN Summer Student Webfest 2014, this game tasks you with building the reputation of your particle accelerator as you discover particles and hire staff to collect data. A historical breakdown accompanies each discovery, allowing you to recreate the timeline of the field.

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.

Gravity Probe B verified that rotating objects drag spacetime

The experiment involved attaching four freely floating gyroscopes to a telescope aimed at a distant star for a year. The observed angular shift of the gyroscopes as a result of Earth's rotation pulling on spacetime was approximately 0.0000012 degrees from the amount predicted by general relativity.

Explore an optical table to create quantum physics experiments

This interactive teaches users to create quantum experiments through lessons resembling video game levels. Users can also use the virtual lab to visualize landmark quantum experiments like entanglement, teleportation, cryptography, and quantum computing, or create their own experiments with drag-and-drop elements.

Simulate the impact of an antimatter bomb, which releases energy from the annihilation of matter with antimatter, around the world.

Grasp the earth's orbit around the sun with this interactive

See why seasons exist with this interactive simulator. Toggle between different views of the earth in relation to the sun and watch as the earth's axis causes the sun to shine at different angles throughout the year, causing the weather to fluctuate and more. This simulator clarifies a process many take for granted.

Simulate how the distribution of dark matter affects stellar orbits within galaxies, which allows astronomers to infer the presence of the invisible substance from the motion of stars.

Try out an online tone generator, which produces sounds of specific frequencies to help tune instruments, test your hearing, and generate beats by overlapping tones.

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.

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.

LIGO boosts its signal sensitivity by manipulating quantum uncertainty

According to the uncertainty principle, the amplitude, or height of a wave's peak, and its phase—the point in time when the first peak is observed—are inversely linked. Based on the frequency of the gravitational waves of interest, scientists sacrifice precision in one optical property to reduce noise in another.

Simulate asteroid impact, mitigation, and defense scenarios

Tens of thousands of asteroids are in orbit in our solar system, and only a fraction come close enough to concern humanity. This simulator allows you to see the big picture scenarios of an asteroid impact, strategies to mitigate its destruction, and how Earth could defend itself.

Play a puzzle game where the speed of light is reduced to everyday speeds

In "Velocity Raptor," space and time warp as you make your way across a room of obstacles at "normal" speeds. Take advantage of these relativistic effects to solve puzzles involving clocks running at different rates and colors shifting as a result of your movement.

Simulate geological activity at plate boundaries using virtual planetary models.

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.

Simulate the precession of Mercury's perihelion—the gradual shift in the closest point of Mercury's path around the sun as its elliptical orbit rotates.

Probability waves govern particles like electrons in the quantum space

Just as waves are extended objects whose heights change across space, quantum waves dictate that the probability of an object existing in different locations across space varies. Like water waves, the probability waves can combine, changing the heights—and therefore probabilities—in different locations.

Within quantum mechanics, the Heisenberg uncertainty principle limits how precisely one can simultaneously know the values of pairs of complementary properties for an object, such as position and momentum.

Build atoms to learn about atomic number, mass number, and charge notations

In this simulation, users can add protons, neutrons, and electrons to create atoms, view their stability and atomic configurations, and identify how the amounts of these subatomic particles are detailed around the abbreviation for the resulting element using AZE notation.

Calibrate to the U.S. government's official clock, synchronized to the atomic standard

The site displays the current time across all American time zones—corrected for network delay—and shows whether your device's clock has drifted. Computer clocks rely on quartz crystals that drift slightly with temperature and age, requiring alignment with atomic clocks (which are so stable that seconds are defined by them).

Why atoms are used to keep track of time and define the meaning of a second

Small variations in the production of human-made devices, such as mechanical watches and pendulums, make them unreliable for the ultraprecise timekeeping required by modern technology, including financial transaction timestamping and GPS. Because atoms obey fundamental laws of nature and never wear out, their light emissions when energized provide a consistent timekeeping signal.

Understanding turbulence, which is characterized by growing chaos at multiple scales

As energy is lost in fluids undergoing turbulent, non-linear flow, particles will collectively form eddies, or whirlpools, of various sizes that mix together. The energy cascade of larger eddies becoming smaller ones before dissipating completely varies based on the initial conditions of the fluid, a characteristic feature of chaotic systems.

Simulate the buildup and discharge of static electricity through friction by dragging the arm and leg of 'John Travoltage' along surfaces.

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.

How neutron stars serve as laboratories for testing Einstein's theory of gravity

Data from a 16-year study of an orbiting pair of spinning neutron stars showed that measurements of gravitational wave energy and the bending of light around the system were consistent with predictions made by general relativity to an accuracy of at least 99.99% across seven tests. (Some readers may experience a paywall.)

Particle interactions follow rules related to specific quantum properties

Like a business whose purchased inventory matches sold inventory, physicists identified specific patterns across decades of particle accelerator data, which became new conservation laws for elementary particles. These laws help identify which particle decays are expected or forbidden.

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.

Some earthquakes can produce smaller tremors called boomerang earthquakes, which travel in the opposite direction and ripple back to the original earthquake's source.

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.

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.

The double-slit experiment is observational evidence for the wave-particle duality

Light and electrons can behave like particles when observed, but they also exhibit wave-like properties in motion, creating patterns through interference. Even larger objects, such as baseballs, have wavelengths, but they are negligibly small compared to their size.

When struck, objects vibrate at unique rates called resonant frequencies, which makes tapping wood sound different from tapping metal, but providing objects with sound at their resonant frequency can cause them to shatter.

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.

Explore a visual and auditory guide to wave properties and waveforms to learn how every sound imaginable can be created by combining multiple vibrations of air.

Watch 1 billion years of tectonic plate movement in 40 seconds

Researchers at the University of Sydney developed the model, incorporating changing plate boundaries rather than just moving continents, to better understand the changes to biological and climatic systems throughout Earth’s history.