A mathematical and visual analysis distinguishing chaos from randomness

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.

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.

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.

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.

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.

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.

Computational and biological systems share fundamental principles, like randomness

Proposed by Alan Turing and John von Neumann, the concept originates from how DNA and computers run by executing encoded instructions in parallel. The advent of AI and the temperature parameter has seen neural networks mimic the randomness of protein encoding.

The universe only evolves toward increasing disorder, providing a direction to time

Within physics, the second law of thermodynamics states that entropy always increases, as seen by a concentrated drop of dye slowly diffusing throughout a cup of water, increasing disorder. Because such processes do not unfold in reverse, entropy may be what provides time with a single direction, unlike space.

Pareidolia: the cognitive bias of finding meaning in randomness

Many people equate "random" with "well-spaced," creating a cognitive bias that influences how people interpret events. This article explains how people tend to assume a generative force behind the patterns produced by randomness. This phenomenon, known as pareidolia, is seen in claims ranging from backward messages hidden in reversed audio recordings to tea leaf readings.

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

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 first algorithmic random number generator was developed in 1946

Although previous methods of randomness involved physical tools with limited outcomes, such as dice, or relied on chaotic variables for number generation, such as electronic noise, John Von Neumann's middle-square method turned the problem into a mathematical computation. It has since been replaced by methods that generate numbers from unique initial values, such as the time on a clock.

Learn why modern scientific patterns have been described as theories rather than laws

After confirming that the accuracy of some laws of physics is limited to certain domains, science became more hesitant to categorize identified mathematical relationships, such as quantum mechanics, as unchanging laws. Instead, these recent, complex frameworks are designated as theories to acknowledge that they are conditional and may be part of a larger, more complete future framework.

In quantum mechanics, matter is not in one location until it's measured

The probability that a particle is located at a given point is described by a wave function, in which peaks correspond to high probability and valleys to low probability. The wave nature of matter introduces uncertainty, allows it to exhibit complex behavior, such as interfering with itself, and also explains the observed structure of atoms.

Laminar flow moves in smooth, parallel layers, while turbulent flow is chaotic

In laminar flow, particles that make up a fluid move in consistent paths that do not cross each other. Contrastingly, differences across the speed of these particles cause them to mix, creating turbulence when viscosity cannot suppress the differences.

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.

Chaos gardening involves scattering seeds without any planning

The trend is intended to produce colorful, dense gardens by mixing various seeds. While encouraging experimentation and lower stakes, the success of these gardens depends on choosing plants with similar needs and incorporating native, nectar-rich flowers.

Time crystals are systems of atoms organized into periodic patterns of motion

Unlike traditional crystals—whose atomic structures cause them to look identical no matter how you look at them in space—time crystals exhibit this behavior through time, where the motion of atoms will bring them to the same positions relative to one another regularly.

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.

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.

There are more chess game variations than observable atoms in the universe

Mathematician Claude Shannon estimated there are 10 to the 120th power variations of chess games possible. Known as the "Shannon number," this is more than the estimated number of atoms in the cosmos.

A visual guide to how bitcoin transactions work and rely on cryptography

The "crypto-" in cryptocurrencies like bitcoin refers to the mathematical functions that generate pairs of public and private keys and encrypt data in blocks. When combined, these systems secure transactions on the network, verify digital asset and token transfers, and contribute to end-user anonymity.

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.

How algorithmic redistricting works

Computational methods are convincing a number of judges that some state district maps are drawn illegally. Here's a simple overview of how it works.

The complexity of classifying a tree

It may seem obvious to us when we see a tree, but very specific characteristics cause scientists to distinguish trees from other types of plants. This video provides a visual and verbal explanation of distinctions that separate trees from other plants like shrubs or herbs. See a scientific breakdown of these traits.

How generative AI differs from machine learning

Compared to machine learning - which is sort of like teaching a toddler how to recognize what a dog looks like by showing them images of one - generative AI is like teaching that same child how to actually draw a dog. Unlike machine learning, generative AI can actually generate. Read this article for a more in-depth comparison between these two types of artificial intelligence.

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.

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.

Radioactive elements can introduce software errors because the ionizing radiation they produce can detach electrons from atoms, and computer code relies on the distribution of electrons.

Gerrymandering works through two techniques: cracking opposing voters across districts to dilute their power, and packing them into a few to waste their votes in lopsided majorities.

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.

Alzheimer's can lead to brain shrinkage, with an average of around 2% of volume reduction per year

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.

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

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.

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.

Fireworks make a loud booming sound after exploding due to air expanding faster than the speed of sound.

Fear or disgust related to clusters of holes and cracks, called trypophobia, may stem from evolutionary processes to help us avoid pathogens.

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

Art theft is the third-largest source of criminal revenues worldwide, thanks in part to the subjective nature of valuing a work of art.

Watch this video that uses Lego bricks to explain how venture capital works.

Pure water does not conduct electricity because this flow of energy requires the movement of charged particles, and water is a neutral molecule.

Explore the known asteroids, comets, and spacecraft involved in asteroid studies within our solar system with 'Eyes on Asteroids,' a 3D visualization from NASA.

In 1945, Surrealist artist Salvador Dalí and legendary animator Walt Disney began collaborating on a short film that was not completed until 2003, years after both men had died.

'Grand Canyon of the Yellowstone,' a work by American painter Thomas Moran, inspired Congress to establish Yellowstone National Park.

The word 'jazz' evolved from 'jasm,' 19th-century American slang for 'energy'

See a 3D reconstruction of Tenochtitlan, the Aztec capital that was home to an estimated population of 200,000 to 400,000.

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.