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

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.

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

Benign species can become invasive if introduced into predator-free habitats

Most ecosystems contain organisms that, over many generations, have evolved into a stable balance with one another and their surroundings. In a new environment, a species may no longer be restricted by the limiting factors it is used to, and the native organisms in the area may not adapt quickly enough to the invader to establish a new balance.

View illustrations of bioluminescent functions in marine organisms

Bioluminescence can startle predators, camouflage, lure prey, confuse rivals, or facilitate mate recognition. One tactic—the burglar alarm—involves an organism releasing light to attract a larger predator to its attacker as a means of defense.

Visualizing the world's food supply chains

The world's food supply chain is complex and lopsided, with much of the world's nutrient-rich food ending up in Europe and North America. This visualization illustrates the paths our food takes around the world, from starchy vegetables in...

Mapping how human population has grown throughout history

It took 200,000 years for our human population to reach 1 billion—and only 200 years to reach 7 billion. When will our global population peak? And how can we minimize our impact on Earth’s resources, even as we approach 11 billion?

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.

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.

See how cholera spreads in a community

This animated video offers a comprehensive look at how cholera spreads. It makes clear why it's so difficult to stamp out this disease in communities around the world. It also details what steps can help control cholera.

Understanding R0, a measure of how contagious a pathogen is

The R0 value is a basic measure of how many people an infected person will transmit a pathogen to, on average. If R0 is less than 1, transmission dies out; if greater than 1, an outbreak is likely. Measles has one of the highest known R0 values at almost 18, meaning that after four transmission steps, more than 100,000 people could be infected.

Mortality tables help determine the probability of death for specific populations

A mortality table is a diagram that helps predict a specific population’s likelihood of death within a particular time period. Life insurance companies often use them to help figure out pricing for their insurance products. They’re sometimes known as actuarial tables or life tables.

Network effects are the economic term for a good or service's popularity

When goods or services become especially popular, firms that provide that good or service become dominant in the market—which can ultimately lead to a monopoly.

How math whizzes exploit sportsbooks

Chris Dirkus is a professional sports bettor who approaches the hobby as a math problem, not from a fan's perspective. He focuses on mispriced odds in popular games where many fans are betting for high liquidity. He's one of a rare segment of sports bettors exploiting slow models and human mistakes.

Simulate the distribution of dark matter within a galaxy

By modifying the dark matter density at different distances from the center of a simulated galaxy, you can plot the resulting orbital speed of stars. Preset data enables you to replicate astrophysicists' work in determining the dark matter concentration that best fits observed structures.

An explanation of dynamic pricing algorithms and the factors that impact them

Rather than static prices set by manufacturers or retailers based on the costs of goods, online businesses or box stores with electronic price tags can change prices in real time based on supply, demand, time, location, and competitor prices. These algorithms can be programmed to maintain or maximize profit margins based on shoppers' willingness to spend.

Explore the efficiency of photosynthesis under various conditions

In this simulation, users can track how the cellular concentrations of ATP, NADPH, glucose, and oxygen vary over time in response to changes in temperature, acidity, carbon dioxide concentration, and the properties of incoming light. Through multiple trials, users can identify the optimal conditions for photosynthetic reactions.

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.

Logic bomb viruses can awaken from a dormant phase without human input

Theoretically, computer viruses can remain idle indefinitely while awaiting specific user actions, as instructed by their code. However, logic bombs—embedded code that triggers when specific conditions are met—can automatically execute viruses, such as when a certain date or time is reached.

See this graphic of the complex malaria transmission lifecycle

Female Anopheles mosquitoes contract malaria by feeding on the blood of a human already infected with the disease-causing parasite. The parasite then develops in the mosquito's gut for one to two weeks before it migrates to the mosquito's salivary glands. Next, when the mosquito bites another person the parasite is delivered into the bloodstream of another human. This graphic further details what happens inside the human body during infection.

Explore the complex relationship between calories and weight

What if you found out that weight loss is actually much more complex than "calories in, calories out" and that the 2,000 calorie-a-day recommendation may not actually be accurate?

A map of the moon, using lidar

When planning to land on the moon, knowing the topography matters. This map of the lunar surface was created by using 6 billion measurements from NASA's Lunar Orbiter Laser Altimeter, a type of lidar. Check out the technically accurate depths of craters and ancient oceans on our beloved lunar neighbor with this fun, impressive map.

The key ingredients to life on Earth—and elsewhere

Life can only arise in incredibly specific conditions, an abundance of certain substances like water and phosphorus, lots of time, and the right proximity to a heat-and-light-providing star. These two experts on evolution of life say above all, life requires luck. Here's a list of eight ingredients to life.

Large language models struggle to play video games, despite coding them easily

As of 2025, given clear tasks, specific feedback, and test results to learn from, LLMs have been successful at coding simple video games that resemble pre-existing ones. However, given the variety of games, the lack of training datasets, and the need for spatial awareness, which LLMs find challenging, some of these models experience greater difficulty navigating and "winning" games.

Compare various El Niño-Southern Oscillations using maps of atmospheric and ocean variables

Because the intensity of the phases in the cycle is sensitive to small variations in atmospheric and ocean conditions and depends on complex interactions between these two systems, no two El Niño or La Niña events are identical. This contributes to the variability in the length of each phase, which is why an ENSO can last as little as two years or as long as seven.

Mapping fiction's imagined breakdowns of the US after apocalypse

Post-apocalyptic fiction in the US is at least a century old, and how the future is envisioned reflects contemporary anxieties and divisions. This piece examines the various creative maps birthed from the tradition from Jack London's 1912 "The Scarlet Plague," a pandemic novel set in 2073, through Stephen King's "The Stand", dystopian "The Hunger Games," and visions of civil war.

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

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

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.

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.

Larger animals like giraffes, elephants, and blue whales have developed a unique defense against cancer, a phenomenon known as Peto's Paradox

See how each of the more than two million known species relates to each other—and to common ancestors—in this zoomable tree of life.

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.

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.

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

Massive blooms of microscopic plankton called dinoflagellates produce the neon blue light seen in glowing tides.

How the most famous Ebola book, The Hot Zone, which sold 3.5 million copies and was translated into 30 languages, got aspects of the disease so wrong.

Use this calculator to find the dollar value of your mental load, from meal planning to coordinating after-school care for your children.

Studies suggest when you sleep is as important as how much you get, and syncing with your circadian rhythms may lead to less fatigue and improved long-term health outcomes

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

Volcanoes 'die' when their supply of magma runs out, if they are shifted away from their underlying magma reservoir, or if they get clogged by collapsing or their magma cooling back into rock.

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

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.

The hormonal disorder PCOS was renamed PMOS in 2026, a move designed to improve care and research by shifting the focus to the hormonal causes of this condition that affects 1 in 8 women.

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.

The syncytiotrophoblast is the outer layer of the placenta and the largest human cell, spanning an area about twice that of a king-sized bedsheet.

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.

While El Niño moves warm Pacific Ocean waters east along the equator, La Niña moves them west, shifting where rainfall will become more common for months.

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.

The Russia-Ukraine conflict is the largest ground war in Europe since World War II in casualties, forces mobilized, and displacement of civilians.