Within mathematics, some infinities are bigger than others

Infinity can come in different sizes and be ordered by these sizes

Russian mathematician Georg Cantor demonstrated that certain sets of numbers (e.g., the natural numbers) form countable infinities, while others cannot be counted (e.g., the real numbers) and are therefore uncountable, representing larger versions of infinity. Through the axiom of choice, German mathematician Ernst Zermelo proved this concept of ordering infinities.

If the universe is infinite, it may be impossible to say what's beyond it

The universe is defined as all that exists, including space and time. If the universe is infinite, as observations currently suggest, there is no boundary from which we can ask what lies beyond.

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...

The Koch snowflake has an infinite perimeter enclosing a finite area

Counterintuitively, although it would be impossible to build a fence along the edges of the shape because it would require an infinite amount of materials, this perimeter can be contained in a larger shape with a finite perimeter and limited area. This is equivalent to enclosing an infinitely long spiral that never reaches the center within a circle.

What is a singularity?

It's hard to imagine a singularity, which are described as an infinitely tiny point, created by gravity, until it has zero volume. The definition feels impossible, though perhaps not at the center of a black hole or at the beginning of the universe. Even then, singularities may be something else entirely. Singularities are a mathematical quandary that challenge us to better understand physics and the universe.

The history of the 10 axioms of Zermelo–Fraenkel set theory with choice, regarded as the most universal foundations of math

Like postulates in geometry—accepted truths that form the logical building blocks for formal proofs of more complex theorems—axioms cannot be proven, but set the underlying logic and rules for systems. The ZFC axioms resulted from painstaking work to identify the underlying rules of mathematics and are built on the premise of sets, or well-defined collections of elements.

An interesting perspective on whether (and how) brain size matters in cognitive ability

Some brains are bigger than others, but what is the relevance, if any, to brain size and intelligence? This Scientific American article explores the question and offers examples. It also visually compares frontal slices of several mammals' brains, showing that humans don't necessarily have the biggest cerebrums on the block.

A scale of the universe, from atoms to cosmos

Can you really appreciate just how small protons are, or how massive the universe is? To attempt to grasp it, watch this short video which zooms out from a young woman in California to the very limits of the cosmos using established scales. Then, dive deep into the atoms of her white blood cells for a fully-rounded effect.

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.

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

The cloud relies on hardware infrastructure with real-world limits

While unlimited storage providers exist, this does not mean online storage is limitless, as it relies on servers, cables, power, and physical space, all of which are finite. However, the cloud seems infinite because the number of servers and data centers built outpaces demand. Storage solutions are also routinely replaced with more efficient options.

How the number 15 solved a longstanding math challenge

The “packing coloring” problem asks how many numbers are needed to fill an infinite grid so that identical numbers never get too close to one another. A new computer-assisted proof finds a surprisingly straightforward answer—only 15. While...

Visualizing the 75 largest underwater animals

With so many fish in the sea, which are the largest? This infographic shows fantastic illustrations of the 75 largest sea animals, ranked by average size. The detailed visual is organized in descending order of size, starting with the blue whale,...

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.

Play Infinite Craft, an AI software-powered game

In this browser-based game, users combine the elements of earth, wind, fire, and water to create an infinite number of new elements. Try your best to make as many natural phenomena as you can as you build up to the multiverse.

Compare the true size of any country

This interactive lets you drag any country across the conventional Mercator projection to compare its true size to others. Africa, which appears similar in size to Greenland on standard maps, is actually close to 15 times larger. It's a great way to recalibrate your assumptions about world geography.

What are college endowments? An expert explains

College endowments are meant to last forever. That's why some of the oldest, most prestigious schools in the US have lasted so long. A higher education finance expert weighs in on how these funds work and offers fascinating tidbits on how they impact university finances.

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.

There may be as many as four types of black holes, categorized by their mass

Three types are known to exist: stellar mass black holes, supermassive black holes, and intermediate black holes. A fourth type—micro black holes—is theoretical and would have the same mass as Mount Everest in a space the size of an atom.

IBM developed a benchmark to measure the usable performance of a quantum computer

Dubbed "quantum volume," the benchmark tests how well a quantum computer runs complex tasks. It stops when the computer can no longer correctly select the most likely answers, called "heavy outputs," more than two-thirds of the time.

The similarities and differences between bitcoin and ethereum

Ethereum is the second-largest cryptocurrency by market cap behind Bitcoin. Usually, that comparison just scratches the surface of the overall market cap. Instead, a visual pits the tokens up against each other on a per-token basis and shows how and why the Ethereum/bitcoin ratio has fluctuated over time.

How 'vibes' factor into scientists' decisions about using certain units

Although there are absolute quantities from which units can be defined, such as the speed of light and the corresponding light year for distance, the usefulness of units comes from their ability to easily communicate values, rather than how fundamental their origins are. For example, while Kelvin measures absolute temperature, Celsius is scaled to the temperatures of boiling and freezing water, making them more practical for everyday usage.

Apeirophobia is an overwhelming fear of living forever, perhaps stemming from the brain's inability to conceptualize an infinite life.

What is the Gini coefficient?

The Gini coefficient is a measure of income inequality on a scale from 0 to 1. A value of 0 indicates perfect equality, where everyone has the same income, while a value of 1 indicates perfect inequality, where one person has all the income and...

Large language models are examples of foundation models

Such models are trained on massive amounts of data, which enable them to perform a variety of tasks, rather than being provided task-specific data to complete a narrow function. Such models come with high compute costs and potential trust issues due to the nature of their unorganized training data.

Inflation is a monetary phenomenon in which consumers see a broad rise in the prices of goods and services across the economy over time

Sustained periods of inflation are often caused by central bank policies including low interest rates or increased money supply. With more money chasing the same number of goods and services, prices tend to rise and the value of a single dollar drops.

Private schools are generally smaller than public schools

As of 2021, 44% of private schools enrolled fewer than 300 students. By comparison, over 60% of public schools enroll more than 500 students.

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

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.

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.

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.

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

The Nasdaq's 'Magnificent 7' companies are responsible for roughly 40% of the Nasdaq 100 index's gains, while the remaining 93 companies account for the other 60%.

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

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

Roughly 43 million US borrowers owe roughly $1.7T in student debt, the second-largest source of consumer debt behind mortgages.

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.

Existentialist writer Jean-Paul Sartre coined the famous phrase 'Hell is—other people' in his play 'No Exit'

In 2026, Finland was named the world's happiest country for the ninth consecutive year based on its wealth, social safety network, and high life expectancy, among other factors.

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.

Warren Buffett's former professor Benjamin Graham is known as the 'father of value investing,' a strategy in which investors seek to buy stocks trading below their intrinsic worth.

Some coin denominations have ridges etched on their sides, partly to prevent counterfeiting, in a practice called 'reeding.'

The rivalry between Alexander Hamilton and Thomas Jefferson centered on federalism—the question of how much power the federal government should hold relative to the states.

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

Against his wishes, Albert Einstein's brain was removed before his cremation, and the pathologist responsible stored it for decades in mason jars within a beer cooler.

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.

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

A small portion of The Winstons' 'Amen, Brother,' an obscure '60s gospel record, is the basis for thousands of hip-hop tracks, making it one of the most sampled records ever that's informally known as 'the Amen break.'

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

Search a massive database of over 4 decades of clues from the TV quiz show 'Jeopardy!'