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

Within mathematics, some infinities are bigger than others

While the concept of infinity refers to a countless set of items, these infinite sets can be compared to one another by matching an item from one set to an item in another, one by one. This analysis shows that certain sets must be larger than others because some infinite sets have leftover, unmatched items.

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.

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.

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.

One model of the universe expresses all constants using only time

About seven base units, such as meters for length, have historically been used to describe everything in our physical reality. However, researchers have shown that, through relativistic space-time, clocks can serve as replacements for rulers, scales, and other instruments, allowing everything to be measured in terms of time.

There may be 74,963 kinds of ice, resulting from water molecules rearranging themselves

The theoretical limit greatly expands the known types of ice, with type Iₕ being the most common on Earth and forming hexagonal crystals. The extreme environments of planets like Uranus and Neptune can create a variety of ice with unique structures and properties, as these conditions can reshape these crystals into new geometries.

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.

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

Ring galaxies, the rarest kind in the universe

Galaxies come in different sizes, but when it comes to shapes there are just four kinds: spiral, elliptical, irregular, and the rarest kind, ring. Ring galaxies only form in the event of a small galaxy passing through a large one, but there's more to them that that. This page explores these rare and beautiful heavenly bodies.

Qubits let quantum computers process massive datasets all at once

The quantum phenomena of superposition and entanglement enable qubits to act in 0 and 1 binary states simultaneously and influence each other instantly. These features give quantum computers incredible power, though keeping them stable is a major challenge.

Figuring out how to get a sofa around a corner has been a math problem since 1966

A sofa with the largest possible area that can make a 90-degree turn is known as Gerver's sofa, named after the mathematician who determined the shape in 1992. In 2025, a 119-page paper provided the proof confirming this shape.

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,...

Turing machines consist of an infinite tape, a read-write mechanism and instructions

Based on the information the mechanism reads on the tape, the machine follows the applicable instruction (e.g., if you read a 1, write a 0) and may proceed to the next cell of data. Given enough time and memory, Alan Turing proposed such a machine could perform any computation.

Imagining a fourth dimension

Famous astronomer Carl Sagan attempts to describe what it's like to think about the fourth dimension using the famous Flatland example from author Edwin Abbott in this classic video.

193 nations are members of the United Nations

The UN's Member States webpage provides an alphabetical list of all 193 member countries, along with the exact dates they were admitted. Take a look at how the UN has expanded over time and understand each nation's place in the global organization.

Taxicab geometry defines displacement as the sum of vertical and horizontal distances

In this mathematical system, pi is valued at four, and circles aren't round—they appear as diamonds with corners instead. The system simplifies pixel distances in digital imaging and helps engineers with robot navigation and chip design.

Try counting like the Maya in this interactive math game

The Maya numerical system used dots for 1, bars for 5, and a shell symbol for zero. This interactive resource lets users practice counting with the Maya base-20 system using the same symbols once common in the Yucatán.

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.

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.

Mathematician David Hilbert developed general relativity equations before Einstein

Although working independently of one another, the two exchanged correspondence in the lead up to November 1915, when Hilbert submitted their gravitational field equations five days before Einstein presented their work. An error in Hilbert's formulation and its nonpublication until March 1916 would leave credit for general relativity to Einstein.

Only 21 million bitcoin will ever be created—what happens then?

Bitcoin is a popular cryptocurrency with a finite supply, and the issuance of new bitcoins is currently offered as a reward for verifying the transactions on the blockchain. Experts say the currency won't reach its limit until the year 2140, and miners may ultimately rely on transaction fees.

'In life, unlike chess, the game continues after checkmate.'

- Isaac Asimov

Inflation was a word before deflation was

The word inflation has Latin roots and technically means an influx of air, first appearing in the English language in the 1890s to describe bike tires. Deflation's first known mention in an economic text was in 1919.

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.

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 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.

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

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

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

While atoms are normally made of protons, neutrons, and electrons, exotic ones have been built with antimatter—alternate versions of normal matter that live for fractions of a second and are about a billion times as rare.

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.

While the first exchange-traded fund (a 'basket' of different assets that can be bought and sold on a stock exchange) in the US only launched in 1993, the global ETF market has grown to more than $10T in the short time since.

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

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.

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

Corals are themselves not coral reefs—the term refers to the tiny marine invertebrates that build the rigid reef.

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

Hurricanes, which rely on air consistently spinning in the same direction, cannot cross the equator because air curves in opposite directions north and south of it.

Waves off the coast of Nazaré, Portugal, can reach over 100 feet and are caused by the area's deep underwater canyon geography

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

In 2025, four executives from large tech companies were given positions in the US Army Reserves—including Palantir Chief Technology Officer Shyam Sankar, which reinforced Palantir's ties with the US military and its role as a leading defense partner.

Sunflowers have long been a symbol of nuclear disarmament because they can extract large amounts of radioactive cesium from soil.

See this overview of the firework rules across all 50 states, ranging from a complete ban on non-professional fireworks in Massachusetts to states with more lax rules like Indiana.

Electrical grids worldwide must double their power delivery capacity by 2050 to meet surging demand from AI and electric vehicles.

A 2019 Facebook event called on attendees to "Storm Area 51, They Can't Stop All of Us," resulting in countless memes, the creation of a music festival, and eventually the arrests of those who did try to approach the area.

Explore a list of the 100 best novels of all time, compiled by some of the world's best living writers, including Salman Rushdie, Stephen King, and Jennifer Egan.

The first horror movie is believed to be George Méliès' 1986 short 'The Haunted Castle,' which pioneered the use of special effects

The most expensive ad of all time was for Chanel No. 5 perfume and cost roughly $33M to produce in 2004—about $57M in today's dollars.