The Koch snowflake has an infinite perimeter enclosing a finite area

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.

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.

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.

Snowflakes form when water vapor turns directly into ice

Every water molecule has a slightly bent shape due to electron repulsion and regularly forms bonds with hydrogen molecules. When they freeze, they form a six-sided (or hexagonal) lattice, resulting in the six-sided symmetry of snowflakes. As the snowflake falls, more water molecules freeze onto its six sides, forming the unique, beautiful geometric designs.

How chemical recycling could create 'infinite' plastic

The way we normally recycle plastics is a downward spiral of waste and degraded materials, but there is another option—turning plastic back into the oil it was made from.

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.

Despite claims of limitless free energy, zero-point energy cannot be harnessed

Because quantum mechanics prevents systems from ever reaching an energy value of zero, a seemingly empty vacuum of space still contains some amount of energy associated with its lowest allowed state. Since descending to even lower states is forbidden, extracting meaningful amounts of energy from an empty vacuum is impossible.

A group claims a man is trapped inside Chicago's 'Cloud Gate'

Although they appear to be performance artists, the group claims they're earnestly protesting the captivity of a man inside the iconic public art piece designed by Anish Kapoor. City officials say there is no man inside the object.

The event horizon is an inescapable sphere around a black hole singularity

The horizon is often surrounded by an accretion disc of superheated material that orbits the black hole while emitting radiation. Light emitted behind a black hole can be bent around it to create the appearance of a ring of material known as a photon sphere.

Wildfire containment requires trenches, natural barriers, or burned land

Containment refers to creating a perimeter around a fire that it is unlikely to spread beyond. However, full containment does not mean the fire is extinguished. Strong winds can cause fires to jump containment lines, including 10- to 12-foot-wide trenches or thin rivers.

Explaining the galaxy-sized gas bubbles hovering around the Milky Way

Depth perception is hard to achieve when looking out into space. For decades, two camps of astrophysicists have disagreed over the nature of the North Polar Spur, a radio-wave emitting arc above the Milky Way galaxy's plane. Was it close and relatively small? Or was it faraway and much more massive? New research utilizing X-ray mapping of the galaxy has helped clarify the existence of both so-called smaller Fermi bubbles and larger X-ray ones.

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.

A look inside neutron stars, which are thought to contain nuclear pasta

Simulations show that beneath the outer crust of these objects, the gravitational force is so extreme that atomic nuclei link together into chains of primarily neutrons and some protons called nuclear spaghetti. Deeper in the star, these links combine to form sheets—nuclear lasagna—a quintillion times stronger than steel, making it the strongest material in the universe.

The Kennan Telegram shaped early US Cold War strategy

In 1946, diplomat George Kennan sent a long telegram from Moscow warning that the Soviet Union was inherently expansionist. His analysis laid the groundwork for the US policy of containment that defined the early Cold War.

Winter Olympic sports highlight various laws and principles of physics

Figure skaters leverage the conservation of angular momentum during spins by tightening their bodies, and the skintight suits used in ski jumping help reduce drag. Centripetal force allows speedskaters to make tight turns, while friction on skis helps ski mountaineers climb slopes. No action can violate the conservation of energy.

Permafrost, explained.

This article explains the role of permafrost—the layer of permanently frozen soil beneath the Arctic tundra—in shaping both the ecosystem and the terrain. During the summer, the permafrost keeps the soil above it moist, which in turn impacts the...

Great circles provide the shortest path between two points on a sphere

On Earth, a great circle is formed by slicing through the planet, such that one cuts through the center alongside the two locations from which the shortest distance needs to be found. The resulting arc along the Earth's surface is responsible for curved routes on flight maps, rather than straight trajectories.

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.

Claude Shannon, for whom Anthropic's AI tools are named, proved the existence of an unbreakable cipher

In the one-time pad approach, the encryption key is randomly generated and of equal length to the plaintext. This randomness ensures that every possible plaintext is equally likely—RBBHN can mean HELLO, SOLVE, CODES, and so on, using a different key—while matching the message and key lengths prevents the appearance of patterns that can be exploited.

Faraday cages protect enclosed objects by redirecting electrical currents

Constructed from conductive materials, the cage allows electrons to distribute themselves evenly throughout its surface and travel to ground, keeping the interior neutrally charged and preventing any internal electricity. The same principle blocks electromagnetic waves by redirecting the electric field component, preventing telecommunication signals from entering the apparatus.

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

Scientific progress works toward absolute truth, but can never reach it

Through the acquisition of scientific knowledge, scientists construct explanations of the world, which are rigorously tested and verified until they narrow down to a leading theory of the natural world. However, because of how science works, these ideas will never become factual, as they must always remain open to be iteratively revision based on new data.

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

In 1978, lower courts ruled that neo-Nazis had the right to march through Skokie, Illinois

Skokie was home to a large community of Holocaust survivors, making the planned march an extreme test of First Amendment principles. The courts ultimately ruled that even deeply offensive speech remains protected by the Constitution.

All-you-can-eat buffets are a practical demonstration of economic ideas

Economists have noticed some restaurant goers prefer a flat pricing model despite it usually being more expensive than per-dish costs. The strategy behind endless servings includes concepts such as economy of scale, cross-subsidization, diminishing returns (of enjoyment of excess food), and more.

The Oort Cloud is a shell of icy objects surrounding the solar system, from which many comets originate

Proposed in 1950 by Jan Oort, the theoretical structure is believed to be the origin of comets with periods of more than 200 years. Leftover planetary building blocks from the early solar system formed the Oort Cloud after being flung out by the gravity of planets like Jupiter.

Explore different living beings' sizes, from DNA to whales and the Pando clone

The Pando is actually a single organism in Utah, comprising more than 47,000 stems connected by a vast root system. It spans roughly 108 acres. Meanwhile, the average human stands at 5 feet and 7 inches. DNA, the genetic instructions for life, stands at a height of 3.5 nanometers.

Wikipedia's puzzle globe logo was developed by a 17-year-old

In 2003, the site held a contest for its users to submit designs for a new logo. Paul Stansifer submitted the winning design, which used the incomplete puzzle as a symbol for the encyclopedia being a work in progress. Symbols and letters would later replace text within Stansifer's colorful design.

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.

Legendary architect Frank Lloyd Wright designed the SC Johnson headquarters in Wisconsin, aiming to make a cathedral-like workplace

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.

After NASA's Voyager 1 captured Earth as a single pixel as it left the solar system in 1990, astronomer Carl Sagan used the image to reflect on humanity's fragility and the responsibility to protect 'the only world we've ever known.'

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

An investigation suggests a Scottish island became awash in forever chemicals via seafoam and seaspray—raising questions about what that means for the safety of other coastal areas.

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

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.

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.

Scientists can determine Earth's past climate conditions by analyzing air trapped in microscopic bubbles within ice cores—columns of compressed snow layers that build up over hundreds of thousands of years.

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.

SpongeBob exemplifies Aristotle's philosophy for living well because he understands that happiness doesn't come from external factors like money, recognition, or power.

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

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

Explore fun facts from the USPS, including unique post office buildings, a stamp that traveled past Pluto, and more.

Georgia O'Keeffe's flower paintings are often assumed to be sexual, though the artist claimed she was simply interested in highlighting the beauty of flowers that most people neglect

Exploring the Larry Bird punch that almost changed March Madness

The power and influence of the American presidency have expanded significantly over time—far beyond what the founders envisioned.

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

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.

Calculate how much of your tax dollars go to each federal program based on your income, filing status, and number of children

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