The Koch snowflake has an infinitely long border but can fit inside a simple circle with a finite area.

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.

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.

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.

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...

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.

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.

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.

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.

In 1900, a Vienna tradesman named Erwin Perzy I was attempting to invent an inexpensive way to improve light amplification in hospital operating rooms. After experimenting with a glass globe filled with water, the idea for the snow globe struck. Industrious as well as inventive, Perzy eventually turned his idea into a successful business. Read the full article to learn how the invention grew to become recognized globally.

Their existence would reveal where Einstein’s theory breaks down and quantum gravity takes over. The prevailing idea—cosmic censorship—suggests these singularities are always hidden behind black hole horizons, shielding the rest of the universe from their unpredictability.

While flat maps can never be as perfectly scaled as a globe, they're far more practical than carrying around a sphere. Researchers set out to minimize six major errors of flat maps, including boundary cuts and skewed proportions, by simply squashing...

At the center of black holes is a theoretical point of infinite density called a singularity. Alternatively, one theory suggests a black hole's core could form a Planck star—a mind-bogglingly tiny object just a trillionth trillionth trillionth of a meter in size.

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.

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.

In pop culture, the analogy has conveyed the impression that the flapping of a butterfly's wings has a cascading effect that directly produces a tornado somewhere else on Earth. However, butterfly-wing flapping alone is insufficient to cause tornadoes, as the complexity of atmospheric systems requires many more conditions to produce them, any of which can be the tipping point to initiate tornadogenesis.

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.

Recurring comets lead astronomers to believe they originate from an extremely distant, icy shell encapsulating the solar system. They call it the "Oort cloud," and this giant swarm of icy objects is as fascinating as it is is massive. Home of comets and a tremendous number of planetesimals, this NASA site introduces you to the Oort.

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.

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 massive model was built across a Mojave Desert lake bed using 800 lights spaced just 10 meters apart. Within the model, the entirety of human civilization appears only in the final centimeter.

Black holes contain singularities where information about the material that fell in is lost, violating quantum mechanics. Instead, modeled as a bundle of strings, information is stored and slowly released as the black hole evaporates.

Although a mathematical possibility according to general relativity, there is no observational evidence for these objects, which would endlessly eject material into the universe via anti-gravity. It is also unclear how they would form or avoid collapsing due to the gravity of the ejected material.

Such objects, also known as Einstein-Rosen bridges, would resemble funnel-like tunnels connecting two regions of spacetime, allowing for instant travel across the universe. However, such structures would be incredibly unstable and require undiscovered materials with negative energy to keep them open.

In light of the challenges of the Treaty of Versailles, which concluded the Great War, a mostly unknown author, and son of the map's printer, PA Maas suggested central Europe be integrated into a single state. With a central capital in Vienna, 24 Kantons, or regions, would radiate outward in pie-slice shapes.

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.

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.

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.

As light from a distant galaxy spreads in all directions, rays that pass around a massive object, such as another galaxy, may be bent around it. If the massive object is positioned between the distant galaxy and our telescopes, we may observe this deflected light forming a ring around it.

Republican consultant Thomas Hofeller quietly shaped US politics for decades by engineering partisan maps nationwide. His files, released after his death, revealed strategies to maximize GOP advantage and diminish Democratic influence.

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...