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.

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.

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...

A naked singularity is an infinitely dense point without a black hole's event horizon

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.

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.

Data suggests that claims about the golden ratio underlying the geometry seen in nature, from mollusk shells to human proportions, are largely exaggerated.

What lies inside a black hole remains one of the universe's greatest mysteries

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.

Containment refers to what percentage of a fire is surrounded by physical barriers, such as rivers or trenches, but it may still be burning and can jump containment with strong winds.

Electricity can flow indefinitely along a ring of superconducting material, which provides zero resistance to the movement of charges due to quantum effects.

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.

Nothing can enter white holes—the opposites of black holes

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.

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.

String theory replaces black hole singularities with fuzzballs of tangled strings

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.

The story of Kryptos, the long unsolved cryptographic sculpture outside the CIA's headquarters

Made by American artist Jim Sanborn, the sculpture features four copper plates engraved with messages written with the 26 letters of the Latin alphabet and question marks. The last of these plates was solved in 2025, but the piece remains of interest to cryptoanalysts.

The mathematics of general relativity does not rule out the existence of wormholes

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.

Even in a perfect vacuum, empty space is full of quantum activity

Scientists have tried to create “nothing” by removing all matter, energy, and heat from a container, but the uncertainty principle and relativity prevent such conditions within any period of time. This means the universe is full of quantum foam, which can produce forces from nothingness.

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.

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.

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.

Even in an infinite universe, its limited age prevents a sky full of starlight

Olbers' paradox asks why the sky would ever appear dark if the universe is full of stars in all directions emitting light everywhere. Light from the farthest stars requires more time to reach our skies than the universe's age.

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.

Einstein and the fourth dimension

How Einstein's greatest theoretical achievement contains a simple but mind-bending proposition—the dimension of time is physically connected to the three observable spatial dimensions.

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.

An explanation distinguishing the butterfly effect from causality, necessity, and sufficiency

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.

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 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.

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.

In mathematics, a 'perfect number' is equal to the sum of the numbers it is divisible by, excluding itself, such as 28, which can be evenly divided by 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 equals 28.

The black hole information paradox comes to an end

Quanta Magazine provides a deep dive into black holes and the black hole information paradox. For a very long time it was believed nothing could escape a black hole. Therein lies the paradox. Per quantum mechanics, whatever falls into a black hole should not necessarily be lost forever, and may eventually return. In a landmark series of calculations, physicists have proved that black holes can shed information.

Permafrost—soil, rocks, and sand that are held together by ice and remain frozen for at least two consecutive years—covers about 15% of the Northern Hemisphere's land area.

Time disappears when encompassing all quantum configurations across the universe

The Wheeler-DeWitt equation, a modified version of the Schödinger equation—the formula that describes the evolution of the quantum configuration of a system—describes the quantum state of space-time across the universe. The transformation causes the variable of time to disappear, suggesting it is not a fundamental, physical parameter of the cosmos.

In 2000, the Clay Mathematics Institute placed million-dollar bounties on seven unsolved problems in mathematics—the Millennium Prize Problems—in the hopes of spurring solutions.

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.

Cooling silica nanoparticles lets physicists push the boundary of how large an object can still behave quantum mechanically.

The Oort cloud, our solar system's theoretical icy shell

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.

Despite the name, infinity mirrors—mirrors that reflect each other—do not produce reflections to infinity because mirrors absorb some of the incoming light with each reflection.

Black holes are formed from massive stars collapsing under their own gravity

This creates a region where spacetime curves infinitely, known as a singularity. Black holes' intense mass and energy allow them to trap light, distort time and space, and present challenges with reconciling Einstein's theory with quantum mechanics.

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.

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.

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.

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.

The ultraviolet catastrophe is the infinite energy prediction of classical physics

When scientists first tried calculating the radiation emitted by a heated object, their assumption was that energy was equally distributed across wavelengths and produced infinite energy at low wavelengths. This contradicted observations, which showed near-zero energy intensity at those values.

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.

The universe expands by stretching space itself, not into anything beyond it

Thanks to Einstein’s relativity and Hubble’s observations, scientists understand that space stretches, like rising bread dough between raisins, with no "outside" to expand into. Some theories suggest our universe might be just one "bubble" in a vast multiverse, possibly floating on a higher-dimensional surface called a brane.

Running a ribbon around the edge of a blade creates curls because it stretches its outer side beyond the yield—the limit at which a material can be deformed but still return to its original shape once the force is removed.

View a gallery of Einstein rings

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.