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.

How many galaxies are there? (Hint: a lot)

The universe is considered by some to be infinite, while others find comfort in the fact that it may have an end. Click for more.

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.

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.

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.

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.

Explore a visualization of the scale of objects across the universe, from an astronaut and planets to stars and galaxy clusters, in this interactive adventure.

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.

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.

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.

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.

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.

A black hole with the mass of the observable universe would be larger than the observable universe, leading some theorists to suggest that we may live inside a black hole.

Calculators convert input values into a form of scientific notation—a fixed set of digits with a decimal multiplied by a base to some exponent—to store and process very large or very small quantities efficiently at the cost of rounding errors.

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,...

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.

An animation illustrating the enormity of the universe

Look around you. Where are you? Where is this place you are occupying? Somewhere in a room, maybe in a city on a continent on a planet orbiting a star in a galaxy among billions. But… where is all of that?

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.

Quantum mechanics and general relativity—two of the most successful explanations of the physics seen across the universe—are logically incompatible with one another because they use contradictory models for space and time.

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.

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.

'Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world.'

- Albert Einstein

Explaining 'impossible' black holes

When stars collapse, the black holes they form can have the mass of either 50 suns or over 120—but current models claim strongly that no black hole should have a mass that falls in between that range. But in 2019, gravitational wave observatories detected two within it: one at 66 solar masses, the other at 85. The discovery disrupted established science and left researchers questioning the Standard Model of physics.

Most of the observable universe's volume consists of voids and supervoids—enormous regions of space with little to no galaxies, stars, or light.

More than 40 quintillion black holes are estimated to exist in the universe, including about 100 million in the Milky Way, each with masses up to hundreds of times that of the sun.

Compound interest has been called the eighth wonder of the world

That's because it has the near-magical quality of helping financial investments grow exponentially. Rather than accruing a standard lump of simple interest annually, compound interest takes the previous year’s yield and reinvests it as part of the principal.

Zooming out from Earth to the edge of space

For perspective on the universe's absolute massiveness, watch this four-minute video. Beginning at a facility in Germany, the "camera" pulls back revealing the surrounding countryside, Germany itself, Europe, the Earth, and so on. At one point stars form galaxies and galaxies, at a distance, become as seemingly "small" as the individual stars.

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.

How string theory explains gravity on the quantum scale, but may be untestable

Combining the general theory of relativity with quantum field theory creates nonsensical mathematical outcomes, such as probabilities of infinity. These are resolved by treating particles as vibrating strings rather than points of matter. However, probing the scale of these strings to validate the theory requires technology equivalent to a particle accelerator the size of the solar system.

Explore over 30,000 Olympiad-level math problems from around the world.

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.

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.

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.

The sophisticated math jokes of 'The Simpsons'

"The Simpsons" has always been a slyly sophisticated show and has featured more than a few references to complex mathematical concepts. That's largely due to its writers' room, which has featured more than one writer who studied math before transitioning to comedy.

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.

Mathematical breakthroughs influenced Shakespeare’s stories

New ideas in math were transforming Europe at the time Shakespeare penned his plays, including newly popular conceptions of zero via philosophers Descartes and Leibniz.

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.

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

The Big Bang became physically plausible thanks to Stephen Hawking’s thesis

Stephen Hawking's PhD thesis mathematically showed that singularities—points of no volume and infinite density—are an expected feature of nature. It laid the theoretical groundwork for a model of the universe where everything originates from a single point.

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.

General relativity describes a singularity as the final moment in time

Using Penrose diagrams—mathematical structures that compress all of space and time—physicists can observe the path of light and which events can affect one another. The space-time maps help visualize the structure of black holes and describe what happens when objects fall in.

The observable universe has a mass of 100 sexdecillion kilograms

The amount of visible matter in the universe is most accurately determined by measuring the cosmic microwave background. Quantities of dark matter and dark energy are calculated through the observed bending of light around galaxies and the expansion of the universe, respectively.