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.

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.

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.

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.

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

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.

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 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.

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...

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

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.

Because some models of the universe do not incorporate time, it may be an emergent property born from our limited perception of the universe, rather than a fundamental property of reality.

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.

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 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.

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.

In a standard deck of 52 cards, shuffling produces one of approximately 8.07 × 10⁶⁷ potential card orders—a number about as large as the number of atoms in the Milky Way.

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.

Abaci, the plural of abacus, allowed individuals to perform arithmetic and tally quantities by moving pebbles called calculi, from which the word 'calculator' is derived.

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.

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

- Albert Einstein

Within quantum physics, the order of events across time may not follow causality

Although time is considered a fixed, ordered mapping of events, where the past affects the future, quantum superposition breaks down causality, suggesting that future events can occur simultaneously with past ones. Within the quantum model, the notion of cause and effect is not a fundamental feature of our universe.

Explore the fundamental constants of the metric system

In 2019, the International System of Units was revised, such that all seven fundamental units are defined in terms of combinations of seven fundamental constants of the universe. These include the speed of light in a vacuum, Planck's constant, Avogadro's number, and the elementary charge of protons and electrons.

Modern physics suggests that many universes with different physical laws could exist

Physicists suggest that quantum fluctuations in the early universe could have given rise to these universes. Ancient Greek philosophers like Chrysippus proposed a version of the multiverse over 2,000 years ago—millennia before its adoption into various theoretical frameworks today.

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.

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.

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.

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.

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.

The universe may have been reborn from a state of emptiness

Some physicists propose the universe might be the latest in an eternal cycle of universes, each emerging from the empty remains of the one before. This model is inspired partly by Norse and Egyptian myth, suggesting the universe is a cosmic ouroboros—a self-consuming serpent representing an eternal return.

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.

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.

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,...

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

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?

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.

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.

Quantum field theory, which has made predictions accurate to 12 decimal places, explained

Instead of being made of indivisible particles, all visible matter in the universe is built from underlying quantum fields—smooth, invisible entities that behave like waves and extend throughout space and time. The theory suggests these fields are fundamental, and particles are disturbances that move through them.

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.

Einstein created a mathematical term for anti-gravity to keep the universe static

General relativity predicted that the gravity of all the matter in the universe would cause it to collapse, contradicting the belief at the time that the universe was constant and unchanging. Albert Einstein added the cosmological constant to his equations, which acted like anti-gravity, to "fix" his math.

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.

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.