Taxicab geometry defines displacement as the sum of vertical and horizontal distances

Pizza boxes are square for practical reasons

Although circular pies have sometimes appeared in circular boxes, they usually travel in square, cardboard boxes. That's because the cardboard industry manufactures them with edges so they can be made in one piece and be easily stacked.

Self-driving car companies like Waymo use LiDAR technology for navigation, which constructs 3D maps of a car's surroundings by emitting light pulses in all directions.

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.

Picasso is known as one of the creators of cubism

In the early 1900s, cubism redefined modern art by breaking objects into geometric shapes and showing multiple perspectives at once. Developed by Pablo Picasso and Georges Braque, it challenged the idea that art should mirror reality, and laid the foundation for many movements that followed.

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.

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.

When you complete a CAPTCHA on a website to prove you are human, you may be unknowingly training algorithms for self-driving cars, whose systems require training data to help identify crosswalks, bicycles, and other outdoor objects.

Explore a visualization of curved spacetime under general relativity

While depictions involving bends in elastic fabrics are common to illustrate gravity, they are circular representations that rely on gravity in the real world to depict gravity in the model. The river model instead shows how bends in spacetime convert speed through time into speed through space, producing acceleration.

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

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.

Picasso is known as one of the creators of cubism

Cubism is an artistic style pioneered by Pablo Picasso and Georges Braque in the early 20th century. Cubist works portray objects and people from multiple perspectives rather than from one fixed position. This helpful explainer from The National Galleries of Scotland breaks down the history of the style and movement, offering some examples of cubist works.

Picasso was one of the 20th century's most influential artists

The artist is recognized as one of the inventors of cubism, which simplified figures and scenes into flattened geometric shapes and imagined arrangements. The movement spurred new styles in response and led to increasingly abstract art. Styles like futurism, dada, De Stijl, and art deco were built on the simple forms, abstract ideas, collage, and found objects of early cubism.

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...

No map of the Earth is perfect because each introduces inaccuracies in the shape, size, or orientation of landmasses and oceans by trying to flatten the planet's curved surface.

Virtual particles are bookkeeping tools used to describe force interactions

Physicists mathematically describe the electromagnetic, weak nuclear, and strong nuclear forces as the result of these particles being exchanged between particles of matter. Despite not being real, observable particles, this method has provided experimental predictions with an accuracy of over 12 decimal places.

The origin of shortest-path algorithms and how they became optimized for navigation apps

In 1956, Dutch programmer Edsger Dijkstra spent about 20 minutes developing an algorithm to demonstrate how computers could quickly determine the shortest route between Rotterdam and Groningen. Since then, modified versions of Dijkstra's formulation comprise the processing backbone of systems such as Google Maps and Apple Maps.

This flat map is still wrong, but it's the least wrong of all flat maps

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...

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.

Color originates from the brain processing light of varying wavelengths

Various geometric models, including Newton's color circle of hue and saturation and the horseshoe-shaped XYZ space for mixing colors via vector mathematics, have been proposed to describe colors accurately. However, these models do not encapsulate the variety of psychological effects that cause perceived colors to differ from mathematical representations.

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.

Photonic integrated circuits use light instead of electrons within circuit components

Instead of processing electric signals, photons in the visible and near-infrared serve as optical signals between chips made of electro-optical crystals. PICs can enable lower power consumption and heat generation by eliminating the friction from electrons moving along wires.

Philip Glass drove a taxi in his 30s and 40s, something he continued even after his critical breakthrough, 'Einstein on the Beach.'

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.

As of 2025, the College Board has banned the use of calculators with computer algebra systems on the SAT and PSAT.

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.

Taxi and ambulance drivers appear to have the lowest risk of dying from Alzheimer's disease out of 443 occupations, which suggests that constant real-time navigation work strengthens the hippocampus and may help protect against the disease.

An AI-built pixel map reimagines New York City as a video game

Isometric NYC is a zoomable pixel-art map that renders the city in game-style perspective (think SimCity), helpfully clarifying the dense geography and architecture. The makers built the project tile-by-tile using satellite data and generative AI. Zoom into building-level views or out to see all five boroughs.

Many Pixar movies reference 'A113,' an allusion to the classroom where many of the studio's founders took animation classes.

The construction of LIGO needed to compensate for Earth's curvature

Across the 4-kilometer (2.5-mile) arms, the height of Earth's surface falls by almost a meter, preventing the lasers traveling within the arms from meeting the mirror at the end of each. Concrete slabs built beneath the arms help keep them level and provide structural support against seismic vibrations.

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.

Learn about how highways are designed and the geometry behind what makes them safe and efficient.

The European Space Agency's Euclid mission is designed to create a 3D map of the large-scale structure of the universe out to 10 billion light-years, helping reveal the nature of dark matter and dark energy.

The internet explained at five levels of difficulty, from protocols to internet flattening

Resembling a roadway with traffic laws that all cars must follow, the internet is the hardware that connects a source to a destination, and these connections must follow specific software rules. Although data had to travel through about 10 networks to reach a destination in the past, data centers increasingly consolidate traffic, just as a highway does for cars, to improve data speeds.

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...

As of 2025, Waymo has rolled out "robotaxi" services in five US cities

The company has been under investigation by the National Highway Traffic Safety Administration following dozens of crashes and incidents that appear to show Waymo vehicles violating traffic laws over the past decade in Phoenix, San Francisco, Los Angeles, Atlanta, and Austin.

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.

Traffic flow can be improved through research into card shuffling

Card-shuffling mathematics has been applied to solving problems in biology to estimate the evolutionary distance between organisms and in computer science to determine the optimal distribution of files and folders in databases.

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.

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.

Try counting like the Maya in this interactive math game

The Maya numerical system used dots for 1, bars for 5, and a shell symbol for zero. This interactive resource lets users practice counting with the Maya base-20 system using the same symbols once common in the Yucatán.

Astronomers use a set of scaffolding methods to measure distances across the universe

Different methods are used depending on how far away an object is—from radar for nearby planets, to parallax for stars, to redshift and supernovae for distant galaxies. These tools allow scientists to chart the universe across unimaginable scales using astronomical units (AU), light-years, and parsecs.

Five charts that changed the world

Data visualization helps us understand the world, providing a powerful way to communicate complicated information in a quick and efficient way. There are many different ways to create graphs, and these five examples—the pie chart, geographic dot...

The European Space Agency's Euclid telescope maps dark matter

As light rays travel near massive objects in space, their paths can be distorted on their way to Earth, causing objects to appear displaced, magnified, shrunken, or sheared. By simulating these effects for various mass distributions, astronomers can match the distortion to a simulated mass and compare it with the observed visible matter to infer the presence of dark matter.

Digital twins create real-time 3D models of smart cities

These re-creations allow city planners to track changes and manage infrastructure more easily by testing strategies in virtual environments. Geospatial technologies, such as mobile mapping and satellite imaging, help cities assess urban layouts, plan for disasters, and manage resources more efficiently.

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.

A mathematical and visual analysis distinguishing chaos from randomness

Random systems lack any underlying pattern or causal relationships, while chaos is mathematically deterministic because the behavior follows established laws of nature. In 3D plots, chaotic systems produce outputs that approximate a mathematical function, while random points appear as statistical noise.

The centrifugal force is associated with the feeling of being pushed outward when moving in a circle, but it is a fictitious force arising from objects' tendency to move in straight lines.

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.

Slide rules—ruler-sized mechanical calculators with a component that slides out to show relationships between numbers—were used by engineers at NASA during the Apollo missions.

How parking infrastructure reveals a region's values

As cars became more popular, the need for city parking increased. In Europe, well-established cities with limited space generally opted for more expensive underground parking beneath public spaces to preserve aesthetics. Available land and decentralized planning in the US saw more parking lots and garages built. These decisions reveal broader policy realities for these regions.