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.

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.

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.

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.

The golden ratio is believed to underlie patterns in nature and beauty

The irrational mathematical constant has been cited in art, architecture, and nature since its origins in ancient Greece. Its connection to the Fibonacci sequence has associated it with spiral patterns in pineapples, pinecones, and other plants.

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.

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.

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.

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.

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.

The US has over 4 million miles of roads. Learn about how highways are designed and the geometry behind what makes them safe and efficient

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.

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.

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...

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.

Since 1983, the meter has been defined based on the speed of light

The 1875 Metre Convention established the unit as the length between two lines inscribed on the surface of bars made by the International Bureau of Weights and Measures. Advancements in the measurement of light redefined it as the distance light travels in 1/299,792,458 of a second.

Objects can be 3D printed with light by projecting 2D computerized tomography (CT) scans onto rotating cylinders of resin, which solidify upon exposure.

The next generation of gravitational wave detectors may include a triangular design

The Einstein Telescope will use three pairs of underground detectors—each with arms separated by 60 degrees—arranged in an equilateral triangle. The European observatory will work alongside the Cosmic Explorer, a pair of new US detectors, to provide an eightfold increase in sensitivity over LIGO and extend the current detection frequency range.

QR codes were developed in Japan in 1994 to fix bar code limitations

These codes store more data than traditional barcodes—up to 1,850 uppercase characters—and can be quickly scanned from any angle thanks to visual indicators like green squares and zebra-striped lines communicating orientation.

Objects can be 3D printed with light by projecting 2D computerized tomography (CT) scans onto rotating cylinders of resin, which solidify upon exposure.

How math whizzes exploit sportsbooks

Chris Dirkus is a professional sports bettor who approaches the hobby as a math problem, not from a fan's perspective. He focuses on mispriced odds in popular games where many fans are betting for high liquidity. He's one of a rare segment of sports bettors exploiting slow models and human mistakes.

Machu Picchu, 101

Machu Picchu was built right on top of two intersecting fault lines, which makes it even more baffling how its temples and terraces have survived for more than 500 years—especially since no mortar was used to hold the precisely cut stones together. As this three-minute NatGeo video explains, this ingenious construction method allows the stones to bounce in place when earthquakes hit, shaking and rattling but rarely collapsing.

How geometry and uneven charge distribution make some molecules greenhouse gases

Molecules vibrate and rotate with different frequencies based on their elemental composition and shape, which can produce uneven distributions of electrons, as seen in carbon dioxide and water. The resulting unbalanced electric field can interact with electromagnetic waves of the same frequency, causing the molecule to absorb the radiation.

A startup is trying to create the diamond equivalent of the gold standard

It's called the Diamond Standard Co, and it uses an algorithm to assign standard values to a variety off small diamonds, grouping them together into individual coins. Historically, diamonds haven't been able to be classified as commodities or standardized like gold because they aren't uniform—factors such as clarity, cut, color, and carat size significantly influence the value of a diamond.

The price of gas is a composite made up of several different factors, with the federal gas tax making up less than 20% of what drivers pay at the pump.

Stanford built a Google Maps for the Roman Empire, revealing how much it cost to travel in the ancient world

All rainbows are 'raincircles' caused by water droplets separating white sunlight into a spectrum of colors, but Earth's surface blocks the bottom half of the circle from our line of sight.

Power lines are dotted with large, brightly colored balls to improve the visibility of these thin wires to helicopters and small planes and help prevent potential crashes.

Fear or disgust related to clusters of holes and cracks, called trypophobia, may stem from evolutionary processes to help us avoid pathogens.

Some coin denominations have ridges etched on their sides, partly to prevent counterfeiting, in a practice called 'reeding.'

The term 'algorithm' originated from the Latinized name of al-Khwārizmī, a 9th-century mathematician whose works introduced Arabic numerals and the decimal-based numeral system to Europe and whose treatise 'Al-Jabr' gave algebra its name.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

After NASA's Voyager 1 captured Earth as a single pixel as it left the solar system in 1990, astronomer Carl Sagan used the image to reflect on humanity's fragility and the responsibility to protect 'the only world we've ever known.'

Domino's Pizza's 30-minute delivery guarantee was responsible for the chain's 1980s surge, but it was discontinued after the company was found liable for delivery driver accidents.

Venture capital is a form of funding provided to early-stage companies with high growth potential, typically in exchange for an equity (or ownership) stake

SpongeBob exemplifies Aristotle's philosophy for living well because he understands that happiness doesn't come from external factors like money, recognition, or power.

Take a 3D tour of the Hajj, Islam's annual pilgrimage to Mecca, undertaken by over 1.5 million Muslims.

Radioactive elements can introduce software errors because the ionizing radiation they produce can detach electrons from atoms, and computer code relies on the distribution of electrons.

Art theft is the third-largest source of criminal revenues worldwide, thanks in part to the subjective nature of valuing a work of art.

Exploring the Larry Bird punch that almost changed March Madness

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

Gerrymandering works through two techniques: cracking opposing voters across districts to dilute their power, and packing them into a few to waste their votes in lopsided majorities.

The concept of phantom limbs, the perception that amputated limbs are still present, dates back to a French military surgeon in the 16th century.

Watch a 1934 short film starring jazz legend Cab Calloway, a major figure in the Harlem Renaissance's music scene

The deadliest aspect of volcanic eruptions is often not lava, but pyroclastic flows—avalanches of ash, gas, and rock that can move more than half the speed of sound.

The identity of the Tank Man in the iconic photo from the 1989 Tiananmen Square protests remains unknown and was possibly killed.