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In Taxicab Geometry, Pi Equals 4 and Circles Aren’t Round

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In Taxicab Geometry, Pi Equals 4 and Circles Aren’t Round

A red Etch A Sketch toy set against a light purple background displays an animation of a simple line drawing.

Math, Revealed

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Welcome to a city where pi equals 4 and circles aren’t round.

Each installment of “Math, Revealed” starts with an object, uncovers the math behind it and follows it to places you wouldn’t expect. Sign up here for the weekly Science Times newsletter for upcoming installments.

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A red Etch A Sketch toy set against a light purple background displays an animation of a simple line drawing.

The Etch A Sketch is a marvel of space-age technology. It’s like a sheet of paper, a pencil, a portable table and an eraser all rolled into one.

One knob draws horizontal lines on the screen. The other produces vertical lines.

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A red Etch A Sketch screen displays an intricate line drawing resembling Van Gogh’s “Starry Night,” set against a light purple background.

By turning both knobs simultaneously, you can draw diagonal lines, smooth curves or even pay homage to Van Gogh, as in this sketch by Princess Etch:

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The Etch A Sketch shakes back and forth and Van Gogh’s “Starry Night” disappears, revealing a clear screen.

From a mathematical perspective, an Etch A Sketch showcases a space in which two directions, horizontal and vertical, are favored above all others.

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Map of Manhattan, NY, showing various neighborhoods like Harlem, Upper West Side, Times Square, and Chelsea, with surrounding bodies of water, against a light purple background.

Anyone who has spent time in Manhattan will be familiar with a space like this. The cityscape is organized around two perpendicular directions: uptown/downtown and crosstown.

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Zoom into the map of Manhattan, and a small toy yellow taxi moves on top of the map.

Indeed, mathematicians use terms like Manhattan geometry or taxicab geometry to describe spaces like these. Here, the distance between two points is defined commonsensically as the sum of their horizontal and vertical separations.

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On the map of Manhattan, two red lines are drawn on the streets to form a right angle. Each of the red lines has a number 1 next to them.

For example, suppose you’re meeting a friend in the city and you have to go a mile crosstown and a mile uptown to get there by cab.

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Then it’s natural to say that you have to travel 1 + 1 = 2 miles by taxi to get there.

On the map of Manhattan, red lines forming a triangle are drawn on the streets, with the two perpendicular sides labeled a and b, and the hypotenuse labeled c.

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Of course, that’s not how you learned to calculate distances in school.

Back then, you used the Pythagorean theorem, the most important result in Euclidean geometry. It says that in a right triangle, the length c of the hypotenuse satisfies a2 + b2 = c2, where a and b are the lengths of the sides:

On the map of Manhattan, red lines forming a triangle are drawn on the streets, with the two perpendicular sides labeled a and b, and the hypotenuse labeled c.

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This math would apply if all directions were equally available to you — say, if you were a crow flying overhead. Then you’d travel a diagonal distance c, equal to the square root of 12 + 12 (or 2), since both a and b equal 1 mile. The square root of 2 is about 1.41 miles — that’s c as the crow flies.

Same red triangle on the Manhattan map, with perpendicular lines labeled a and b, and the hypotenuse labeled c.

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But on a grid ruled by taxicab geometry, where the roads are what matter, distance becomes much simpler: a + b = c.

Same red triangle on the Manhattan map, with perpendicular lines labeled a and b, and the hypotenuse labeled c.

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That boils down to 1 + 1 = 2 miles traveled by taxi, just as before.

A yellow toy taxi with a checkered roof sits atop a map of Manhattan, positioned over the Times Square and Midtown West areas.

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You have to admit: Taxicab geometry has its advantages!

Close-up of a yellow toy taxi, showing checkered stripes, “TAXI” on the roof sign, and a logo with checkered flags on the door, against a purple background.

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But it also leads to surprises.

A wooden checkerboard with alternating black and light wood squares, centered on a light purple background.

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For instance, what does a circle of radius 3 look like in this grid-based geometry?

Same wooden checkerboard against a light purple background, with four red checkers, equally spaced, forming a diamond shape and one black checker in the center.

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To find out, let’s start by drawing four red dots, each 3 units away from a central black dot, as measured horizontally or vertically.

Same wooden checkerboard against a light purple background, with 12 red checkers, equally spaced, forming a diamond shape and one black checker in the center.

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Those aren’t the only points that are 3 units away from the center. All the new points shown also qualify since they’re 1 + 2 = 3 units away.

Same wooden checkerboard against a light purple background, with four red lines of equal length forming a diamond.

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Points with horizontal plus vertical separations like 1.38 + 1.62 would also work, as long as the two numbers add up to 3.

Connecting all the dots, we discover that a circle in taxicab geometry looks like a diamond. It has corners, and it’s not round. One of my students shouted in protest when she realized this.

Same wooden checkerboard against a light purple background, with a red diamond and red dashes across the center connecting the right and left corners of the diamond.

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Even more surprising is the value of pi in this strange, non-Euclidean geometry.

Recall that pi is defined as the ratio of the circumference of a circle to its diameter.

To find the circumference, observe that our circle of radius 3 is composed of four arcs, the four sides of the diamond. Each arc is 6 taxicab units long, since it extends 3 units horizontally and 3 units vertically.

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Same wooden checkerboard against a light purple background, with a red diamond and red dashes across the center and two numeral 6s next to one side of the diamond and the center dashed line.

Taken together, those four arcs yield a circle of circumference 4 × 6 = 24. The diameter, for its part, is 6 units long, as shown by the red dashed line. Thus, the circumference divided by the diameter equals 24/6, so pi equals 4 in taxicab geometry.

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A wooden checkerboard with alternating black and light wood squares, shown at an angle against a light purple background.

By now, you’re probably wondering why anybody would use this weird geometry. There are at least two reasons.

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Same wooden checkerboard on a light purple background, with a small, retro-style toy robot moving across it.

In some real-world settings, taxicab geometry is more convenient, and more relevant, than Euclidean geometry. Engineers use it when planning the most efficient paths for robots to take when navigating a grid of rails in a shipping fulfillment warehouse.

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Same wooden checkerboard on a light purple background, with a small, retro-style toy robot moving in a square formation on the board.

In the design of computer chips, taxicab geometry makes it easier to estimate the length of wire connecting electronic components; that’s important for optimizing chip layout. Likewise, in digital image processing, taxicab distance provides the simplest way to measure how far apart pixels are. This is essential for finding outlines and grouping similar parts of the image together.

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A red Etch A Sketch screen displays a line drawing of a checkered taxi cab, set against a light purple background.

Beyond its practical uses, taxicab geometry upends our assumptions about space by reimagining circles as angular shapes.

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A red Etch A Sketch screen displays a line drawing of a checkered taxi cab, set against a light purple background.

It’s a topsy-turvy take on the Etch A Sketch’s lesson: that a simple toy, seemingly confined to making straight lines, can defy that limitation and produce curves through sheer ingenuity.

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In math and in play, the human spirit expresses itself beyond the lines.

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China Launches Reusable Rocket in Race With SpaceX

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Video released by Chinese state media shows a state-owned aerospace company launching a rocket and recovering part of it on Friday. The successful launch of a reusable rocket was a major step for China toward challenging SpaceX’s satellite internet dominance.

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Nobel Prize winner leaving UC Berkeley for new role in China

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Nobel Prize winner leaving UC Berkeley for new role in China

Nobel Prize recipient Omar Yaghi is leaving his role at UC Berkeley to lead the development of a new artificial intelligence institute at Tsinghua University in Beijing, the Chinese university announced.

Yaghi will head the AI Chemistry and Materials Research Institute at Tsinghua, where he was appointed an honorary professor in 2022. Known as AIMATRY (AI × Materials × Chemistry), the new center will focus on material design and synthesis through artificial intelligence, according to a statement from the university.

In 2025, Yaghi shared the Nobel Prize in chemistry with Susumu Kitagawa of Kyoto University and Richard Robson of the University of Melbourne for their development of metal-organic frameworks, a type of super-porous material in which metal ions and carbon-based molecules combine to form crystals with exceptionally large surface areas.

The material has the potential to combat climate change by capturing and storing carbon or other pollutants, and by extracting water from the atmosphere in water-scarce areas. Upon awarding the prize, a member of the Nobel committee likened the technology’s ability to store enormous amounts of stuff in seemingly compact spaces to Hermione Granger’s enchanted handbag in the Harry Potter series.

Yaghi’s Irvine-based company, Atoco, has said it will start taking orders later this year for its technology that harvests water from the air.

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A representative for Yaghi said he was not yet available to respond to questions.

China is one of several countries that has been actively recruiting scientists from the U.S., where the Trump administration has slashed science funding, suspended research grants, fired science advisors and tightened immigration restrictions.

“For many, many years, our funding was very competitive; if you worked hard and you were doing good research, you would get funding,” Yaghi said of the U.S. in an interview with Scientific American earlier this year. “The current state is not so encouraging because of the cutting back on grants and support of science by the very agencies that many university researchers rely on.”

Yaghi was born in Jordan to Palestinian refugees, and immigrated to the U.S. when he was 15 to study.

“We’ve learned over and over in human civilization that scholars can move across borders,” Yaghi told the New York Times last year. “This is how knowledge spread and how vast regions of the world lifted themselves out of poverty.”

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Trump administration seeks to limit federal funding that doesn’t ‘advance’ presidential policies

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Trump administration seeks to limit federal funding that doesn’t ‘advance’ presidential policies

A new rule proposed by the White House Office of Management and Budget would fundamentally overhaul the way federal grants are awarded and overseen — a sweeping change that one scientific society said “would all but end the use of scientific merit in the selection of grants and programs across the government.”

Proposed in late May, the rule would give political appointees unprecedented control over federal grants for research, education and infrastructure, and specifies that government funds can only be spent on projects “aligned with administration policies and priorities,” according to a copy of the proposed rule.

The rule would also restrict research topics, limit U.S. scientists’ ability to collaborate with colleagues in other countries and make it easier for the government to suspend or cancel grants at any time.

The changes are intended to improve “transparency, accountability, and oversight for Federal awards” while “ensuring that American tax dollars are not wasted or misused,” according to the White House office.

But critics say that if the rule is implemented, the final sign-off for grants will no longer be in the hands of subject-matter experts within individual agencies, but in those of political appointees.

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“This touches all parts of American life,” said Dr. Eric Rafla-Yuan, a psychiatrist who practices at the Veterans Administration and San Diego County’s psychiatric hospital.

“Control of how all of the federal grants and programs are funded will fall under a small group of highly partisan individuals who would have very few limits on how they spend these billions of taxpayer dollars,” said Rafla-Yuan, who also chairs the Committee to Protect Public Mental Health advocacy group. “This touches everyone’s life, even if they don’t realize it.”

OMB published the proposed rule May 29, opening a 45-day comment period that closes July 13.

Opposition to the proposed rule has mobilized multiple sectors of society. Professional groups representing cancer researchers, civil engineers, county governments, medical schools, housing agencies, city and municipal governments, nonprofits and others have publicly expressed concerns about potential consequences.

By midday Thursday, the Federal Register logged nearly 100,000 comments about the proposal, many of them expressing concern.

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“I understand the need for oversight, fiscal responsibility, and accountability. That is not the issue,” wrote Jack Feldman, a neuroscientist who holds the David Geffen School of Medicine Chair in Neuroscience at UCLA. “The issue is whether scientific research is to be judged by scientific merit, or whether it can be approved, denied, or terminated according to broad political criteria that may change from one administration to the next.”

Crucially, the rule converts policies governing federal grants from “guidance” into binding regulations that all agencies would be required to follow. It would give political appointees power to override federal agencies’ merit-based reviews and mandate that a political appointee review decisions to ensure that all awards “demonstrably advance the President’s policy priorities.”

The elevation of political appointees in what were previously merit-based decisions has alarmed many scientists.

“The proposed rule changes would all but end the use of scientific merit in the selection of grants and programs across the government,” read a statement from the Planetary Society, a nonprofit dedicated to space research.

Researchers and science groups have also expressed concern about a section of the rule prohibiting the promotion of “theories of disparate-impact liability” — a legal concept that refers to policies that appear neutral but cause disproportionate harm to certain groups.

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The section’s vague language and many loopholes could have a chilling effect on any research that studies the effects of a disease, policy or public health intervention on any specific group of people, Rafla-Yuan said.

As an example, he said, “if there’s a specific age range that is at higher risk for suicide, and we want to figure out, well, what’s going on with people that are aged 14 to 19 … we can’t do that under the wording in this rule.”

New restrictions on collaborations with scientists in other countries would hinder opportunities for U.S. researchers and limit innovation, said Joanne Padrón Carney, chief government relations officer for the American Assn. for the Advancement of Science.

“Science is a global enterprise. Especially in biomedical and public health fields, diseases don’t care about borders or government policies,” she said.

California’s congressional delegation sent a letter Wednesday asking OMB to rescind the proposal, outlining concerns about its impact on scientific innovation, U.S. competitiveness and the fiscal stability of local governments, many of which rely on federal grants for local services.

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The proposed rule grants the federal government broad powers to suspend or cancel grants for any reason, introducing “unprecedented unpredictability into local governance,” the lawmakers wrote, “leaving vital infrastructure projects unfinished and abandoning vulnerable populations who rely on these services.”

Republican Sen. Susan Collins has also asked the White House to withdraw certain parts of the letter and extend the public comment period, saying the proposed rule as written would “harm small and rural communities, undermine scientific and biomedical research, and conflict with Congress’ control over the federal funding process.”

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