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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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More middle-class Californians cancel health coverage after losing federal aid

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More middle-class Californians cancel health coverage after losing federal aid

Facing higher premiums and the loss of federal subsidies, 374,000 people with health insurance from the state marketplace known as Covered California canceled their coverage in the first three months of the year, according to government statistics.

The cancellations amount to 19% of those who had renewed their policies on the state marketplace during open enrollment, state officials said. Those cancellations are higher than in the past three years when they ranged from 13% to 15% of those who renewed.

Jessica Altman, executive director of Covered California, attributed the jump in cancellations to the expiration of enhanced federal subsidies that caused the cost of a plan to leap for most middle-class Californians.

“We expect coverage losses to increase through the year,” she said.

Overall, Covered California had 1.8 million enrollees in February, down from 1.94 million the year before — a decline of 7%.

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Altman said monthly enrollment numbers are delayed because consumers have a three-month grace period to resume their premium payments before the insurance carriers end their coverage for nonpayment.

This year, many middle-class Californians who depend on the state-run insurance marketplace created under the Affordable Care Act faced annual costs that were hundreds of dollars higher than last year because of the end of enhanced federal subsidies that began during the COVID-19 pandemic.

In 2021, Congress voted to temporarily boost the amount of subsidies Americans could receive for an ACA plan.

The law also expanded the program to families who had more money. Before that 2021 vote, only Americans with incomes below 400% of the federal poverty level — currently $62,600 a year for a single person or $128,600 for a family of four — were eligible for ACA subsidies. The 2021 vote eliminated the income cap and limited the cost of premiums for those higher-earning families to no more than 8.5% of their income.

On top of the loss of the enhanced federal subsidies, the average premium charged by insurers this year for a Covered California plan rose by more than 10% because of fast-rising medical costs.

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The decline in ACA plan enrollees, however, has been greater in some other states. California has tried to keep people insured by using state tax money to fill in the gap for lower-income families.

This year, the state budgeted $190 million for premium subsidies for people with incomes of up to 165% of the federal poverty level.

In his budget plan, Gov. Gavin Newsom proposed spending $300 million on those state subsidies in 2027. That would expand the subsidies to enrollees with incomes up to 200% of the federal poverty level, or $31,920 for an individual or $66,000 for a family of four.

“We may actually see a number of Covered California enrollees paying less in 2027” because of the additional state subsidies, Altman said.

In May, Newsom also proposed in his budget that an additional $27 million in state money be used to help enrollees pay for the cost of gender-affirming care. That amount is an increase to the $30 million that he earlier proposed be spent this year and next to defray those costs for Covered California enrollees, according to state officials.

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Last year, federal health officials enacted a rule that said the federally subsidized ACA plans could no longer cover gender-affirming care because it was no longer considered an “essential health benefit.”

Newsom’s proposed budget still faces debate in Sacramento and approval by the state Legislature.

The state marketplaces, created by the Affordable Care Act, also known as Obamacare, were meant to help those who don’t have access to an employer’s health insurance plan and have incomes too high to qualify for Medi-Cal, the government-paid insurance for the poor and disabled.

Because of the higher cost this year, more people are choosing the lower-priced Bronze plans. Those plans have higher co-pays and deductibles than the more expensive plans.

“We’re very concerned with the large shift to Bronze,” Altman said. “When you have higher cost-sharing, you’re more likely to defer care.”

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Political play or budget fix? Competition for JPL’s management comes at a fraught moment

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Political play or budget fix? Competition for JPL’s management comes at a fraught moment

Weeks after Trump administration officials announced that management of NASA’s Jet Propulsion Laboratory would open to competitive bidding for the first time, questions remain as to why Caltech could lose control of the lab its researchers founded in 1936.

On one hand, observers note, high-profile delays and cost overruns on significant recent JPL projects earned sharp criticism from NASA even before the 2024 presidential election.

On the other, the second Trump administration’s record of squeezing scientific funding and attacking institutions in Democrat-led states make it difficult to consider any action separate from the charged political atmosphere, analysts say.

“My first instinct is that this [competition] isn’t necessarily a bad thing. It’s not written in stone that Caltech must run JPL, and it wouldn’t be the worst thing to have some competition for running the place,” said Casey Dreier, chief of space policy at the non-profit Planetary Society.

“That said, that requires this contract evaluation to be fair and unbiased, and this administration has no credibility in such things,” he added. “The responsibility is on NASA to earn the trust and ensure such an evaluation is open and free from political meddling. That’s almost impossible.”

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JPL became part of NASA when the space agency was formed in 1958, and Caltech has been awarded the contract to run the institution outright ever since.

Its current 10-year contract with NASA, which is valued at up to $30 billion, runs through Sept. 30, 2028.

NASA Administrator Jared Isaacman announced the competition on May 22 as part of a slate of sweeping organizational changes at the space agency.

“When you step back, it is worth considering how many additional missions we could have undertaken with the resources lost to program cancellations and cost overruns over the years,” Isaacman wrote in a memo to staff. “That is the problem we must fix, so the American taxpayer and space-loving community can receive the highest scientific return on every dollar we spend at NASA.”

Competing the contract for JPL, the lone Federally Funded Research and Development Center (FFRDC) in NASA’s portfolio, was an effort to address cost-efficiency concerns, Isaacman wrote.

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“This process will take several years, and I do not anticipate it having any impact on the projects underway or the location of the facilities,” he wrote. “It does, however, provide an opportunity to evaluate management costs, overhead burdens, and ideally find ways to get after the science faster and more affordably.”

In a joint statement, Caltech President Thomas F. Rosenbaum and JPL Director Dave Gallagher said the competition was “no surprise” and that a team was already in place “to ensure we are positioned for success.”

In July, NASA’s Office of Procurement held an informational event for companies and institutions interested in the upcoming FFRDC contract.

The dozens of registered attendees included universities like USC, Texas A&M University and Georgia Tech, aerospace companies such as Boeing and Lockheed Martin and nonprofit corporations like MITRE, which manages several FFRDCs, and Universities Space Research Association, a university consortium founded by the National Academy of Sciences in 1969. (SpaceX, which has been awarded more than $13 billion in NASA contracts in the last decade, was not on the list.)

“Lockheed Martin has more than 50 years of deep space exploration success with JPL, supporting landmark missions to Jupiter, Venus, Saturn, Pluto, including nearly a dozen missions to Mars,” said Bob Behnken, VP of Exploration and Technology Strategy. “We look forward to building on that unmatched partnership in the years ahead. We are closely following NASA’s review and will continue to assess how we can best contribute to the agency’s mission.”

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Other attendees contacted by The Times declined to discuss their involvement.

Isaacman indicated that JPL could come under scrutiny even before he took over NASA. The billionaire entrepreneur referenced high costs at the La Cañada Flintridge institution in a memo prepared in advance of his confirmation hearings on his priorities for the space agency.

“Contract structure: Very expensive,” Isaacman wrote of JPL in a table outlining organizational issues at each of NASA’s centers. “Must increase the output and ‘time-to-science’ KPI.”

The institution has recently suffered a number of high-profile management stumbles.

After the JPL-managed Psyche mission to a metal-rich asteroid failed to meet its 2022 launch date, NASA commissioned an independent review that said internal reorganizations and personnel changes created distracted and uninformed managers and burned-out, stretched-thin staffers.

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After a 2023 independent review found there was “near zero probability” of the JPL-managed Mars Sample Return mission making its proposed 2028 launch date, and “no credible” way to bring rocks back from the Red Planet within the stated budget, Isaacman’s predecessor Bill Nelson put out a call for proposals to industry and all other NASA centers, forcing JPL to compete for its own project.

After Trump’s election, Nelson announced that the final decision would be in the next administration’s hands.

The White House pushed for massive cuts to NASA’s 2026 budget that Congress overturned, and has lobbied for similarly steep cuts again this year. JPL has instituted painful cost-cutting measures of its own, reducing staffing from roughly 6,500 employees in 2023 to 4,500 last year through layoffs and attrition.

Its struggles come at a point when NASA is enthusiastically embracing private industry. Last month the agency awarded several key contracts for its upcoming lunar missions to Jeff Bezos’s Blue Origin and other private companies.

Trump has also made no secret of his willingness to punish states that haven’t voted for him through job losses. In announcing his decision to move U.S. Space Command from Colorado to Alabama, Trump acknowledged that his loss in Colorado in three presidential elections played a part in the move.

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It’s impossible to consider any decision on JPL’s future separate from the administration’s track record of politically-motivated decisions, Dreier said.

“At the heart of this is why? Why now? If this is not just some rank political attack on California, what do they hope to gain from this?” Dreier said. “That deserves explanation, because the administration otherwise has no credibility here.”

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Dive Into a Very Noisy Sea With Some Very Rare Whales

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Dive Into a Very Noisy Sea With Some Very Rare Whales

The Gulf of Mexico, which the Trump administration calls the Gulf of America, is one of the noisiest bodies of water in the United States. Air gun blasts are the loudest element there, according to research by scientists who monitor underwater acoustics. Shipping traffic is another major contributor.

The noise could affect the ability of Rice’s whales to find food and mates, scientists say. The chronic stress of living in a loud environment could be detrimental to their health.

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