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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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What to plant (and what to remove) in California’s new ‘Zone Zero’ fire-safety proposal

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What to plant (and what to remove) in California’s new ‘Zone Zero’ fire-safety proposal

After years of heated debates among fire officials, scientists and local advocates, California’s Board of Forestry and Fire Protection released new proposed landscaping rules for fire-prone areas Friday that outline what residents can and can’t do within the first 5 feet of their homes.

Many of these proposed rules — designed to reduce the risk of a home burning down amid a wildfire — have wide support (or at least acceptance); however, the most contentious by far has been whether the state would allow healthy plants in the zone.

Many fire officials and safety advocates have essentially argued anything that can burn, will burn and have supported removing virtually anything capable of combustion from this zone within 5 feet of houses, dubbed “Zone Zero.” They point to the string of devastating urban wildfires in recent years as reason to move quickly.

Yet, researchers who study the array of benefits shade and extra foliage can bring to neighborhoods — and local advocates who are worried about the money and labor needed to comply with the regulations — have argued that this approach goes beyond what current science shows is effective. They have, instead, generally been in favor of allowing green, healthy plants within the zone.

The new draft regulations attempt to bridge the gap. They outline more stringent requirements to remove all plants in a new “Safety Zone” within a foot of the house and within a bigger buffer around potential vulnerabilities in a home’s wildfire armor, including windows that can shatter in extreme heat and wooden decks that can easily burst into flames. Everywhere else, the rules would allow residents to maintain some plants, although still with significant restrictions.

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The rules generally do not require the removal of healthy trees — instead, they require giving these trees routine haircuts.

Once the state adopts a final version of the rules, homeowners would have three years to get their landscaping in order and up to five years for the bigger asks, including removing all vegetation from the Safety Zone and updating combustible fencing and sheds within 5 feet of the home. New constructions would have to comply immediately.

The rules only apply to areas with notable fire hazard, including urban areas that Cal Fire has determined have “very high” fire hazard and rural wildlands.

Officials with the Board will meet in Calabasas on Thursday from 1 p.m. to 7 p.m. to discuss the new proposal and hear from residents.

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Some L.A. residents are championing a proposed fire-safety rule, referred to as “Zone Zero,” requiring the clearance of flammable material within the first five feet of homes. Others are skeptical of its value.

Where is the Safety Zone?

The proposed Safety Zone with stricter requirements to remove all vegetation would extend 1 foot from the exterior walls of a house.

In a few areas with heightened vulnerabilities to wildfires, it extends further.

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The Safety Zone covers any land under the overhang of roofs. If the overhang extends 3 feet, so does the Safety Zone in that area. It also extends 2 feet out from any windows, doors and vents, as well as 5 feet out from attached decks.

What plants would be allowed in the Safety Zone?

Generally, nothing that can burn can sit in the Safety Zone. This includes mulch, green grass, bushes and flowers.

What plants would be allowed in the rest of Zone Zero?

Homeowners can keep grasses (and other ground-covers, like moss) in this area, as long as it’s trimmed down to no taller than 3 inches.

The rules also allow small plants — from begonias to succulents — up to 18 inches tall as long as they are spaced out in groups. Residents can also keep spaced-out potted plants under this height, as long as they’re easily movable.

What about fences, trees and gates?

Any sheds or other outbuildings would need noncombustible exterior walls and roofs in Zone Zero — Safety Zone or not.

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Residents would have to replace the first five feet of any combustible fencing or gates attached to their house with something made out of a noncombustible material, such as metal.

Trees generally would be allowed in Zone Zero. Homeowners would need to keep any branches one foot away from the walls, five feet above the roof and 10 feet from chimneys.

Residents would also have to remove any branches from the lower third of the tree (or up to 6 feet, whichever is shorter) to prevent fires on the ground from climbing into the canopy.

Some trees with trunks directly up against a house in this 1-foot buffer or under the roof’s overhang might need to go — since keeping branches away from the home could prove difficult (or impossible).

However, the board stressed it wants to avoid the removal of trees whenever feasible and encouraged homeowners to work with their local fire department’s inspectors to find case-by-case solutions.

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What’s new and what’s not

Some of the rules discussed in Zone Zero are not new — they’ve been on the books for years, classified as requirements for Zone One, extending 30 feet from the home with generally less strict rules, and Zone Two, extending 100 feet from the house with the least strict rules.

For example, homeowners are already required to remove any dead or dying grasses, plants and trees. They also have to remove leaves, twigs and needles from gutters, and they already cannot keep exposed firewood in piles next to their house.

Residents are also already required to keep grasses shorter than 4 inches; Zone Zero lowers this by an inch.

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Video: Rescuers Mount a Likely Final Push to Save a Stranded Whale

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Video: Rescuers Mount a Likely Final Push to Save a Stranded Whale

new video loaded: Rescuers Mount a Likely Final Push to Save a Stranded Whale

Rescue crews mounted a likely final push to save a stranded humpback whale off the coast of Northern Germany on Friday. The large mammal, nicknamed “Timmy,” captivated the nation after it was stranded in shallow waters for weeks.

By Jorge Mitssunaga

April 17, 2026

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1,200% jump in kratom-related calls to poison control centers in last decade, analysis shows

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1,200% jump in kratom-related calls to poison control centers in last decade, analysis shows

Over the last decade, poison control centers around the country have received tens of thousands of calls from consumers of kratom products reporting adverse and life-threatening health effects, with researchers saying reports in 2025 reached a new level. California’s poison center is reporting similar findings.

Last month, researchers analyzed information from the National Poison Data System and found that between 2015 and 2025, poison control centers across the nation received 14,449 calls related to kratom. More than 23% of those calls, or 3,434, were made last year, according to a published report in the Centers for Disease Control and Prevention. That represents a more than 1,200% increase from 2015, when only 258 calls were reported.

Officers gather illegally grown kratom plants in 2019 in Phang Nha province, Thailand. The country decriminalized the possession and sale of kratom in 2021.

(Associated Press)

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Kratom is derived from the leaves of Mitragyna speciosa, a tree native to Southeast Asia. It has a long history of being used for chronic pain or to boost energy and in the U.S., research points to Americans also using it to alleviate anxiety. In low doses, kratom appears to act as a stimulant but in high doses, it can have effects more like opioids.

But in the last few years, a synthetic form of kratom refined for its psychoactive compound, 7-hydroxymitragynine or 7-OH, has entered the market that is highly concentrated and not clearly labeled, leading to confusion and problems for consumers. The synthetic form gaining momentum in the market is sparking concern among public health officials because of its ability to bind to opioid receptors in the body, causing it to have a higher potential for abuse.

Los Angeles County leaders, meanwhile, have grappled with differentiating the two and regulating the products that come in the form of powder, capsules and drinks and have been linked to six county deaths. Sales of kratom and 7-OH products were banned in the county in November.

In reviewing the data, which did not differentiate whether callers had consumed natural or synthetic kratom, researchers set out to understand the effect of what they believe is a “rapidly evolving kratom market,” and highlight the role poison centers can play as an early warning surveillance system to detect new trends.

National Poison Data System findings

The data showed that over the last 10 years, 62% of the kratom-related calls to poison control centers were from people who said they consumed the drug by itself, and the other 38% were from people who combined it with another substance or substances.

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Those who consumed kratom with another substance combined it most frequently with one or a combination of the following: alcohol, opioids, benzodiazepines (like Xanax or Valium), cannabis and cannabinoids, stimulants and antidepressants.

The data also broke down hospitalizations related to kratom — adults who took it alone or in combination and experienced “adverse” health effects; and adults who took it alone or in combination and experienced more serious “moderate” or “major” health effects, including death.

Kratom powder products are displayed at a smoke shop.

Kratom powder products are displayed in a smoke shop in Los Angeles in 2024.

(Michael Blackshire/Los Angeles Times)

Hospitalizations for adults who had consumed kratom alone and experienced adverse effects increased from 43 in 2015 to 538 in 2025. For those who took it in combination and were hospitalized with an adverse health effect, the total jumped from 40 in 2015 to 549 last year.

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The numbers were even higher for hospitalizations where the health effects were more serious or fatal.

In 2015, there were 76 reports of people being hospitalized after taking kratom alone and experiencing a serious health effect or dying. By last year, that number had climbed to 919. The reports of serious health effects, including death, for those who took kratom in combination with another substance grew from 51 in 2015 to 725 last year.

The research does not break down kratom-related deaths by year but states that there were 233 deaths over the 10-year study period, or just over 3% of all 7,287 serious medical outcomes. Of the total number of kratom-related deaths, 184 cases involved the consumption of multiple substances.

What California’s poison control system found in its state data

The California Poison Control System is currently reviewing its data concerning kratom-related calls but an initial analysis shows parallels to the national report, said Rais Vohra, medical director of the state poison control system.

“We have about 10% of the national population and about 10% of the national call volume with poison control,” Vohra said. “And so, not surprisingly, we were able to identify over 900 cases of calls related to kratom in that same period.”

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Local researchers are still deciphering the state data but they too have found that kratom-related calls are climbing.

“It’s accelerating, which I think is one of the main points of the [published] report,” Vohra said.

A majority of calls received by poison control come from healthcare facilities where “presumably someone has a problem … severe enough to warrant calling 911 or going to the emergency room, and that’s when our agency gets involved,” Vohra said.

Kait Brown, clinical managing director for America’s Poison Control Centers, said the fact that kratom and 7-OH are federally unregulated products sold online, in gas stations and smoke shops gives people across the country easy access.

And while kratom enthusiasts maintain that it has been used in its natural form for hundreds of years, “there are new formulations that are a little bit different than how people have used it, at least historically,” said William Eggleston, a pharmacist and the assistant clinical director of the Upstate New York Poison Center in Syracuse.

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People are no longer consuming kratom only as a powder or capsule but also in the form of an energy shot or extract; it’s similar for synthetic, more concentrated 7-OH products.

When regional poison centers compare their findings and experiences with the analysis of calls in the National Poison Data System, Eggleston said, “undeniably there is an increase in calls related to kratom.”

“But when you put it in the bigger perspective of all the calls … this is still a very small percentage of what we’re dealing with on a day to day basis,” he said.

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