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What the Golden Ratio Says About Your Bellybutton

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What the Golden Ratio Says About Your Bellybutton

A yellowish-green apple with a short brown stem, centered on a light blue background.

Math, Revealed

What an apple, a pentagram and a bellybutton have in common.

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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June 16, 2025

A yellowish-green apple with a short brown stem, centered on a light blue background.

Hiding inside every apple

A yellowish-green apple, seen from directly above, with its stem prominent, on a light blue background.

is a little bit of secret geometry.

A yellowish-green apple, cut in half to show its core and seeds arranged in a star shape, on a light blue background.

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To reveal it, cut the apple sideways, straight through the core, like this:

A close-up of a halved apple’s core, showing five dark seeds in a star shape, on a light blue background.

There you’ll find five seeds in the shape of a star.

Five yellow pencils are arranged in a five-pointed star shape on a light blue background.

In its idealized form, this kind of five-pointed star is known as a pentagram.

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In mathematics, the pentagram is a poster child for “self-similarity,” a symmetry that’s like worlds within worlds. The shape contains infinitely many smaller and smaller copies of itself, like nested Russian dolls.

A five-pointed star made of yellow pencils is centered on a light blue background. Five shorter black pencils appear one at a time connecting the points of the yellow star.

To see self-similarity in action, imagine connecting the star’s points with straight lines.

A five-pointed star is formed by yellow pencils with five black pencils connecting the points. The pentagram formed in the center of the star by its overlapping lines is highlighted in blue. All on a light blue background.

The newly created pentagon immediately calls attention to a smaller pentagon nestled inside itself.

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A five-pointed star is formed by yellow pencils with five black pencils connecting the points. The pentagram formed in the center of the star by its overlapping lines is highlighted in blue. All on a light blue background.

A famous number known as the golden ratio describes the proportions of the smaller parts to the whole.

The parts add up: a blue segment and a yellow segment, laid end to end, are exactly as long as a black segment. Or: small plus medium equals large.

A five-pointed star is formed by yellow pencils with five black pencils connecting the points. Below the star formation, two additional pencils—one all black and one yellow and blue—are arranged horizontally.

Moreover, the parts are in the same proportion: medium is to small as large is to medium.

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That common proportion defines the golden ratio, which is approximately 1.618.

Illustration of a red and grey skeletal dodecahedron on aged paper, hanging from a red pushpin, set against a blue background.

In 1509, an Italian mathematician named Luca Pacioli ascribed cosmic significance to the golden ratio in his book “On the Divine Proportion.”

The book included many illustrations by his friend Leonardo da Vinci, including this 3-D shape (a dodecahedron) made from identical pentagons.

Leonardo da Vinci’s “Vitruvian Man” illustration on aged paper, depicting a male figure in two overlaid stances with arms and legs apart, inscribed within a circle and a square. Pinned with a blue pushpin to a light blue background.

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Years earlier, Leonardo had performed his own study of proportions, based on the theories of the ancient Roman architect Vitruvius.

This led him to draw an iconic image known as “Vitruvian Man.”

Did he hide the golden ratio in it?

Illustration of Leonardo da Vinci’s “Vitruvian Man” showing a single standing figure with outstretched arms touching the edges of a square, pinned to a blue background.

This much we know: With his arms spread wide, Vitruvian Man fits perfectly in a square — his wingspan equals his height.

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Illustration of Leonardo da Vinci’s “Vitruvian Man” with arms and legs spread, inscribed in a circle, pinned to a blue background.

With arms and legs splayed, he also stands comfortably on the circumference of a circle, which his middle fingers extend to touch.

At the center of it all is his navel.

Illustration of Leonardo da Vinci’s “Vitruvian Man” with arms and legs spread, inscribed in a circle, pinned to a blue background.

Leonardo’s handwritten notes specify many of the man’s proportions as fractions of his height:

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“From the hairline to the bottom of the chin is one-tenth of the height of the man.”

“From below the chin to the top of the head is one-eighth of the height of the man.”

And so on.

Leonardo da Vinci’s “Vitruvian Man” illustration on aged paper, depicting a male figure in two overlaid stances with arms and legs apart, inscribed within a circle and a square. Pinned with a blue pushpin to a light blue background.

Yet nowhere does Leonardo quantify the location of the man’s navel.

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This omission seems surprising, given the navel’s centrality in his scheme.

Some commentators claim that Leonardo positioned the navel to divide the man’s height according to the golden ratio.

But Leonardo doesn’t mention the golden ratio, either in the drawing or in his notebooks.

Leonardo da Vinci’s famed “Vitruvian Man” illustration on aged paper, depicting a male figure in two overlaid stances with arms and legs apart, inscribed within a circle and a square. Pinned with a blue pushpin to a light blue background.

In fact, a 2015 analysis by the architect Vitor Murtinho found that the placement of the Vitruvian man’s navel does not quite comport with it.

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The man’s height in the drawing is 181.5 millimeters and the height of the navel is 110 millimeters, for a ratio of 1.65 to 1.

That’s close to the golden ratio (1.618 to 1), but surely Leonardo could have come closer if he’d meant to.

A round, bright pink balloon with its tied end facing forward, centered on a light blue background.

Is it possible that Leonardo really thought 1.65 was the correct anatomical proportion for a well-shaped human being?

Zoomed into the same round, bright pink balloon with its tied end facing forward, centered on a light blue background.

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And what, in fact, is the typical number? Presumably, it falls on a bell curve — or should we say bellybutton curve.

An inflated pink balloon fills the screen. The gathered, stretched latex of the balloon’s opening forms a textured knot in the center.

As for what’s most desirable, proponents of the golden ratio insist on a divine proportion. The ideal navel should divide the upper abdomen from the lower abdomen, not in half but in a ratio of 1.618 to 1.

A close-up of a white mannequin’s navel, an “innie,” on its smooth torso.

Indeed, a 2015 study published in the journal Aesthetic Plastic Surgery asked participants to select the most attractive navel position on digitally altered pictures of bikini models, and found that the golden ratio was ideal.

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A white mannequin’s hand holds a yellowish-green apple in front of its torso, against a light blue background.

On the other hand, a 2022 eye-tracking study in the Journal of Plastic Surgery and Hand Surgery, in which volunteers looked at digitally altered images of a female patient, found that a 2 to 1 ratio of upper to lower abdomen was more pleasing than the golden ratio.

A white mannequin’s hand holds a yellowish-green apple in front of its torso, against a light blue background.

Divine proportions? Meet diverse ones — they can be beautiful too.

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Health concerns mount as Boyle Heights warehouse fire stretches into a week

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Health concerns mount as Boyle Heights warehouse fire stretches into a week

Tens of thousands of people in southeast Los Angeles County have been engulfed in a dense cloud of smoke for nearly a week as a fire continues to tear through a massive refrigerated warehouse in Boyle Heights. Toxic air has covered the San Gabriel Valley and beyond at times, as the fire continues to burn and the wind shifts the pall in different directions.

People have reason to be concerned about their loved ones breathing in the plume, experts say.

“There’s no safe level of exposure to particle pollution,” said Will Barrett, assistant vice president for nationwide clean air policy at the American Lung Assn.

Soot can be deadly. The charred microscopic particles can travel deep into a person’s lungs and bloodstream, causing swelling and triggering heart attacks and strokes.

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People aren’t just being exposed for hours. They’ve been exposed for days in Boyle Heights, unincorporated East Los Angeles, Maywood, Montebello and Bell, according to the South Coast Air Quality Management District.

“There are some pollutants where just breathing in a little bit of it can cause some serious issues for people,” said Dr. Afif El-Hasan, a pediatrician with Kaiser Permanente. He said he’s most concerned about particles, carbon monoxide, volatile organic compounds and chemical gases from incinerated insulation, plastics and paint in the smoke.

“Those chemicals can cause irritation in the lungs, they can cause long-term lung damage, and sometimes they can even cause cancer,” he said. “I also worry about children, because children breathe in more air per volume of their body than adults do and they tend to be more active.”

“People also need to remember that even if you are healthy, these chemicals are going to put you at risk. It’s not just people who are vulnerable, anyone is in danger.”

The fact that the smoke continues to billow into the sky for a sixth day matters, said Jill Johnston, associate professor of environmental and occupational health at UC Irvine. “The longer the exposure time, the more dose you’re getting, or the more potential chemicals that you’re inhaling. So you’re gonna be increasing a potential risk,” she said.

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Pregnant women and their babies in utero are known to be vulnerable to smoke from wildfires, she said. But less is known about city fires. “We see increased risk of low birth weight and preterm birth connected to exposure to wildfire smoke. This isn’t exactly the same composition of smoke, but would anticipate … there could be potentially similar risk.”

A fire like this can leave people with no good choices. They can stay home with an air filter if they have one. But homes need “fresh” air, and a fire can make getting that impossible.

For that reason, some people believe that the official response to the gravity of the fire at Lineage Logistics has been inadequate. Jane Williams, executive director of California Communities Against Toxics, is among several activists who criticized the Los Angeles Fire Department and city officials who appeared to downplay health risks from prolonged smoke, and ultimately decided against evacuating these areas. They think many more people should have been evacuated.

“They always under-warn, they under-evacuate, they bring people back too fast,” Williams said. “I get that there’s a societal desire to return to normalcy.”

Local officials have opened a pair of shelters to house residents who want to temporarily relocate. The Los Angeles Unified School District also canceled summer programming for schools in the smoke-affected communities.

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But “there is nothing in the air that is so dangerous that we have to do evacuations or even shelter in place,” LAFD Chief Jaime Moore said. Asked at a recent news conference whether the air was dangerous, Mayor Karen Bass said, “not to the extent that required a mandatory evacuation.”

Yet Williams pointed to the burning chemical-laden insulation foam inside the building, which could release several other highly toxic gases, including hydrogen cyanide, an asphyxiating gas, and isocyanates, chemical vapors that can cause serious lung damage.

“It’s about what you value and who you value,” Williams said. “If you value truth, you cannot sit there in front of a burning building and say the air is safe.”

A Fire Department spokesperson declined to comment when asked why the department considered a shelter-in-place order more appropriate than issuing an evacuation. It’s not clear that evacuation would have been purely a city responsibility. Lineage Logistics sits along the city boundary, with unincorporated Los Angeles County and other cities nearby.

mark! Lopez, a community organizer with East Yard Communities for Environmental Justice, also said the recently lifted shelter-in-place orders were not enough to protect residents from the heavy smoke and potential chemical releases. Residents, he said, have complained about smoke seeping into homes through cracks in doorways and windows, giving them sore throats and breathing problems.

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Lopez said many of the smoke-affected communities have long suffered from poor air quality from decades of heavy polluting industrial facilities, highway traffic and rail yards. He said the public statements from Fire Department and elected officials that cast doubt on the risks from smoke were unacceptable.

“This is what happens when the Fire Department says there’s not a threat to human health. … The LAFD, they aren’t public health experts.”

Times staff writer David Zahniser contributed to this report.

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Here’s why the Lincoln Memorial Reflecting Pool went green so fast

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Here’s why the Lincoln Memorial Reflecting Pool went green so fast

Just days after the Trump administration completed millions of dollars in renovations on the Lincoln Memorial Reflecting Pool to make it American flag-blue, residents and online users noted it had turned a phosphorescent green.

Here’s why:

The calm, still waters of the Reflecting Pool make it an ideal nursery for algae growth. Algae need nitrogen and phosphorus to grow, and the Reflecting Pool is primarily fed by the Potomac River, which gets heavy doses of those nutrients from nearby urban and agricultural lands.

The Potomac also absorbed one of the largest sewage spills in U.S. history earlier this year when a pipe burst five miles upstream of Washington, although that event probably happened too long ago to contribute to the algal bloom today.

Untreated sewage is high in nitrogen and phosphorus. When nutrient levels are high, feasting algae can quickly reproduce.

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The Department of the Interior said when the algae first appeared that it was “residual,” from the supply lines to the pool.

Experts also speculate that the darker blue color may be helping the Reflecting Pool absorb more heat. The higher temperatures promote algae growth by allowing their metabolisms to shift into overdrive.

Summer temperatures in D.C. aren’t helping. This week, temperatures are as high as 95 degrees in the city, prompting a heat alert.

The combination probably explains the excessive growth, turning the water surface an opaque green and preventing onlookers from seeing the new blue hue of the concrete basin.

Algae are important and beneficial organisms when the ecosystem is in balance. They’re the base of the aquatic food chain, fed on by herbivores of all shapes and sizes, including shrimp and juvenile fish, which in turn feed organisms higher up the food chain. The single-celled organisms use the power of the sun to produce energy through photosynthesis, similar to houseplants on your balcony.

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In an effort combat the algae in the Reflecting Pool, employees of the National Park Service were seen pouring in gallons of hydrogen peroxide, a chemical commonly used in pool maintenance.

The Department of the Interior also is employing a “high-tech nanobubble ozone technology” to destroy the cells of the algae.

Ozone — yes, the same irritant that is in smog — is a gas composed of three oxygen molecules, and the small size of the bubbles allow the most gas transfer into the water, where it can damage algal cells, similar to how it irritates our lungs.

This only treats the symptoms, however. Generally, ozone nanobubbling is effective as a temporary solution for algae blooms. Longer-term fixes would have to address what makes the Reflecting Pool so ideal for algae, such as its depth, darker color and inflow of nitrogen and phosphorus.

In California, ozone nanobubbles also have been used in a project to improve water quality in the Tijuana River. The 120-mile river that runs near the border in northern Mexico and Southern California was the site of a pilot study in 2025. The U.S. section of the International Boundary and Water Commission reported that the nanobubbling reduced “odors and bacteria,” but the project concluded prematurely after a flood swept some of the instrumentation into the river.

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This plant extract can make a lethal drug cocktail. Can it also treat opioid addiction?

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This plant extract can make a lethal drug cocktail. Can it also treat opioid addiction?

A plant extract that’s gaining popularity as a pain cure-all and has been associated with multiple California deaths in its concentrated, synthetic form has been approved for research as a treatment for opioid addiction by the federal government.

Kratom is derived from the leaves of Mitragyna speciosa, a tree native to Southeast Asia, and is commonly made into a powder or pill.

Researchers say people in the U.S. are using kratom to alleviate anxiety, treat chronic pain or as a remedy for the symptoms associated with quitting opioids, due to its ability to bind with opioid receptors in the body. But recently, public health officials have raised alarms about a component of the leaf called 7-hydroxymitragynine, also known as 7-OH, an alkaloid that has the potential for abuse and addiction in high doses.

Last year, the Los Angeles County Public Health Department linked the deaths of six county residents to the use of 7-OH mixed with other substances. The toxicology screens for some of the deceased revealed both kratom and 7-OH, leading to a countywide crackdown of products with either compound because they’re unregulated.

Although there is no scientific consensus on whether kratom has therapeutic value, the Food and Drug Administration has recommended that its potent 7-OH form be classified as a controlled substance. Consumers who use 7-OH as a pain reliever expecting an experience similar to consuming kratom are at risk, said Dr. Mason Turner, president-elect of the California Society of Addiction Medicine.

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“I have a couple of patients that I work with who use 7-OH for chronic pain management, not realizing the potential of the medication, and then developed an opioid use disorder,” Turner said. “I think in that case it was very clear they were seeking it for the chronic pain, not to get high, not to have some kind of experience, but really to reduce their pain.”

About two decades ago, Turner said, the healthcare industry started acknowledging the limits and risks of prescribing opioids for chronic pain. Some doctors pulled back on prescriptions, recognizing the potential for abuse.

That led some patients to find alternative solutions, he said.

“Maybe they don’t get a good benefit, or maybe the benefit from some of the other treatments is not as robust as what they got from opioids,” Turner said. “So they seek out some of these illicit products … or they look for kratom or 7-OH to be able to mitigate the pain.”

Turner said he supports further research into kratom and regulation because “it could be worth exploring as a treatment for chronic pain.”

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On June 1, the National Institutes of Health announced that researchers from the University of Florida would begin the first phase of clinical trials on kratom to evaluate it as a potential treatment for opioid addiction. The research would be done with the FDA’s approval, according to officials.

“This … is a major step toward expanding treatment options for the millions of Americans struggling with opioid use disorder, which has contributed to historically high overdose mortality rates,” said Dr. Nora Volkow, director of NIH’s National Institute on Drug Abuse, in a statement.

Interest in kratom surged in the last couple of years as users have reported consuming the compound in the form of a pill, powder or tea to treat various ailments. A John Hopkins survey conducted in 2020 reported that 91% of respondents used kratom to treat chronic pain, 67% to treat anxiety, 64% for depression and 41% to treat opioid dependence.

A more recent study by the University of Michigan and Texas State University found that more than 5 million people in the U.S., including more than 100,000 children ages 12 to 17, have used kratom, the compound experts say is growing in popularity with young adults.

In the study, which analyzed data from the National Survey on Drug Use and Health collected between 2021 and 2024, researchers say that despite numerous state-level bans on kratom across the nation, its use is at an all-time high and is increasing.

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People between the ages of 21 and 34 said they used kratom at least once and 1% said they used it in the last year. The share of children ages 12 and older who said they had used kratom increased from 1.6% in 2021 to 1.9% in 2024.

The FDA has stated that neither kratom nor 7-OH are approved as drug products, dietary supplements or food additives, but that hasn’t stopped storefronts and companies from selling them as such.

Up until November you could find kratom and 7-OH products in smoke shops and specialty stores in California, but that has stopped.

“Until kratom and its pharmacologically active key ingredients mitragynine and 7-OH are approved for use, they will remain classified as adulterants in drugs, dietary supplements and foods,” the California Department of Public Health told The Times via email.

Kratom “Feel Free Classic” liquid products are displayed at a smoke shop in Los Angeles in 2024 before they were banned.

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(Michael Blackshire / Los Angeles Times)

In May, the California Department of Public Health and Atty. Gen. Rob Bonta filed a complaint against Ashlynn Marketing Group Inc., accusing the company of repeatedly flouting the state’s regulations on kratom products.

The filing, submitted in the San Diego County Superior Court, seeks a judge’s order to condemn and destroy the embargoed kratom products, halt ongoing unlawful manufacturing and impose civil penalties.

The California Department of Public Health “is pursuing legal action because Ashlynn’s continued manufacture and sale of these products pose a clear and preventable public‑health risk and violates state and federal law,” said Dr. Erica Pan, the department’s director and state public health officer. “7-OH and kratom-derived products have been associated with addiction, serious health harms, overdose and death.”

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The state is alleging its inspectors visited Ashlynn Marketing Group’s facility in Santee in May 2025 and found kratom powders, capsules, liquids and chewable tablets being manufactured and held for sale.

During the visit, inspectors issued an embargo to prohibit the sale and distribution of all kratom-related materials on-site, according to the complaint.

Public health inspectors conducted follow-up visits at the facility in October and April, “collecting evidence at both inspections that indicated embargoed kratom products had been moved, tampered with and repackaged,” according to public health officials.

“In addition, investigators observed evidence of continued manufacturing and distribution of kratom materials,” officials said. “The firm’s owner continues to manufacture kratom products and ships orders weekly.”

To date, the California Department of Public Health has seized more than $5 million worth of kratom and 7-OH products, a spokesperson for the department told The Times.

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California and Los Angeles County are considering whether to tighten regulations or ban the compounds altogether.

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