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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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The share of Americans medically obese is projected to rise to almost 50% by 2035

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The share of Americans medically obese is projected to rise to almost 50% by 2035

On Wednesday, a new study published in JAMA by researchers at the University of Washington in Seattle projected that by 2035, nearly half of all American adults, about 126 million individuals, will be living with obesity. The study draws on data from more than 11 million participants via the U.S. Centers for Disease Control and Prevention’s National Health and Nutrition Examination and Behavioral Risk Factor Surveillance System, and from the independent Gallup Daily Survey.

The projections show a striking increase in the prevalence of obesity over the past few decades in the U.S. In 1990, only 19.3% of U.S. adults were obese, according to the study. That figure more than doubled to 42.5% by 2022, and is forecast to reach 46.9% by 2035.

The study highlights significant disparities across states, ages, and racial and ethnic groups. While every state is expected to see increases, the sharpest rises are projected for Midwestern and Southern states.

For example, nationwide, by 2035, the study projects that 60% (11.5 million adults) of Black women and 54% (14.5 million) of Latino women will suffer from obesity when compared with 47% (36.5 million) of white women. Similarly, 48% (13.2 million) of Latino men will suffer from the disease compared with 45% (34.4 million) of white men and 43% (7.61 million) of Black men.

The findings say California will see similar trends in gender and racial disparities. The study projects that by 2035, obesity rates among Latino and Black women in California will reach nearly 60%, compared with nearly 40% for their white counterparts. Additionally, Latino men in California could see rates over 50%, compared with nearly 40% for their white counterparts.

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“These numbers are not surprising, given the systemic inequalities that exist,” in many California cities, said Dr. Amanda Velazquez, director of obesity medicine at Cedars-Sinai Hospital, pointing to economic instability, chronic stress and the car-dependency of Los Angeles and other California metro areas. “There are challenges for access to nutritious foods, depending on where you’re at in the city,” Velazquez said. ”There’s also disparities in the access to healthcare, especially to treatment for obesity.”

That’s recently become more of a challenge, since changes in Medi-Cal plans that went into effect at the beginning of this year mean obesity medication and treatment are no longer covered for hundreds of thousands of low-income Californians. “To take that away is devastating,” said Velazquez.

Despite these disparities, California is projected to fare better than most other states, with its rates of obesity growing more slowly than the national average.

“There are statewide and local policies that influence food, nutrition and social determinants of health for individuals,” said Velazquez.

Church pointed to measures such as SB 12 and SB 677, passed in the mid 2000s, which set strict nutritional standards for schools, existing menu labeling laws at both the state and federal levels requiring restaurants to provide nutritional facts on menu items, and cities like Berkeley and Oakland imposing local soda taxes as key local and statewide initiatives to keep obesity at bay.

To keep up this momentum, both doctors stressed that California must continue to strengthen school nutrition standards, expand transportation infrastructure that encourages walking instead of driving, maintain and expand economic disincentives to unhealthy foods, such as beverage taxes, and address food deserts by incentivizing new grocery stores and farmers’ markets in underserved neighborhoods.

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Future efforts, Church says, should prioritize the Black and Latino populations identified by the study as most affected.

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Pediatricians urge Americans to stick with previous vaccine schedule despite CDC’s changes

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Pediatricians urge Americans to stick with previous vaccine schedule despite CDC’s changes

For decades, the American Academy of Pediatrics and the U.S. Centers for Disease Control and Prevention spoke with a single voice when advising the nation’s families on when to vaccinate their children.

Since 1995, the two organizations worked together to publish a single vaccine schedule for parents and healthcare providers that clearly laid out which vaccines children should get and exactly when they should get them.

Today, that united front has fractured. This month, the Department of Health and Human Services announced drastic changes to the CDC’s vaccine schedule, slashing the number of diseases that it recommends U.S. children be routinely vaccinated against to 11 from 17. That follows the CDC’s decision last year to reverse its recommendation that all kids get the COVID-19 vaccine.

On Monday, the AAP released its own immunization guidelines, which now look very different from the federal government’s. The organization, which represents most of the nation’s primary care and specialty doctors for children, recommends that children continue to be routinely vaccinated against 18 diseases, just as the CDC did before Robert F. Kennedy Jr. took over the nation’s health agencies.

Endorsed by a dozen medical groups, the AAP schedule is far and away the preferred version for most healthcare practitioners. California’s public health department recommends that families and physicians follow the AAP schedule.

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“As there is a lot of confusion going on with the constant new recommendations coming out of the federal government, it is important that we have a stable, trusted, evidence-based immunization schedule to follow and that’s the AAP schedule,” said Dr. Pia Pannaraj, a member of AAP’s infectious disease committee and professor of pediatrics at UC San Diego.

Both schedules recommend that all children be vaccinated against measles, mumps, rubella, polio, pertussis, tetanus, diphtheria, Haemophilus influenzae type B (Hib), pneumococcal disease, human papillomavirus (HPV) and varicella (better known as chickenpox).

AAP urges families to also routinely vaccinate their kids against hepatitis A and B, COVID-19, rotavirus, flu, meningococcal disease and respiratory syncytial virus (RSV).

The CDC, on the other hand, now says these shots are optional for most kids, though it still recommends them for those in certain high-risk groups.

The schedules also vary in the recommended timing of certain shots. AAP advises that children get two doses of HPV vaccine starting at ages 9 to12, while the CDC recommends one dose at age 11 or 12. The AAP advocates starting the vaccine sooner, as younger immune systems produce more antibodies. While several recent studies found that a single dose of the vaccine confers as much protection as two, there is no single-dose HPV vaccine licensed in the U.S. yet.

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The pediatricians’ group also continues to recommend the long-standing practice of a single shot combining the measles, mumps and rubella (MMR) and varicella vaccines in order to limit the number of jabs children get. In September, a key CDC advisory panel stocked with hand-picked Kennedy appointees recommended that the MMR and varicella vaccines be given as separate shots, a move that confounded public health experts for its seeming lack of scientific basis.

The AAP is one of several medical groups suing HHS. The AAP’s suit describes as “arbitrary and capricious” Kennedy’s alterations to the nation’s vaccine policy, most of which have been made without the thorough scientific review that previously preceded changes.

Days before AAP released its new guidelines, it was hit with a lawsuit from Children’s Health Defense, the anti-vaccine group Kennedy founded and previously led, alleging that its vaccine guidance over the years amounted to a form of racketeering.

The CDC’s efforts to collect the data that typically inform public health policy have noticeably slowed under Kennedy’s leadership at HHS. A review published Monday found that of 82 CDC databases previously updated at least once a month, 38 had unexplained interruptions, with most of those pauses lasting six months or longer. Nearly 90% of the paused databases included vaccination information.

“The evidence is damning: The administration’s anti-vaccine stance has interrupted the reliable flow of the data we need to keep Americans safe from preventable infections,” Dr. Jeanne Marrazzo wrote in an editorial for Annals of Internal Medicine, a scientific journal. Marrazzo, an infectious disease specialist, was fired last year as head of the National Institute of Allergy and Infectious Diseases after speaking out against the administration’s public health policies.

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‘We’re not going away’: Rob Caughlan, fierce defender of the coastline and Surfrider leader, dies at the age of 82

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‘We’re not going away’: Rob Caughlan, fierce defender of the coastline and Surfrider leader, dies at the age of 82

Known by friends and colleagues as a “planetary patriot,” a “happy warrior” and the “Golden State Eco-Warrior,” Rob Caughlan, a political operative, savvy public relations specialist and one of the early leaders of the Surfrider Foundation, died at his home in San Mateo, on Jan. 17. He was 82.

His wife of nearly 62 years, Diana, died four days earlier, from lung cancer.

Environmentalists, political operatives and friends responded to his death with grief but also joy as they recalled his passion, talent and sense of humor — and his drive not only to make the world a better place, but to have fun doing it.

“He’d always say that the real winner in a surfing contest was the guy who had the most fun,” said Lennie Roberts, a conservationist in San Mateo County and longtime friend of Caughlan’s. “He was true to that. It’s the way he lived.”

“When he walked into a room, he’d have a big smile on his face. He was a great — a gifted — people person,” said Dan Young, one of the original five founders of the Surfrider Foundation. The organization was cobbled together in the early 1980s by a group of Southern California surfers who felt called to protect the coastline — and their waves.

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They also wanted to dispel the stereotype that surfers are lackadaisical stoners — and show the world that surfers could get organized and fight for just causes, said Roberts, citing Caughlan’s 2020 memoir, “The Surfer in the White House and Other Salty Yarns.”

Before joining Surfrider in 1986, Caughlan was a political operative who worked as an environmental adviser in the Carter administration. According to Warner Chabot, an old friend and recently retired executive director of the an Francisco Estuary Institute, Caughlan got his start during the early 1970s when he and his friend, David Oke, formed the Sam Ervin Fan Club, which supported the Southern senator’s efforts to lead the Watergate investigation of President Nixon.

According to Chabot, Caughlan organized the printing of T-shirts with Ervin’s face on them, underneath the text “I Trust Uncle Sam.”

“He was an early social influencer — par extraordinaire,” he said.

Glenn Hening, a surfer, former Jet Propulsion Laboratory space software engineer and another original founder of the Surfrider Foundation, said one of the group’s initial fights was against the city of Malibu, which in the early 1980s was periodically digging up sand in the lagoon right offshore and destroying the waves at one of their favorite surf spots.

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According to Hening, it was Caughlin’s unique ability to persuade and charm politicians and donors that put Surfrider’s efforts on the map.

Caughlan served as the foundation’s president from 1986 to 1992.

The foundation grabbed the national spotlight in 1989 when it went after two large paper mills in Humboldt Bay that were discharging toxic wastewater into an excellent surfspot in Northern California. The foundation took aim and in 1991 filed suit alongside the U.S. Environmental Protection Agency; the paper mills settled for $5.8 million.

Hening said the victory would never have happened without Caughlan.

The mills had tried to brush off the suit by offering a donation to the foundation, Hening said. But Caughlan and Mark Massara — an environmental lawyer with the organization — rebuffed the gesture.

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“The paper mill guys said, ‘Well, what can we do here? How can we make this go away?’” said Hening, recalling the conversation. “And Rob said, ‘It’s not going to go away. We’re not going away. We’re surfers.”

Roberts said Caughlan’s legacy can be felt by anyone who has ever spent time on the San Mateo County coastline. In the 1980s, the two spearheaded a successful ballot measure still protects the coast from non-agricultural development and ensured access to the beaches and bluffs. It also prohibits onshore oil facilities for off-shore facilities.

The two also worked on a county measure that led to the development of the Devil’s Slide tunnels on Highway 1 between Pacifica and Montara, designed to make that formerly treacherous path safer for travelers.

The state had wanted to build a six-lane highway over the steep hills in the area. “It would have been dangerous because of the steep slopes, and it would be going up into the fog bank and then back down out of the fog. So it was inherently dangerous,” Roberts said.

Chad Nelsen, the current president of the Surfrider Foundation, said he was first drawn into Caughlan’s orbit in 2010 when Surfrider got involved with a lawsuit pertaining to a beach in San Mateo County. Silicon Valley venture capitalist Vinod Khosla purchased 53 acres of Northern California coastline for $32.5 million and closed off access to the public — including a popular stretch known as Martin’s Beach — so Surfrider sued.

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Nelsen said that although Caughlan had left the organization about 20 years before, he reappeared with a “sort of unbridled enthusiasm and commitment to the cause,” and the organization ultimately prevailed — the public can once again access the beach “thanks to ‘Birdlegs.’”

Birdlegs was Caughlan’s nickname, and according to Nelsen, it was probably coined in the 1970s by his fellow surfers.

“He had notoriously spindly legs, I guess,” Nelsen said.

Robert Willis Caughlan was born in Alliance, Ohio, on Feb. 27, 1943. His father, who was a parachute instructor with the U.S. Army, died when Caughlan was 4. In 1950, Caughlan moved with his mother and younger brother to San Mateo, where he saw the ocean for the first time.

He rode his his first wave in 1959, at the age of 16, from the breakwater at Half Moon Bay.

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