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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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China Launches Reusable Rocket in Race With SpaceX

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

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Nobel Prize winner leaving UC Berkeley for new role in China

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Nobel Prize winner leaving UC Berkeley for new role in China

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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