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Hidden Detail in Crotch Solves 500-Year-Old Leonardo Da Vinci Mystery : ScienceAlert

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  • Geometric Mystery Solved: A London dentist, Rory Mac Sweeney, believes he has discovered the reason behind Leonardo da Vinci's specific proportions in the Vitruvian Man drawing.

  • Equilateral Triangle: Mac Sweeney identified a hidden equilateral triangle in the crotch of the Vitruvian Man, which he suggests explains the proportions.

  • Tetrahedral Ratio: Calculations based on this triangle reveal a ratio close to 1.633, known as the tetrahedral ratio, which is associated with optimal packing and balance.

  • Connection to Dentistry: This ratio is also found in Bonwill's triangle, a principle used in dentistry since 1864 for optimal jaw function, suggesting a potential link to anatomical efficiency.

  • Universal Principles: Mac Sweeney proposes that da Vinci may have intuited fundamental geometric principles governing optimal spatial organization, reflected in both human anatomy and universal structures.


Leonardo da Vinci, the famous Italian polymath who painted the Mona Lisa, had a sophisticated geometric understanding way ahead of his time.

To draw the Vitruvian Man in 1490 – an illustration of the 'ideal' human body – the Renaissance man may have relied on a mathematical ratio not formally established until the 19th century.

It's one of the most iconic images of all time, and yet for more than 500 years, no one could figure out why da Vinci chose such specific proportions for the arms and legs.

A London dentist thinks he's solved the mystery at last.

Related: Four Ways Leonardo Da Vinci Was Well Ahead of His Time

Rory Mac Sweeney has found a crucial hidden detail, tucked in the Vitruvian Man's crotch: an equilateral triangle that he thinks may explain "one of the most analyzed yet cryptic works in art history."

The Vitruvian Man is partly inspired by the writings of Roman architect Vitruvius, who argued that the perfect human body should fit inside a circle and square.

Da Vinci's drawing uses a square to precisely contain a 'cruciform pose', with arms outstretched and legs in. The circle, meanwhile, encompasses a posture where the arms are raised and the legs are spread.

A popular explanation is that da Vinci chose the Vitruvian Man's proportions based on the Golden Ratio Theory, but the measurements don't quite match up.

According to Mac Sweeney, "the solution to this geometric mystery has been hiding in plain sight".

"If you open your legs… and raise your hands enough that your extended fingers touch the line of the top of your head… the space between the legs will be an equilateral triangle," da Vinci wrote in his notes for the Vitruvian Man.

When Mac Sweeney did the math on this triangle, he found that the spread of the man's feet and the height of his navel created a ratio of around 1.64 to 1.65.

That's very close to the tetrahedral ratio of 1.633 – a uniquely balanced geometric form, officially established in 1917.

The ratio is used to establish the optimum way to pack spheres. If four spheres are connected as closely as possible into a pyramid shape, for instance, then the height to base ratio from their centers will be 1.633.

Perhaps Mac Sweeney recognized the significance of that number because of a similar triangular principle used in dentistry.

Imagined on the human jaw, Bonwill's triangle dictates the optimal positioning for jaw function, used since 1864. Its ratio is also 1.633.

Mac Sweeney doesn't think that's a coincidence.

Similar to minerals, crystals, and other biological packing systems found in nature, Mac Sweeney thinks the human jaw naturally organizes around tetrahedral geometries, which maximize mechanical efficiency.

If the tetrahedral ratio is repeated around our bodies, Mac Sweeney thinks that is because "human anatomy has evolved according to geometric principles that govern optimal spatial organization throughout the universe."

If Mac Sweeney is right, Da Vinci may have stumbled across a universal principle while drawing the Vitruvian Man.

"The same geometric relationships that appear in optimal crystal structures, biological architectures, and Fuller's coordinate systems seem to be encoded in human proportions," writes Mac Sweeney, "suggesting that Leonardo intuited fundamental truths about the mathematical nature of reality itself."

Whether other scientists agree with Mac Sweeney remains to be seen, but the fact that da Vinci mentioned the equilateral triangle in his notes suggests that what lies between the Vitruvian man's legs is important.

The study was published in the Journal of Mathematics and the Arts.

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Contrary To The Poltical Narrative, A Lot Of Government Science Funding Doesn't Go To Science | Science 2.0

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  • Government vs. Private Sector: The article argues that government is not the primary driver of American scientific leadership, contrary to common media narratives.

  • Inefficient Government Spending: It criticizes government science funding for prioritizing self-promotion and administrative costs over actual research, citing examples of wasted funds on public relations and non-scientific projects.

  • Private Sector Innovation: The piece highlights that significant scientific advancements, such as the COVID-19 vaccine, were largely driven by the private sector, with individuals like Katalin Karikó facing initial lack of government support.

  • Funding Process Flaws: The author suggests that the government funding process is flawed, partisan, and often fails to recognize or support revolutionary scientific ideas, leading talented researchers away from academia.

  • Media Bias: The article contends that media reporting on science funding and political impact is heavily biased, often exaggerating the negative consequences of budget cuts, particularly concerning Republican administrations.


Every day we read a new headline warning us that American leadership is about to erode because of budget cuts to 'science.'

We have been told tuberculosis was about to be eliminated by a vaccine but a grant got cut and, gosh darn it, now Republicans ruined it. We have been told we'll be set back for generations. 

If you don't want to spend a lot of time reading more, I can tell you plainly that after two decades of covering science in a way 'post whatever is new every day to sell advertising' corporate media will not; government is never why America leads in science. Government is barely even why America leads in Nobel Prizes.



Here are the reasons we could be fine with even less funding and a Republican president. Some of these are even why academic scientists should take off their political blinders and support reforms - because broad cuts followed by justifying restoration of grants would lead to more funding for actual science.

A lot more.

1. Government has spent billions in science funding to engage in self-promotion to scientists. That should stop and the money should go toward science.

Government does not fund some science because the private sector won't, that's the ridiculous NPR/PBS argument.(1) As you will read in Item 2 they instead won't fund revolutionary science, except by mistake. Government instead funds science to guide scientists into funding what their government panels want, and to make that easier have spent $10 billion that should have gone to labs to engage in public relations campaigns alleging that only government-funded science was real science and if you go to that stinky private sector, you have to be unethical or hide inconvenient findings.

They have continued to do it knowing that for every academic scientist that gets a grant, seven will not. They will instead have earned a Ph.D. but will work in someone else's lab as a post-doctoral researcher for low pay. The government 'stay in school' campaigns have worked so well even some post-doc jobs stipulate you have to bring your own funding source. And they still get filled.

Most labs are run rather lean when it comes to the science itself. Government self-promotion used science funding that could have supported over 3,000 labs.

2. We won't lose leadership. Government has never funded the bulk of science research, it doesn't even fund the majority of basic research. What government does do that has value is give out visas to scientists, which is why America laps the field when it comes to Nobel prizes

If government wants to improve science leadership without wasting money, get out of the way. The original Trump presidency was actually quite good for science - we got a COVID-19 vaccine in record time thanks to the "DOGE" efforts of FDA Commissioner Scott Gottlieb, MD, to get rid of junk mechanisms that blocked progress. There was never any public benefit for FDA needing 18 months to approve a font color change on a drug label but it routinely happened.(4)

The COVID-19 vaccine came out fast but a key researcher behind it, Katalin Karikó, Ph.D., didn't have her own lab when she developed it. She couldn't get a government grant. She was working as a post-doc because the government didn't think her mRNA work was useful. The University of Pennsylvania not only refused to give her tenure, they demoted her because she believed in mRNA but the federal government did not.(2) She had to accept the pay cut and make less than the bench techs sitting near her because she had cancer and needed two surgeries when they told her she was being demoted.(3) They hoped humiliating her would make her quit. 

She instead has now won a Nobel prize while the University of Pennsylvania and the NIH that should have looked stupid and incompetent used science funding to take her work right to the publicity bank. They now do their best to take credit for it.

She is not special. The National Science Foundation also claims they basically invented Google.

She eventually left academia due to being jaded by the lack of interest in real science by government science funding panels. It is a pattern followed by many others who went on to happier lives. CNN nonetheless claimed we were going to lose leadership in science because disgruntled scientists sold a pipe dream by government and other academics were going to the private sector. Nearly a year before Trump took office

American science saved the world during COVID-19 but government had only an indirect relationship to that. The private sector did it while the Trump administration made government get out of the way.

We won't lose leadership with less science funding when most of it wasn't used for science anyway.

3. Much of government science funding does not go to science and never has. A lot of science funding is actually tied up in government administration on the front end and universities treating science labs they way they do undergraduate student loans on the back end; like piggy banks. Academic science labs are basically small businesses. A scientist does the work to get a grant and he pays for graduate students and techs and equipment plus nebulous university costs like electricity. Like sports, schools want science to be a profit center. Johns Hopkins spends an alarming amount of money they take from academics who got grants to recruit more academics who are proficient at getting government grants.

That doesn't help the public or even scientists.

Worse, they fund a lot of things you know aren't science. When Ebola was a concern 10 years ago, I wrote about how the NIH had refused to fund a vaccine trial for a modest amount. They didn't have the budget to do anything about Ebola at all, NIH Director Dr. Francis Collins told Congress, they would need hundreds of millions of dollars more. Yet with the over $300 billion they'd already gotten just since the Bush years, when they couldn't spend $10 million to fund an Ebola vaccine, they spent $10 million on a video game that was supposed to teach kids to eat less. The last thing an obese child needs is to play a video game about eating too much pizza. It didn't matter, the $10 million was wasted. No one has ever played the government-funded vaporware. Yet the NSF lists it as one of their success stories.

In 2010, the National Science Foundation once even gave two academics $500,000 to create a "Science 2.0." Despite us being a registered trademark in existence for years, NSF does so little thinking about grants it paid academics to violate United States copyright law. 


When I wrote the duo to ask about what their grant would accomplish the one who replied said he didn't know very much about Science 2.0 anyway, and the "expert" refused to respond.

Are we really losing out when government wastes taxpayer money that way?

No, nor do we need 'science' funding so sociologists can write about why people play Farmville (actual science funding went to that) or chelation therapy ("") or why political candidates make vague statements ("") or any of the hundreds and hundreds more examples funded each year.

We lose nothing losing garbage that isn't science but media only talks about high-profile, obvious work caught in the broad net before being restored.

But even if funding declines we won't lose leadership. It has happened before.

4. Science survived with less funding and a Democratic president who distrusted science a lot more than Trump; Bill Clinton. He is the president who gutted nuclear energy because his constituents hated science. He gave supplements a free pass. He diverted NIH funding from science to provide a 1000% budget increase to an alternative medicine outfit, National Center for Complementary and Alternative Medicine. He forced the US Department of Agriculture to create a marketing seal for Organic food, then made his own government exempt this new panel from USDA oversight. He prevented the Army Corps of Engineers from fixing the levees they wanted to fix because environmentalists objected, an anti-science decision that resulted in disaster when Hurricane Katrina hit.

NASA took a budget hit, so did the NIH.

All while he tightened the noose around medical companies. 

We didn't lose leadership because government didn't fund most science then any more than they do now. His team could ban nuclear research  and with a ban in place there was no sane reason for the private sector to pursue it, but that is the only reason why we haven't seen a giant leap in nuclear plants the way we did cell phones.

Government held nuclear back with a ban, not by funding too little. Government has instead spent the GDP of a small country subsidizing and mandating solar panels and they are no better than 15 years ago. 

America didn't want to fund the Large Hadron Collider because it was too limited. Academics and media insisted America was going to lose leadership in science due to Republicans then as well. Since it went online, 10 of the 15 Nobel Prizes in physics have gone to Americans. Letting Europe be the leader on the LHC clearly didn't ruin American physics.

5. The best thing for government-funded scientists is more awareness of how partisan and flawed the funding process really is. 

Fixing the government's problems, starting with, as Daniel Sarewitz, co-director of the Consortium for Science, Policy and Outcomes at Arizona State University wrote in Nature, realizing that institutional arrangements are often more important than peer review in claims about the social value of science, would double the funding that goes to science. Yet awareness of government control of academic science remains a challenge as long as academia is partisan, corporate media is partisan, and social media algorithms remain easy for well-funded political groups to manipulate. What is gelatin wrestling in Antarctica really accomplishing for the world? The private sector can handle researching group dynamics in EverQuest 2, government doesn't need to fund "studies" of that, or why people are on <a href="http://Match.com" rel="nofollow">Match.com</a> either. The company knows why they are there.

There is no indication that President Trump will fix the problems as long as Secretary Kennedy manages to not fall out of favor with him, government will instead shift to false problems he wanted to highlight when he was a lawyer opposed to science at Natural Resources Defense Council, like food coloring and pasteurized milk, but it's not even been a year so there is hope.

You just won't find much of  hope in newspapers. Doom sells. 

6. Media simply cannot be trusted when it comes to Republicans. Was tuberculosis about to be cured before Trump took office? It was if you believe corporate journalism and Democratic pundits.

We have heard the supernatural 'about to be cured if Republicans would get out of the way' claims many times before, like with human embryonic stem cells (hESCs). Democrats trotted out a grief-stricken former First Lady, Nancy Reagan, to claim if Bush would just stop banning hESC research, her beloved Ronnie's Alzheimer's could be cured.

There was one big problem with that, and it wasn't just that they exploited an elderly woman whom they had previously derided as a moron for believing in astrology. Clinton had banned hESCs, not Bush. The technology to make an hESC was created without federal funding because Clinton had banned NIH funding for anything modifying an embryo. When the technology was invented anyway, using state and private funds, he didn't want to create any more votes against him during his impeachment hearings his bipartisan ban he had just signed three years earlier was popular. So he stalled requests about using hESCs to let Al Gore deny it funding.

Except that didn't happen. George W. Bush won. President Bush thought it over, listened to all sides, and told the NIH to fund hESC research, but only using the existing lines while the ethics were debated and its legal standing under Clinton's law. Despite being the president who funded it, media claimed during the 2004 election he banned it. 

Even in 2008 with him out of the picture they couldn't be trusted. They washed away Senator Obama's anti-vaccine statements and his decision to renege on limiting himself to public financing the way his opponent did, they dismissed his desire to put Robert F. Kennedy, Jr. in charge of EPA and that he surrounded himself with conspiracy theorists. When he finally got into office in 2009 he did to hESCs what he did about Guantanamo Bay after making lots of campaign promises; basically nothing. His advisors said it was a very bad idea to undo the Clinton law so he added a few more lines to the ones Bush had already funded.(5)

Media are not suddenly going to become diverse any more than academic faculties are, one political side is too entrenched and it would take a generation of Affirmative Action for Republicans In Universities for them to get enough presence they could have a voice.

That is the beauty of science. That Harvard loses some science funding will not make a bit of difference because they do shockingly little science with it. We would actually benefit from less of what they do produce; papers using bad epidemiology to promote Miracle Diets or Scary Chemical narratives no one literate among the public believes.

NOTES:

(1) PBS and NPR simultaneously claim their government funding is too minor to be worried about and they are apolitical while telling advertisers they have the wealthiest, most liberal, best educated audience in media.

(2) Her academic boss before that, at Temple, got mad at her and threatened to have her deported. Academics are liberal when it comes to surveys but when it comes to their own labs, they are like any Republican construction company owner. 

(3) So much for the superior culture of academia. Her mental situation at Penn eventually improved, though not the financial one. Universities recruit "rainmakers" just like law firms do, and eventually they got Drew Weissman, and he got to know her and funded her drastically-reduced work from his own grant.

In 2005 they published a paper showing how an mRNA vaccine could work. Government panels still ignored them. She left academia, hoping she could monetize the patent. He was the co-recipient of their Nobel Prize. 

(4) Bureaucratic mechanisms still block out small companies with good products who can't raise enough money for a Phase III clinical trial due to FDA added costs that don't improve safety. So they sell to large companies. Epi-Pens should be $1 but despite needles and epinephrine being well-understood for 100 years, few want to spend $1.5 billion and 10 years just to sell cheaper than Mylan. The government keeps large pharmaceutical companies in place because small competitors can't afford the cost of government. 

(5) And it made no difference. hESCs never went anywhere while the induced pluripotent stem cells (iPSCs) the private sector embraced have led to lots of breakthroughs.
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A New Geometry for Einstein’s Theory of Relativity | Quanta Magazine

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  • Encounter: A chance meeting between mathematicians Clemens Sämann and Michael Kunzinger on a flight sparked a discussion.

  • Context: They were discussing limitations of Einstein's general theory of relativity.

  • Problem: Einstein's theory doesn't work where space-time isn't smooth, such as at black hole singularities.

  • Limitation: The breakdown is caused by the mathematics used, primarily differentiation.

  • Goal: The mathematicians aimed to develop alternative methods to overcome the limitations of calculus in non-smooth space-time environments.


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The sadist assault on the ‘Coldplay couple’ // We need a new ethic of privacy

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  • The incident involving the CEO and HR chief at a concert highlights a cultural obsession with public shaming and surveillance, amplified by social media.
  • This trend of leaking and doxing private citizens creates a pervasive atmosphere of surveillance, making everyone vulnerable to public judgment.
  • The author argues that focusing on such "contentless" stories distracts from more significant societal issues and powerful entities.
  • New technology enables and exacerbates pre-existing human tendencies for gossip and ridicule, leading to a moral warping effect.
  • There is a critical need for new societal norms and codes of conduct around technology use, emphasizing privacy and self-restraint to counteract digital "stonings."

I wish I didn’t know who Andy Byron and Kristin Cabot are. As you no doubt know by now, the Astronomer CEO and his firm’s HR chief were caught on the jumbotron at a Coldplay concert. He tried to sneak away, while she covered her face, but it was too late. Someone posted the footage online. A pile-up ensued. And their lives were turned upside down.

As I say, I wish I didn’t know who they are. And that’s because any of us might, for any particular reason, become Andy Byron or Kristin Cabot: flawed people in the wrong place at the wrong time. Anybody can make a bad-but-irrelevant-to-the-rest-of-us mistake on camera and face personal destruction. What such episodes reveal is a pathology in the social body, and the desperate need for a new ethic of privacy.

The “Coldplay-concert-couple” story is fundamentally irrelevant and effectively contentless, telling us nothing about the world, about ourselves; it isn’t particularly funny or tragic or profound. There is no pathos, no insight. True, the current culture on X and other social-media apps is perhaps less immediately destructive than it was during peak woke (circa 2017 to 2021). Still, the viral potential of these kinds of stories is a warning sign that our culture is obsessed with shame, surveillance, and control. An obsession with other people’s private lives is a sickness.

Leaking, doxing, spying on private citizens on Reddit, on private Facebook groups, on X, on Instagram, on TikTok — regardless of the moral quality of the subjects under consideration — hurts everyone. Because it makes everyone subject to this omnipresent, collectivized panopticon, and the more these images circulate, the more normal it becomes.

Group chats, dating apps, emails — or in Byron’s and Cabot’s case, simply being in public with no intention to be filmed — should not be subject to public judgement. This species of story is a distraction from serious issues, and runs cover for the genuinely powerful individuals and entities who materially impact our lives.

Clearly, there is something in human nature that is primed to gossip, shame, and ridicule (thoughtlessly, automatically). But new technology allows us to give vent to these sordid tendencies in historically unprecedented ways. We should take seriously the moral warping effect that such stories, and our obsession with them, have on our souls.

The global village is like any village — scolding, punitive, consumed by hearsay, with nothing better to do. While turning someone into a meme might seem fun, it’s severely undignified, and discards the customs and safeguards of a liberal society in order to participate in a sadistic pile-on. So while the particular example at hand might not seem worth this much deliberation, that is my point.

There might be a story here about the hypocrisy of a CEO and an HR officer having an in-house affair that might get lesser-ranking employees fired. But what I see surfacing in the discursive arena is not intelligent critique of HR departments and their culture, but a glee at the immiseration of a random person.

If these trends continue — if algorithms continue to work the way they do, and I can’t imagine they won’t — then more and more victims will be dragged out of anonymity for mimetic sacrifice. And as a consequence, meaningful taboos about privacy, shame, and respect have disappeared, and everyone’s lives and behaviors have become fodder for comment and ridicule.

We sorely need new taboos — not against adultery or sexual immorality or interpersonal deception, because clearly we have plenty of those already. Rather, we need taboos against aggregating ourselves into idiotic, loathsome mobs, incapable of thinking about what the subjects of our ridicule and these digital stonings might suffer. A wiser perspective would admit that we all, at the wrong moment, are liars, fools, cheaters, charlatans, hypocrites. Let who has not sinned click the “like” or “repost” button.

“If our natural response to constant mutual surveillance is not disgust, then it should be.”

We all sin, and we might have received grace or mercy under the theocracies of older liberal and Enlightenment arrangements. Algorithmic justice and judgment, however, contains neither the virtues of theocratic society — you could at least avail yourself of religious law, with its exceptions and casuistry — nor of classically liberal society: there is no due process, no appeal mechanism. There’s no mediating structure in the global village. It is instinctive, animalistic, rash, cruel.

The force and direction of this collective online judgment should be turned against itself: We should punish not those caught in the panopticon, but the anonymous posters, Redditors, and the like; all who take joy in random acts of scorn. If we want to live in a tolerant, sane society, we should turn off our cameras, step away from mimetic digital village stonings, and reflect on our own haphazard and imperfect lives and selves.

This “should” will not come from above: there is no sign that tech companies will change their products to disincentivize this kind of behavior. Self-regulation, at scale, is, as of now, the only answer. The way we talk, write, act, and respond matters. Culture matters insofar as culture is a layer of software that at least partially governs how we use our phones. We sorely lack codes of conduct surrounding technology and our ability to record, dox, and harass each other.

If our natural response to constant mutual surveillance is not disgust, then it should be; phones come with cameras, but cameras linked to the global cortex are weapons. Using them to film strangers should not be normalized — telling people to put their phones away should. Piling on viral posts about random people who had no intention of going viral should not be normal, either. If Big Tech won’t take them down, then self-restraint is called for. Institutions can help uphold and reinforce these new norms by prohibiting phone usage in as many public spaces as possible.

There are secondary benefits, too: public spaces are also just better without everyone being on their phones; 50,000 people all filming the same concert, for instance, aside from having unintended consequences, is stupid and unaesthetic. Each of us doesn’t need to take our own badly framed photos of Gustav Klimt’s “The Kiss” when we visit Vienna. There are professional photos and postcards widely available. We could employ the faculty of memory, too.

Codes of conduct are what we might call counter-mimetics; they do have to be injected into social interaction and discourse, artificially at first, before they catch on. That’s what I’m proposing here. We might think something is amusing and worth commenting on or bad and worth scolding, and maybe, in isolation it is. But on the whole, we’re making life intolerable, anxious, and graceless.

It took several centuries for mounted knights to develop the code of conduct known as chivalry to regulate their unparalleled ability to kill or threaten anyone who didn’t have a horse, mail, and sword; let’s hope we can develop codes of conduct around our phones or whatever next-generation AI devices replace phones.

Privacy is a hard-won social good; give it back to others to protect your own.


Matthew Gasda is a playwright, author, and columnist for UnHerd, based in New York City.

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Iran Is Out to Assassinate Trump - WSJ

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  • Iran's officials regularly threaten President Trump's life; as long as the regime exists, so will the threat.

  • Ayatollah Ali Khamenei restated that 'Death to America' means death to Trump.

  • Mohammad-Javad Larijani stated that Trump has taken actions that will make it impossible for him 'to sunbathe in Mar-a-Lago'.

  • Following the drone strike that killed Qassem Soleimani, Iranian officials issued an arrest warrant and offered a bounty for Trump's death.

  • Leading Shiite clerics have labeled Trump with terms that invite violence, and after Trump mocked Khamenei, he was condemned as a mohareb who merits execution.


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The Hidden War Over Ukraine’s Lost Children | TIME

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  • Main topic: The article discusses the issue of Ukrainian children abducted and taken to Russia during the ongoing war.

  • Putin's response: Putin showed little interest in the list of missing children and the Kremlin views their abduction as not urgent.

  • Return efforts: While some children have returned, thousands remain in Russia, with efforts hampered by Russian propaganda and political complexities.

  • U.S. involvement: The U.S. initially supported tracking the children but cut funding, and there are differing views on whether to make the children's return a condition of peace talks.

  • International response: The International Criminal Court has issued an arrest warrant for Putin, and Zelensky continues to push for the children's return through various means.


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