Latest news with #geometry
Gizmodo
3 days ago
- Science
- Gizmodo
Meet Meschers, MIT's Tool for Building Paradoxical Digital Objects
Meet 'impossibagel,' a physically impossible bagel that mathematicians use to resolve intricate geometry problems. But impossibagel—and other 'impossible objects' in mathematics—is notoriously difficult to replicate, and researchers haven't been able to fully tap into their mathematical potential. That may no longer be a problem, thanks to a new tool. On Monday, researchers at MIT's Computer Science and Artificial Intelligence Laboratory (CSAIL) announced 'Meschers,' software capable of visualizing an intricate, 2.5-dimensional representation of impossible objects. In addition to creating aesthetically quirky objects, Meschers could eventually assist in research across geometry, thermodynamics, and even art and architecture, according to the researchers. The paper, set for publication in ACM Transactions on Graphics, will be presented at the SIGGRAPH Conference next week. 'Impossible objects are constructions which cannot exist in real life and instead only exist in our perception,' Ana Dodik, study lead author and PhD student at MIT, told Gizmodo in an email. 'They are interesting because our visual system can make sense of them to some extent, but they cannot exist in real life without bending or cutting them.' Meschers (an amalgam of 'meshes' and the artist M.C. Escher) offers a way to represent impossible shapes that is 'consistent with our perception and that lets us do familiar '3D' processing operations on them, despite them not being 3D,' she explained. For the program, Dodik and her colleagues wove together calculus and certain aspects of the human visual system. For instance, when looking at an impossible object like the Penrose Triangle, our eyes search for something called 'local consistency,' or, simply, the parts of this inconsistent shape that 'make sense' to us. For the Penrose Triangle, those parts are its three L-shaped corners. Separately, these sections make sense, but when we try to connect them as a globally consistent shape, things don't quite add up. As such, recreating impossible objects in real life requires the object to be cut or bent, Dodik said. For computational purposes, this gave rise to issues that potentially interfered with 'geometry operations, such as distance computation,' according to the researchers. Meschers addresses this complication by significantly relaxing the consistency requirement on the global scale. Instead, the program focuses on replicating the locally consistent subsections of the object. The program also supports different lighting conditions, which can influence how well the impossible depths of these objects are represented in computational renders. The resulting shape, according to the paper, is sufficient for 'a wide variety of classic geometry processing algorithms…in a way that aligns with our perceptual intuitions.' For instance, the impossibagel would be an ideal structure for geometry researchers calculating the distance between two points on an incomprehensible surface. Alternatively, it could be used to determine how heat spreads over curved surfaces. Meschers also allows the user to inversely render an impossible object, such as deforming a torus (donut-like shape) into a Penrose Triangle, making it easier for artists or architects to play around with these unusual shapes for their own purposes. Perhaps most importantly, it's a way to do math that's both fun and insightful. And it's also great for decorating coffee mugs, as study senior author Justin Solomon shared with Gizmodo. 'Our research group at MIT is the Geometric Data Processing (GDP) group, so I had this image printed on coffee mugs for all our team members,' he said. 'It's a perfect logo for our group!' Meschers is a publicly available resource, and the code will be released shortly, the researchers say. So yes, you'll soon be able to make an impossibagel for yourself—not to eat, but to stimulate the mathematical part of your brain!
Yahoo
14-07-2025
- Health
- Yahoo
Da Vinci's Vitruvian Man Encodes the Blueprint of Human Evolution, Says London Dentist
New research reveals a hidden geometric ratio in Leonardo's drawing that may define the endpoint of human evolution toward upright posturePhoto Courtesy of Rory Mac Sweeney LONDON, July 14, 2025 (GLOBE NEWSWIRE) -- A London-based dental surgeon and researcher, Dr. Rory Mac Sweeney of Precision Endodontics, has ignited international interest with a provocative reinterpretation of Leonardo da Vinci's Vitruvian Man. His newly published research suggests the Renaissance masterpiece encodes a precise geometric ratio that reflects the biomechanical endpoint of human evolution. Dr. Mac Sweeney's theory, published in the Journal of Mathematics and the Arts, identifies a hidden equilateral triangle embedded in Vitruvian Man. He connects this geometry to Bonwill's Triangle—a fundamental dental structure first described in the 19th century, which governs optimal jaw alignment and function. According to Mac Sweeney, this triangular structure appears throughout the body and is mathematically anchored by the ratio √8/3, or approximately 1.633. 'Leonardo's drawing isn't just a study in proportion—it's a map of tension,' said Dr. Mac Sweeney. 'The 1.633 ratio appears in the jaw, the spine, and the skull. It reflects a state known as vector equilibrium, where structural tension and compression are perfectly balanced. I believe this marks the final step in the human journey toward full upright posture.' This ratio, derived from the geometry of the cuboctahedron, is widely recognized in biomechanics and architecture as a hallmark of tensegrity—the balance of forces within a stable form. Dr. Mac Sweeney contends that this geometry defines the Vitruvian Morphotype: a form that nature has converged on through evolutionary pressures—not because it is aesthetically pleasing, but because it is structurally optimal. 'Human evolution has been a long progression toward uprightness,' he explained. 'The 1.633 ratio may represent our evolutionary omega point—a structural threshold beyond which no further anatomical adaptation is needed to stand, move, and balance efficiently in gravity.' Mac Sweeney suggests that fossil evidence should reveal a slow convergence toward this geometric configuration, particularly in the jaw. He highlights the emergence of Class I occlusion in the fossil record—also known as the overbite/overjet 'step'—around 8,000 years ago as a key moment. While small variations remain, he argues that modern Homo sapiens are the first species to fully express this morphotype. 'It's like the hydrodynamic form of a dolphin,' he said. 'Nature solves gravity the way it solves water. Vitruvian Man is the first full sketch of what that solution looks like.' Dr. Mac Sweeney's theory is now attracting attention from experts in evolutionary biology, bioengineering, and anatomical design. He is currently conducting interviews, public talks, and media appearances to discuss what he calls the Vitruvian Ratio—and its broader implications for anthropology, architecture, and even consciousness. 'Leonardo, somehow, saw it coming,' Mac Sweeney added. 'Vitruvian Man may be the only anatomical diagram ever created that captures not just what a human is—but what a human is becoming.' About Dr. Rory Mac Sweeney: Dr. Mac Sweeney is a London-based endodontist and researcher known for his interdisciplinary work in dental anatomy, geometry, and human evolution. His work explores the convergence of structural biology and classical design, and he is the author of the book The Paradox of Lucid Dreaming. @luciddreamyoga Contact Information: Contact person: Dr. Rory Mac SweeneyCompany name: Precision EndodonticsWebsite: [ Contact: rorymacsweeney@ A photo accompanying this announcement is available at in retrieving data Sign in to access your portfolio Error in retrieving data Error in retrieving data Error in retrieving data Error in retrieving data
Irish Times
06-07-2025
- Science
- Irish Times
The Da Vinci dentist: west Cork man finds secret of 500-year-old drawing in the jaw
A vitally important triangle, hidden in plain sight in Leonardo da Vinci's Vitruvian Man, has been revealed and explained, more than half a millennium after the Renaissance genius drew it. Is there an Irish angle to this, you ask? Yes there is: it was dentist from west Cork who found it. Sometimes these things write themselves. Rory MacSweeney, who practices in London but comes originally from Dunmanway, explained his discovery and the relevance of the tooth trade in an article published in the Journal of Mathematics and the Arts. It goes something like this: everyone knows the famous Da Vinci drawing situates a man within a circle and a square and shows ideal proportions. But there was a third shape hiding in the drawing – an equilateral triangle created by the man's legs, and mentioned by Da Vinci himself. This shape has gone largely ignored by scholars, but MacSweeney says it was no accident and adds insight to our understanding of the drawing's meaning. READ MORE The equilateral triangle made by the figure's legs recall similar proportions in the jaw It's the same shape – stay with us here – as the triangle between the corners of the jaw and the place the teeth meet. Hence the dentist angle. It also happens to also be the ratio you see in superstrong crystals and other things in nature. Da Vinci, ever the engineer, found an 'architectural common denominator', MacSweeney told Overheard. MacSweeney's interest comes back to the question of evolution. Very few other animals have crooked teeth like humans do – 'something went wrong', he said, and he's interested in the maths of how we evolved. 'Humans have evolved over various iterations,' he said. 'We've tried out various versions of bipedalism. We reached a point when we're fully erect – it's when you reach the 1.633 ratio.' 'It's not just that we're the only humans to survive. There's an X factor with homosapiens. We're hugely successful, more dominant than anything else in nature.' The ratio isn't just an ideal, from an engineering perspective – 'it's a destination', the optimal situation our evolution has worked towards. Curiously, MacSweeney points out, the Vitruvian Man as drawn could spin on an X, Y or Z axis – which is to say, after a fashion, that he can breakdance. Plugged into high places Martin Naughton, founder of Glen Dimplex. Photograph: Alan Betson It is easier for a camel to pass through the eye of a needle than for a rich man to enter the kingdom of God, as someone once said. But it looks like it can be done: Dundalk electrical appliance mogul Martin Naughton, the owner of Glen Dimplex, has been awarded the Order of St Gregory the Great, adding a Papal knighthood to his British KBE and his French Legion d'Honneur. The industrialist was recognised alongside his wife Carmel for their philanthropy in education and the arts, including scholarships and supports for university students at home and abroad. The conferral was one of the last acts of Pope Francis, 'who himself studied in Ireland at one point and for whom peace and education were especially close to his heart', said John McCaffrey, president of the Association of Papal Orders in Ireland. The Naughtons join the starry ranks of the knighthood alongside British Tory-turned-Reform politician Ann Widdecombe, Argentinian caudillo Juan Perón and Australian-American media magnate Rupert Murdoch. They are among about 60 Irish people who currently hold one of the five orders. Such honours were perhaps more prominent in Irish life in the past, with names such as Hume, Hillery and Lemass adorned by holy acronyms. The tenor John McCormack had three papal knighthoods as well as being a count. But the Naughtons should be warned: Fine Gael TD Oliver J Flanagan's efforts to style himself Sir Oliver after receiving the honour in 1978 prompted consternation from the government – which considered doing so potentially unconstitutional – and the Church itself, which confirmed that the 'sir' stuff is more of a British thing anyway. RTÉ retires Conor McGregor Conor McGregor: RTÉ News has referred to him as a 'former MMA fighter' Another blow to Conor McGregor in the High Court this week as the decorated-but-dormant UFC star was forced into retirement by the state broadcaster. In broadcasts and online, RTÉ News referred prominently to him as a 'former MMA fighter'. McGregor is appealing the High Court's finding that he is civilly liable for the rape of Nikita Hand. The 36-year-old hasn't fought since 2021, when he was stopped by Dustin Poirier at UFC 264. A comeback in the summer of 2024 was cancelled due to a training camp toe injury. After some talk of a 2025 return, UFC head honcho Dana White said in May that he wouldn't be fighting for him 'any time soon', and with the High Court appeal and an apparent desire to run for president occupying McGregor's time, there are no independent extravaganzas in the books either. But if there was an announcement that he is formally retired, we missed it. Perhaps RTÉ simply saw the writing on the wall. It's unlikely he would have given them the scoop: he once described them as 'robbing, lying, fake hypocrites', 'robbers of the Irish public' and even 'ooh laa laa heads' (snobs) in an online rant. Golf diplomacy Edward S Walsh: The new US ambassador to Ireland Whoever inherits the Áras will have a neighbour who's also new to the area: Edward S Walsh, United States ambassador to Ireland. The businessman and member of the Trump Bedminster golf club in New Jersey presented his credentials to the Irish Government this week, introducing himself to the Irish public with a video. What did we learn? He's Irish-American, like every ambassador since Lyndon Johnson sent a polo-playing second cousin of Winston Churchill's. His roots are in Clare. He doesn't read phonetic renderings of the phrase 'go raibh maith agat' especially well, although that will come with practice. And he likes golf. [ New US ambassador to Ireland takes office, thanking friend Donald Trump Opens in new window ] In fact, he tells us in the video that 'as a lifelong golfer, I've learned a lot about fairness, diligence and building relationships, values that I will bring with me into this position'. It will be interesting to see which aspects of golf help him keep the Irish-US relationship special while the two nations go in opposite directions on Israel. 'Ireland, while often a valuable US partner, is on a hateful, anti-Semitic path that will only lead to self-inflicted economic suffering,' the Senate Foreign Relations Committee chairman Jim Risch threatened on social media website X this week in the wake of coverage of the Government's Occupied Territories Bill. 'If this legislation is implemented, America will have to seriously reconsider its deep and ongoing economic ties.' Ambassador Walsh might have dreamt of cherishing Irish links, so to speak, but he faces a period in the rough first.
Daily Mail
03-07-2025
- Science
- Daily Mail
The real Da Vinci Code is SOLVED after 500 years: Dentist cracks geometric secret hidden in Vitruvian Man
Some 500 years ago, Leonardo da Vinci sketched what he believed was the perfectly proportioned male body. The drawing, called the Vitruvian Man, is one of the most famous anatomical drawings in the world. The complex interplay of art, mathematics and human anatomy has puzzled scientists for hundreds of years. But now, a London-based dentist claims to have worked out the secret to how da Vinci perfectly placed the human figure inside a circle and a square. Dr Rory Mac Sweeney, a qualified dentist with a degree in genetics, says the key to unlocking the drawing's geometric code lies in the use of an 'equilateral triangle' between the man's legs, mentioned in manuscript notes that accompany the drawing. The researcher discovered this isn't just a random shape – and in fact reflects the same design blueprint frequently found in nature. Analysis reveals this shape corresponds to Bonwill's triangle, an imaginary equilateral triangle in dental anatomy that governs the optimal performance of the human jaw. This suggests da Vinci understood the ideal design of the human body centuries before modern science, Dr Sweeney said. Dr Sweeney said the key to unlocking the drawing's geometric code lies in the specific mention of an 'equilateral triangle' drawn between the man's legs (left) which corresponds to a design blueprint found in nature - including the human jaw (right) When this triangle is used to construct the drawing it produces a specific ratio between the size of the square and the circle. Dr Sweeney has discovered that this ratio – 1.64 – is almost identical to a 'special blueprint number' – 1.6333 – that appears over and over again in nature for building the strongest, most efficient structures. This same number is found in the geometry of a perfectly functioning human jaw, the unique proportions of the human skull, the atomic structure of super-strong crystals and the tightest way to pack spheres. 'We've all been looking for a complicated answer, but the key was in Leonardo's own words,' Dr Sweeney, who graduated from the School of Dental Science at Trinity College in Dublin, said. 'He was pointing to this triangle all along. What's truly amazing is that this one drawing encapsulates a universal rule of design. 'It shows that the same "blueprint" nature uses for efficient design is at work in the ideal human body. 'Leonardo knew, or sensed, that our bodies are built with the same mathematical elegance as the universe around us.' According to the dentist, the discovery is significant because it shows that Vitruvian Man is far more than just a beautiful piece of art. Leonardo da Vinci is also known for his magnificent artworks such as the Mona Lisa, which hangs at the Louvre Museum in Paris (pictured) Who was Leonardo da Vinci? Leonardo di ser Piero da Vinci, more commonly Leonardo da Vinci, is one of the greatest minds of the last millennium. The polymath was a driving force behind the Renaissance and dabbled in invention, painting, sculpting, architecture, science, music, mathematics, engineering, literature, anatomy, geology, astronomy, botany, writing, history, and cartography. He has been attributed with the development of the parachute, helicopter and tank. He was born in Italy in 1452 and died at the age of 67 in France. After being born out of wedlock, the visionary worked in Milan, Rome, Bologna and Venice. His most recognisable works include the Mona Lisa, the Last Supper, Vitruvian Man. Another piece of artwork, dubbed the Salvator Mund, sold for a world record $450.3 million (£343 million) at a Christie's auction in New York in 2017. Instead, it is the work of 'scientific genius that was centuries ahead of its time', he said. The pen-and-ink drawing of the nude male in two different poses, with arms and legs enclosed within a circle and square, was created by the Renaissance polymath around 1490. It was partly influenced by the writings of Roman architect Marcus Vitruvius Pollio, who believed the human body has harmonious proportions. He proposed that a human figure could fit perfectly inside a circle and a square but provided no mathematical framework for achieving this geometric relationship. Da Vinci, who solved the puzzle, never explicitly explained how. For the last 500 years scientists have come up with numerous theories and ideas but none have matched the actual measurements. The study, published in the Journal of Mathematics and the Arts, reads: 'For over 500 years, Leonardo da Vinci's geometric system for establishing the precise relationship between the circle and square in his Vitruvian Man drawing has remained a mystery. 'This paper demonstrates that Leonardo's explicit textual reference to 'an equilateral triangle' between the figure's legs provides his construction method and reveals the anatomical foundation for his proportional choices. 'The analysis shows that Leonardo's equilateral triangle corresponds to Bonwill's triangle in dental anatomy—the foundational geometric relationship governing optimal human jaw function.' It concludes: 'The findings position Vitruvian Man as both artistic masterpiece and prescient scientific hypothesis about the mathematical relationships governing ideal human proportional design.' Scientists have previously compared the Vitruvian Man with nearly 64,000 physically fit men and women and discovered da Vinci was to anatomical measurements of the modern-day human. The team found the groin height, shoulder width and thigh length of today's measurements were 10 percent within those of the Vitruvian Man. However, the head height, arm span, chest and knee height are slightly more than da Vinci's estimates. THE MANY ACCOMPLISHMENTS OF LEONARDO DA VINCI Leonardo da Vinci is best known for his stunning artwork but the Italian Renaissance painter had many talents. He was also a sculptor, architect, musician, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist, and writer. The Mona Lisa is his most famous and most parodied portrait while his painting of The Last Supper is the most reproduced religious painting of all time. Leonardo's drawing of the Vitruvian Man is also regarded as a cultural icon - being reproduced on items as varied as the euro coin, textbooks, and T-shirts. Only around fifteen of his paintings survive because of his constant, and frequently disastrous, experimentation with new techniques, and his chronic procrastination. Leonardo is also revered for his technological ingenuity. He conceptualised a helicopter, a tank, concentrated solar power, a calculator, and the double hull, also outlining a rudimentary theory of plate tectonics. Relatively few of his designs were constructed or were even feasible during his lifetime, but some of his smaller inventions, such as an automated bobbin winder and a machine for testing the tensile strength of wire, entered the world of manufacturing unheralded. He made important discoveries in anatomy, civil engineering, optics, and hydrodynamics, but he did not publish his findings and they had no direct influence on later science.
The Independent
30-06-2025
- Science
- The Independent
London dentist discovers hidden detail in Da Vinci's Vitruvian Man
A London -based dentist, Rory Mac Sweeney, discovered a hidden equilateral triangle within Leonardo da Vinci 's 'Vitruvian Man' drawing. This triangle, found between the man's legs, matches Bonwill's triangle, a modern anatomical concept explaining efficient human jaw mechanics. The discovery, published in the Journal of Mathematics and the Arts, suggests da Vinci understood ideal human body design centuries before modern science. The triangle's presence helps create a 1.64 ratio between the square and circle in the artwork, closely mirroring a natural 'blueprint number' of 1.633. This finding indicates 'Vitruvian Man' is a scientific work demonstrating a universal rule of design, linking human anatomy to the mathematical elegance found in nature.
