Famously, at the quantum scale, particles can be in multiple possible locations at once. A particle’s state spreads out like a wave, peaking where the particle is likely to be found. When you measure its position, this spread-out state, known as a wave function, transforms into a single definite location.

The full shape of the wave function long evaded detection, since trying to measure it will destroy it. But in the 1980s, physicists began developing methods to measure and control the wave functions of simple systems — advances that have since formed the foundation of quantum computing. And in the last few years, a new approach is enabling physicists to go further and learn about the wave functions of entire materials.

“We are in the second quantum revolution,” said Riccardo Comin, an experimental physicist at the Massachusetts Institute of Technology who is one of the leaders of this work. “Now, we have the tools to really explore the wave function of quantum particles.”

The new framework describes a wave function as an object moving around a hidden landscape — a space referred to as the material’s “quantum geometry.” The hills and valleys of this invisible world dictate how the wave function of a given material can change, and what states the material can be found in.

“You can get a lot of insight into what’s happening in quantum materials, and that may speed up the discovery of new phenomena,” said Marc Bockrath, a physicist at Ohio State University who has also led the charge on quantum geometry.

Comin and collaborators recently measured the full quantum geometry of a crystal — a first peek into the wave function of a real material.

Let’s explore the hidden landscapes that are coming into view.

Secret Geometry

Physicists often think of a particle’s wave function as an arrow. If the particle can be found in two possible states, they represent these options as opposite directions for the arrow to point in — say, up and down. If the particle is in a combination of both states, then the arrow points somewhere around a sphere, with the two states corresponding to the poles.

Mark Belan/Quanta Magazine

The arrow’s direction captures the relative likelihood of each possibility. Measuring the particle will snap the arrow into pointing exactly up or down, and the chance of each outcome depends on which pole it’s closest to.

Many particles have more than two possible states, in which case the arrow occupies a high-dimensional space. That’s impossible to visualize, but the math gives physicists a sense of the particle’s wave function at a given moment.

For materials made of many particles, a single high-dimensional arrow can represent the combined state of all the electrons inside it. This collective arrow will swing around as you change the material’s ambient conditions — its temperature, for example, or the strength of the magnetic field around it. To control a material, physicists need to know how the arrow will rotate as they turn these various knobs.

To keep track, they build a map. Imagine, for example, that you change the strength of the magnetic field that you apply to the material. On your map, you’ll make the east-west direction correspond to that magnetic field strength. When the field is weak (corresponding to west on the map), the wave function of your electrons will be in some state, which you can represent with an arrow. With stronger magnetic fields, you’re farther east, and the wave function assumes a different state. As you move from west to east across your map, the arrow will rotate, showing how the wave function of the electrons changes as you dial the field up or down.

(© Tiravad – stock.adobe.com)

In a nutshell A physicist proposes that time has three dimensions instead of one, which could finally unite quantum mechanics and Einstein’s relativity into a single theory.

The theory makes specific, testable predictions about particle masses, gravitational waves, and cosmic behavior that experiments over the next decade can verify or disprove.

If correct, this would mean matter and energy are actually made from the bending and flowing of three-dimensional time itself—fundamentally changing how we understand reality.

FAIRBANKS, Alaska — What if everything you thought you knew about time was completely wrong? A physicist is now proposing that time itself isn’t the simple, one-way flow we experience, but actually has three separate dimensions. This wild idea might even finally solve some of the biggest mysteries in science.

Gunther Kletetschka from the University of Alaska Fairbanks has developed a mathematical model suggesting that our familiar sense of time ticking forward is like seeing only the tip of an iceberg. Beneath the surface, he argues, time has a hidden three-dimensional structure that could explain everything from why certain particles exist to how the entire universe works.

Right now, physics has a major problem. Scientists have two incredibly successful theories that describe how the universe works, but they contradict each other. Einstein’s relativity explains big things like planets and black holes perfectly. Quantum mechanics explains tiny particles flawlessly. But when scientists try to combine them (like when studying what happens inside a black hole) the math breaks down completely. It’s like having two different instruction manuals for the same machine, and they give you opposite directions.

Kletetschka’s three-dimensional time theory, published in World Scientific Connect, could be the missing piece that makes both instruction manuals work together.

Does the universe carry a third dimension of time? (Credit: © Irfanbm03 | Dreamstime.com)

How Three-Dimensional Time Actually Works

Think of time like a braided rope. From far away, it looks like a single strand moving in one direction. But up close, you can see it’s actually made of three separate cords twisted together. That’s essentially what Kletetschka is proposing about time itself.

In his model, time has three different “directions” that operate at completely different scales:

The first time dimension controls the tiniest processes in the universe — things happening inside atoms in less than a trillionth of a trillionth of a second. This is where quantum mechanics rules. The second time dimension acts like a bridge, connecting the microscopic quantum world to the everyday world we can see and touch. The third time dimension governs the slowest, most massive changes in the universe, like how galaxies form and evolve over billions of years. This is where Einstein’s gravity takes over.

We only experience one dimension of time because the other two only matter at extremes we never encounter in daily life. It’s like living in a house and only noticing the ground floor, while the basement and attic exist but don’t affect your daily routine.

The Theory Makes Stunning Predictions, And They’re Right

Unlike many physics theories that are too abstract to test, Kletetschka’s model makes specific predictions about the real world. When scientists check those predictions against actual measurements, they match almost perfectly.

Take subatomic particles, the building blocks of everything in the universe. These particles come in three distinct “families” or generations, kind of like three different sizes of the same basic tool. Scientists have known about this pattern for decades, but nobody could explain why there are exactly three families, or why their weights follow such specific patterns.

Kletetschka’s theory says this happens because of the three time dimensions. It predicts that particles in these three families should have weight ratios of roughly 1 to 4.5 to 21. Here’s another way to make sense of it: if the lightest particle in a family weighs as much as a paperclip, the middle one should weigh like a smartphone, and the heaviest should weigh like a large textbook.

This pattern shows up consistently across different types of particles, and the theory says it’s not a coincidence. It’s a direct result of how three-dimensional time is structured.

The theory gets the exact measurements right with incredible precision. Scientists measure particle weights using special units called GeV and MeV (think of them like very precise scales for weighing things smaller than atoms). The theory predicted the top quark (the heaviest fundamental particle we know) should weigh 173.21 units. The actual measured weight? 173.2 units.

Even more impressive, it predicted the weight of the muon (a heavier cousin of the electron that makes up atoms) correctly to seven decimal places. In the world of physics, that kind of accuracy really is like hitting a bullseye from miles away.

Why Some Forces Act Weird—And How Time Explains It

The theory also explains one of nature’s strangest behaviors. There’s a force called the weak nuclear force that governs radioactive decay, which is the process that makes some atoms unstable and break apart over time. This force has a bizarre quirk: it only interacts with particles that “spin” in one direction, like a cosmic preference for left-handed screws over right-handed ones.

Scientists call this “parity violation,” and it’s like discovering that all the locks in the universe only turn clockwise, never counterclockwise. Nobody really understood why nature has this preference.

Kletetschka’s model suggests the answer lies in the geometry of three-dimensional time itself. Just like a spiral staircase naturally curves in one direction, the structure of time creates this built-in asymmetry. It’s not an arbitrary rule, but rather a fundamental feature of how time is shaped.

The theory also makes predictions about gravitational waves. These are ripples in space and time caused by massive cosmic events, like when two black holes crash into each other. These waves were only detected for the first time in 2015, confirming one of Einstein’s predictions about gravity.

According to three-dimensional time theory, these waves should travel at slightly different speeds than light, being off by only 1.5 parts in a quadrillion. To put that in perspective, that’s like measuring the distance from New York to Los Angeles and being off by less than the width of a human hair. It’s an incredibly tiny difference, but our most sensitive detectors might be able to measure it.

Testing the Theory: What Scientists Will Look For

The beauty of this theory is that it doesn’t just make vague philosophical claims. It actually tells scientists exactly what to look for in their experiments.

New particles: The theory predicts that scientists should discover specific new particles when they smash protons together at very high energies—energies about 2,000 to 4,000 times heavier than a proton. The Large Hadron Collider, the world’s most powerful particle accelerator, is being upgraded and might be able to find these particles in the next few years.

Gravitational wave differences: Space-based detectors like LISA (Laser Interferometer Space Antenna) should be sensitive enough to measure those tiny speed differences in gravitational waves. If the theory is right, these waves should arrive at detectors just a few trillionths of a second later than light from the same cosmic event.

Dark energy changes: Dark energy is the mysterious force causing our universe to expand faster and faster—like an invisible hand stretching space itself. Nobody knows what it is, but it makes up about 70% of everything in the universe. Kletetschka’s theory predicts that dark energy should change its behavior in a specific pattern over cosmic time. New space telescopes like Euclid are powerful enough to detect these changes by studying how galaxies are distributed across the universe.

Neutrino masses: Neutrinos are ghost-like particles that barely interact with anything—trillions of them are passing through your body right now without you noticing. The theory makes precise predictions about how much these particles should weigh. Experiments like DUNE (Deep Underground Neutrino Experiment), which uses detectors buried deep underground, are designed to measure these masses with unprecedented precision.

What This Means for Our Understanding of Reality

If this theory proves correct, it would fundamentally change how we think about existence itself. Instead of matter existing within time, the theory suggests that matter is actually made from time.

As Kletetschka puts it in his paper, “what we perceive as mass and energy are manifestations of temporal curvature and dynamics.” In simpler terms, the particles that make up your body, the energy that powers your brain, and even the gravity holding you to Earth might all be different expressions of how time bends and flows in three dimensions.

This is a radically different way of thinking about reality. It’s the kind of paradigm shift that would make every physics textbook obsolete overnight, similar to how Einstein’s relativity overturned Newton’s clockwork universe, or how the discovery that Earth orbits the sun revolutionized astronomy.

The Road Ahead: Proof or Disproof

Of course, extraordinary claims require extraordinary evidence. The physics community will rightly demand rigorous proof before accepting such a radical reimagining of time itself.

But unlike many “theories of everything” that make untestable predictions, this one gives scientists a clear roadmap for verification. Over the next decade, experiments will be able to definitively prove whether three-dimensional time is real or just an elegant mathematical fiction.

The Large Hadron Collider’s upcoming high-luminosity upgrade will probe energy ranges where the predicted new particles should appear. Advanced gravitational wave detectors will become sensitive enough to measure the tiny speed variations the theory predicts. Space telescopes will map dark energy’s behavior across cosmic history with unprecedented precision.

Perhaps most importantly, the theory makes specific numerical predictions that leave little room for ambiguity. Either the neutrinos have exactly the masses it predicts, or they don’t. Either the new particles appear at the predicted energies, or they don’t. Either gravitational waves show the predicted speed differences, or they don’t.

In science, theories live or die by their predictions. And this theory has given scientists plenty of targets to aim for.

If experiments confirm these predictions, we’ll need to rethink our place in a universe where time itself has hidden dimensions we never imagined. If the predictions fail, it will join the long list of beautiful ideas that couldn’t survive contact with reality. Either way, we’re about to learn something profound about the nature of time itself.

Mathematics plays a crucial role in shaping the patterns and structures found in nature. This concept becomes evident as we explore the presence of geometry within natural forms. Two key aspects to consider are fractals in plants and landscapes, and symmetry in biological forms.

Fractals in Plants and Landscapes

Fractals are complex geometric structures characterized by self-similarity, where a shape can be split into parts, each of which is a reduced-scale copy of the whole. This concept is prominently observed in various natural elements.

Fern leaves and Romanesco broccoli exhibit clear fractal patterns, where each smaller component mirrors the larger structure. This recursive nature allows plants to maximize exposure to resources like sunlight and nutrients efficiently.

In landscapes, fractal geometry is evident in coastlines, mountain ranges, and river networks. These irregular and fragmented shapes are more accurately described using fractal dimensions rather than traditional Euclidean geometry. These fractal patterns in nature help scientists model and understand complex systems such as erosion, sedimentation, and ecological networks.

Symmetry in Biological Forms

Symmetry pervades biological forms, providing aesthetic balance and functional efficiency. Reflective symmetry is common, where one half of an organism is a mirror image of the other. Butterflies and human faces are prime examples, showcasing this bilateral symmetry which aids in movement and sensory function.

Rotational symmetry occurs when an organism’s structure looks the same after partial rotation. Starfish and flowers like the coneflower are notable examples. This form of symmetry can contribute to reproductive efficiency and structural integrity.

In addition, radial symmetry, where structures are arranged around a central point, is seen in organisms like jellyfish and sea anemones. This geometric arrangement enhances the organism’s interaction with the environment, optimizing movement, feeding, and sensory reception from all directions.

For more than five centuries, Leonardo da Vinci’s Vitruvian Man has invited admiration but also bafflement. With its harmonious symmetry of limbs outstretched in both a square and a circle, the drawing seems to promise a key to nature’s secrets. But one question has lingered in scholarly circles: how exactly did Leonardo construct it?

A new study may finally have an answer. According to a London-based dentist named Rory Mac Sweeney, the key to understanding the Vitruvian Man, it turns out, was hiding between the figure’s legs. According to Sweeney, the figure’s legs form an equilateral triangle, matching another equilateral triangle that draws the ideal human jaw.

The Enduring Genius of the Vitruvian Man

Leonardo da Vinci’s Vitruvian Man — drawn around 1490 — has become one of the most reproduced, revered, and reinterpreted images in history. More than a sketch, it’s a thesis about humanity’s place in the cosmos.

At the time, artists and architects were enthralled by the writings of Vitruvius, a Roman engineer who claimed that the ideal human body could be used as a model for perfect buildings. His idea was simple: just as a well-designed temple has harmonious proportions, so too does the human form. But Vitruvius left one crucial question unanswered — how, exactly, could a body be placed inside both a circle and a square?

Leonardo did more than illustrate the idea. He solved it.

Instead of aligning both shapes to the same center, Leonardo offset them. He placed the center of the circle at the man’s navel — his axis of rotation — and the square’s center at his groin. That subtle shift allowed the limbs to stretch into both forms. The result wasn’t just geometrically clever. It felt true.

A Triangle in Plain Sight

In his notes around the drawing (sometimes called the Canon of Proportions or the Proportions of Man), Leonardo wrote something that has long been overlooked: “the space between the legs will be an equilateral triangle.” That small phrase — long dismissed as poetic flourish or compositional guidance — has now been identified as a literal clue.

“I think we’ve all been looking for a complicated answer,” Rory Mac Sweeney, a dentist and independent researcher based in London, told The Independent. “But the key was in Leonardo’s own words. He was pointing to this triangle all along.”

Mac Sweeney’s peer-reviewed study, published in the Journal of Mathematics and the Arts, proposes that the triangle Leonardo described matches a precise anatomical feature discovered nearly 400 years later: Bonwill’s Triangle.

Bonwill’s equilateral triangle in mandibular anatomy, demonstrating the foundational geometric relationship governing optimal human jaw function established in 1864. Credit: Journal of Mathematics and the Arts, 2025.

This geometric figure — first identified by dentist William Bonwill in 1864 — is formed by connecting the two points where the lower jaw hinges with the midpoint between the front lower teeth. In most adults, each side of the triangle measures about 10 centimeters.

In modern dentistry, Bonwill’s Triangle is foundational. It’s used to build dentures and understand how the jaw moves efficiently. But Leonardo, it seems, may have foretold its geometry long before X-rays or surgical tools could confirm it.

A Ratio Rooted in Nature

Leonardo’s construction demonstrates the same tetrahedral principles found in craniofacial architecture. The equilateral triangle (green) that Leonardo explicitly referenced between the figure’s legs, when replicated six times around the navel, creates the hexagonal pattern (yellow) that generates the measured ratio of approximately 1.64 between the square’s side and circle’s radius. This geometric relationship may represent Leonardo’s intuitive recognition. Credit: Journal of Mathematics and the Arts, 2025.

By placing this equilateral triangle between the legs of the figure, Leonardo achieves a specific proportion in the drawing. The side of the square compared to the radius of the circle yields a ratio of roughly 1.64–1.65.

Mac Sweeney points out that this ratio closely mirrors what mathematicians call the “tetrahedral ratio” — 1.633. This is the proportion that arises in the geometry of the simplest three-dimensional shape: the tetrahedron, a pyramid made of four equilateral triangles. It’s also the same ratio found in the most efficient way to stack spheres (think oranges in a supermarket crate) or in the atomic structure of certain metals.

“Leonardo’s systematic construction yields a ratio… matching both published measurements of the original drawing and the tetrahedral ratio of 1.633,” the study notes.

“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,” Mac Sweeney argues.

Geometry, Biology, and the Human Face

The implications go even further. In the 20th century, dental researchers like George Monson expanded on Bonwill’s work to suggest that the teeth align along a spherical surface, centered near the forehead. When the jaw is modeled in three dimensions, it forms a tetrahedron with Bonwill’s triangle at the base.

Monson’s spherical theory and tetrahedral configuration. This three-dimensional dental geometry demonstrates the tetrahedral relationships that yield the 1.633 ratio between Monson’s sphere and the circumsphere of the tetrahedral unit. Credit: Journal of Mathematics and the Arts, 2025.

Whether in dental geometry, the curve of the human skull, or Leonardo’s drawing, 1.633 seems to be a recurring theme.

Broader anatomical studies seem to confirm this insight. A 2019 study of 100 human skulls found a consistent cranial ratio of 1.64 ± 0.04 — again remarkably close to the tetrahedral constant.

This evidence from art, anatomy, and geometry converges to the idea that Leonardo wasn’t just sketching a handsome man. He was, consciously or not, illustrating a principle of natural efficiency.

“Leonardo knew, or sensed, that our bodies are built with the same mathematical elegance as the universe around us,” said Mac Sweeney.

A New Chapter in an Old Masterwork

Historians have long known that Leonardo dissected cadavers, filling notebooks with observations of muscles, bones, and organs. But this new analysis suggests he may also have grasped structural relationships that modern science only formalized centuries later.

Leonardo’s construction method, Mac Sweeney argues, wasn’t based on symbolic mysticism or golden ratios alone, as previously suggested. Instead, it reflected an empirical approach grounded in geometry and observation.

Leonardo’s hexagonal-circle pattern studies from the Codex Atlanticus (folios 307v and 309v). These drawings provide direct evidence that Leonardo was actively exploring the same principles of efficient spatial packing that modern mathematics recognizes as fundamental to optimal organization, supporting the argument that his Vitruvian Man construction was based on sophisticated geometric understanding. Credit: Journal of Mathematics and the Arts, 2025.

It’s not the first time the Vitruvian Man has attracted geometric speculation. Previous researchers proposed that Leonardo used polygons like heptagons or octagons. Others assumed he had relied on the golden ratio—1.618—a number famously associated with beauty in art and nature.

But those models never quite fit the drawing’s measurements. They also failed to answer a basic question: why would Leonardo have used those shapes in the first place?

Nature’s Geometry

If the findings hold up under further scrutiny, they may shift the way art historians and scientists interpret the Vitruvian Man. The drawing may no longer be seen simply as a Renaissance icon, but also as a conceptual prototype for biomechanical understanding.

It would not be the first time that nature’s geometry has inspired design. Architect Buckminster Fuller, who developed the concept of “tensegrity” structures — describing structural systems as ‘islands of compression in an ocean of tension’ — also found the tetrahedral ratio at the heart of his work.

In his view, natural forms like dental occlusions and crystals reflect a balance between tension and compression. The same might be said of the human body, as modeled in Leonardo’s pen strokes.

From dental labs to molecular lattices, the same geometric patterns seem to remarkably recur. And now, thanks to a triangle hidden in plain sight, we may finally see that Leonardo had spotted the same pattern too.

From four-dimensional hexagons to the mind-bending amplituhedron, geometrical shapes are wilder than we learn at school – and they are a crucial tool for understanding reality

Yu Siang-Teo/Unsplash

Can you imagine the imprint a four-dimensional hexagon might leave as it passes through your three-dimensional kitchen table? Probably not, but some people can.

One such person was mathematician Alicia Boole Stott, daughter of logician George Boole. Early in the 20th century, she made models of the shapes four-dimensional objects would create when passing through three-dimensional objects. Decades later, when mathematicians could check such things using computer programs, they found Boole Stott had possessed an uncanny gift for getting these shapes right.

This story is part of our Concepts Special, in which we reveal how experts think about some of the most mind-blowing ideas in science. Read more here

For most of us, geometry conjures up thoughts of pencils, rulers, triangles and circles. It means those complicated questions you got asked at school involving parallel lines and angles. But as Boole Stott’s story shows, researchers have been taking geometry way beyond this for some time.

Geometry can stray far from the comprehensible world of two- and three-dimensional shapes – and in so doing, it can be extremely illuminating. Perhaps the best example is general relativity, Albert Einstein’s theory of gravity, which joins the three dimensions of space with time, creating a four-dimensional stage on which everything in the universe plays out.

But geometry can also make use of dimensions that aren’t physically real. Think of meteorology, for instance, where a point in the atmosphere can have many “dimensions” – latitude, longitude, temperature, pressure, wind speed and so on.

Researchers map these dimensions as shapes that extend into higher dimensions to help understand the workings of the atmosphere. “From things like this, you can apply mathematical models and actually work out what happens to [those properties] in many dimensions,” says mathematician Snezana Lawrence at Middlesex University in London.

For theoretical physicists, extra dimensions seem to be a necessary part of any full description of the universe, with some proposing that our reality is a “projection” from a higher dimension, for instance. This may sound outlandish, but if physicists make certain simplifying assumptions related to this idea, it suddenly makes it possible to carry out calculations to do with fundamental particles and black holes that are otherwise impossible.

Some physicists are banking on even stranger geometrical ideas being a route to a “theory of everything”, a single framework that explains the cosmos and everything in it. One of these is the “amplituhedron”, a mathematical object developed by Jaroslav Trnka at the University of California, Davis, and Nima Arkani-Hamed at the Institute for Advanced Study, New Jersey. Think of this as an abstract, multi-dimensional crystal, the properties of which provide an alternative way of describing the fundamentals of particle physics.

Or there is “causal dynamical triangulation”, developed by Renate Loll at Radboud University in the Netherlands. This stitches together an ensemble of geometrical shapes to create a description of space-time that seems to have some of the properties of both quantum theory and general relativity – two ideas that are normally incompatible. It is, she says, not just an abstract geometrical notion, but a testable reflection of the universe’s real properties that could be mirrored in our observations of the cosmic microwave background radiation that fills the whole of space.

Neither of these ideas yet amount to a theory of everything. But some suspect that to have any hope of finding one, we need a fresh vision for physics – and there is an increasing sense that this might be written in the language of geometry. Whether that is true or not, geometry is definitely more than hexagons – even four-dimensional ones.