The agency's Atmospheric Waves Experiment (AWE) captured concentric bands of atmospheric gravity waves stretching across the Southeast as the hurricane progressed miles away.

"Like rings of water spreading from a drop in a pond, circular waves from Helene are seen billowing westward from Florida's northwest coast," AWE principal investigator Ludger Scherliess, a physicist at Utah State University, said in a statement.

Atmospheric gravity waves are vertical ripples that move through quiet areas of the atmosphere, dividing the air into peaks and troughs. According to NASA, these waves can be created by large thunderstorms, wind bursts, hurricanes, tornadoes and even tsunamis. (They are different from gravitational waves, which are ripples in the fabric of space-time that result from violent cosmic events, such as black hole collisions.)

Related: Hurricane season 2024: How long it lasts and what to expect

The AWE instrument is mounted on the International Space Station and detects these waves by measuring airglow — a faint light given off by gasses in the mesosphere, the third layer of Earth's atmosphere. The mesosphere ranges from 31 to 53 miles (50 to 85 kilometers) above Earth's surface. Most weather occurs in the first layer of Earth's atmosphere, the troposphere, though cloud tops can rise into the second layer, the stratosphere, in very strong storms. (These are called "overshooting cloud tops.")

AWE started observing in November 2023, and the Helene gravity-wave images are among the first AWE images that NASA has released publicly. One of the project's goals is to help scientists understand how weather on Earth's surface can affect space weather, the disturbances in the upper atmosphere caused by interactions with charged cosmic particles.

Hurricane Helene was a Category 4 storm with winds up to 140 mph (225 km/h) when it made landfall near Perry, Florida. The storm subsequently moved inland, stalling over eastern Tennessee and western North Carolina, where it caused massive flooding. More than 230 people were killed, according to the Associated Press.

These black hole explosions, powered by Hawking radiation — a quantum process where black holes generate particles from the vacuum due to their intense gravitational fields — could be detected by upcoming telescopes, physicists suggest in a new study. And, once spotted, these exotic explosions could reveal whether our universe contains previously undiscovered particles.

Black holes from the dawn of time

There's already plenty of evidence for the existence of black holes ranging from a few times the mass of the sun to billions of times the sun's mass. These black holes have been directly detected through the gravitational waves they emit during the mergers that help them grow. Some black holes, such as the Milky Way's Sagittarius A*, have even been directly imaged as "shadows" by the Event Horizon Telescope.

PBHs, first proposed by Yakov Zeldovich and Igor Novikov in 1967, are thought to have formed within the first fractions of a second after the Big Bang and may have been as small as subatomic particles, according to NASA. Unlike their larger counterparts, which form from the collapse of massive stars and galaxies, PBHs might have emerged from the collapse of ultradense regions in the extremely hot "primeval soup" of particles in the early universe.

If they exist, these compact objects could provide a natural explanation for dark matter, the invisible entity that makes up about 85% of the matter in the universe. However, PBHs remain elusive. Their theoretical existence is supported by a combination of cosmological models, but they have yet to be directly observed.

The Hawking radiation effect

One of the most interesting aspects of PBHs is their connection to Hawking radiation. According to quantum theory, black holes aren't completely "black"; they can emit radiation and slowly lose mass through a process first theorized by Stephen Hawking. This emission, known as Hawking radiation, occurs when virtual particle pairs pop in and out of the vacuum of space near a black hole's edge — its "event horizon." While these pairs normally annihilate each other, if one falls into the black hole, the other particle can escape as radiation. Over time, this leads to the black hole's gradual evaporation.

"For black holes with masses larger than a few times that of the Sun, Hawking radiation is nearly undetectable," Marco Calzà, a theoretical physicist at the University of Coimbra in Portugal and co-author of the study, told Live Science in an email. "But lighter black holes — such as PBHs — would be much hotter and emit far more radiation, potentially allowing us to detect this process. This radiation can include a variety of particles, from photons to electrons to neutrinos."

Related: Stephen Hawking's black hole radiation paradox could finally be solved — if black holes aren't what they seem

As the PBH evaporates, it loses mass, becoming hotter and emitting more radiation in a feedback loop. Eventually, the black hole should explode in a powerful burst of radiation — a process that existing gamma-ray and neutrino telescopes are actively searching for. Although no definitive PBH explosions have been detected yet, the new study suggests these rare events could be the key to unlocking new physics.

Probing the final moments of a PBH

In their recent study, published in the Journal of High Energy Physics, Calzà and study co-author João G. Rosa, also a theoretical physicist at the University of Coimbra, introduced innovative methods for studying PBHs during their final stages of evaporation. By analyzing the properties of their Hawking radiation, the duo developed tools to estimate a PBH's mass and spin.

"Tracking a PBH's mass and spin as it evaporates could provide valuable clues about its formation and evolution," Rosa told Live Science in an email.

Their work has significant implications for fundamental physics. In a previous study, Rosa, Calzà and collaborator John March-Russell of the University of Oxford explored how string theory — an attempt to unify the fundamental forces of nature within a single quantum theory — could affect an evaporating PBH. String theory predicts the existence of numerous low-mass particles called axions, which have no intrinsic spin. Their research suggested that axion emission could actually spin up a PBH, contrary to Hawking's predictions.

"A spinning PBH would provide compelling evidence for these exotic axions, potentially revolutionizing our understanding of particle physics," Calzà said.

Furthermore, the study suggests that analyzing the evolution of a PBH's mass and spin in its final moments could reveal the existence of other new particles. By tracking the spectrum of Hawking radiation, scientists might be able to distinguish between high-energy particle physics models. Neutrino telescopes, such as IceCube, could even help uncover these new particles as PBHs explode in space.

"If we can catch just one exploding PBH and measure its Hawking radiation, we could learn a tremendous amount about new particles and potentially guide the design of future particle accelerators," Rosa said.

Although no exploding PBH has been detected yet, the tools and methods developed by Calzà and Rosa's team could pave the way for future discoveries. The researchers emphasized that dedicated experiments may not be necessary, as several new gamma-ray and neutrino telescopes with unprecedented sensitivity are already in development.

"Upcoming telescopes could easily spot one if it explodes nearby. If we're lucky enough to detect an exploding PBH, it could change everything we know about the fundamental laws of nature," Rosa said.

While still in high school, Ne'Kiya Jackson and Calcea Johnson from Louisiana used trigonometry to prove the 2,000-year-old Pythagorean theorem, which states that the sum of the squares of a right triangle's two shorter sides are equal to the square of the triangle's longest side (the hypotenuse). Mathematicians had long thought that using trigonometry to prove the theorem was unworkable, given that the fundamental formulas for trigonometry are based on the assumption that the theorem is true.

Jackson and Johnson came up with their "impossible" proof in answer to a bonus question in a school math contest. They presented their work at an American Mathematical Society meeting in 2023, but the proof hadn't been thoroughly scrutinized at that point. Now, a new paper published Monday (Oct. 28) in the journal American Mathematical Monthly shows their solution held up to peer review. Not only that, but the two students also outlined nine more proofs to the Pythagorean theorem using trigonometry.

"To have a paper published at such a young age — it's really mind-blowing," Johnson, who is now studying environmental engineering at Louisiana State University, said in a statement emailed to Live Science. "I am very proud that we are both able to be such a positive influence in showing that young women and women of color can do these things."

Related: Largest known prime number, spanning 41 million digits, discovered by amateur mathematician using free software

By proving Pythagoras' theorem using trigonometry, but without using the theorem itself, the two young women overcame a failure of logic known as circular reasoning. Trigonometry is a branch of mathematics that lays out how the sides, lengths and angles in a triangle are related, and as such, the discipline often includes expressions of the Pythagorean theorem. But Jackson and Johnson managed to prove the theorem using a result of trigonometry called the Law of Sines, dodging circular reasoning.

In the new study, and on top of their initial proof, the young mathematicians described four new ways to prove Pythagoras' theorem using trigonometry, as well as a novel method that revealed five more proofs, totaling 10 proofs.

Jackson and Johnson are only the third and fourth people known to have proven the Pythagorean theorem using trigonometry and without resorting to circular reasoning. The two other people were professional mathematicians, according to the statement.

"I didn't think it would go this far," Jackson, who currently studies pharmacology at the Xavier University of Louisiana, said in the statement. "I was pretty surprised to be published."

In the paper, Jackson and Johnson say there are two ways to present trigonometry and its functions sine and cosine, but these versions are often conflated into one. Sine and cosine are ratios that are defined in the context of a triangle's right angle, and they can be presented according to either the trigonometric method or a method that uses polynomials of complex numbers, according to the paper.

The conflation means that "trying to make sense of trigonometry can be like trying to make sense of a picture where two different images have been printed on top of each other," Jackson and Johnson wrote.

By teasing the two methods apart, researchers can discover "a large collection of new proofs of the Pythagorean theorem," the young mathematicians added.

If you liked reading this story, here are some mathematics books you might also enjoy:

The new number is 2^136,279,841 – 1, which beats the previous title holder (2^82,589,933 – 1) by more than 16 million digits.

Prime numbers, described by mathematicians as the "atoms of integers," are numbers that are divisible only by themselves and 1. The smallest prime numbers are 2, 3, 5, 7 and 11. Technically, prime numbers run to infinity, but finding them becomes significantly harder the bigger they get.

To find the new prime, Luke Durant used a free program called the Great Internet Mersenne Prime Search, or GIMPS, to sift through the possibilities with an algorithm. His efforts required the harnessing of thousands of graphics processing units (GPUs) across 24 data centers in 17 countries — a feat that "ends the 28-year reign of ordinary personal computers finding these huge prime numbers," according to a statement released on the GIMPS website.

The newly confirmed prime number contains 41,024,320 decimal digits, according to the statement.

Related: Pi calculated to 105 trillion digits, smashing world record

The new prime number is also the 52nd known Mersenne prime — a series named after Marin Mersenne, a French monk and polymath who devised a formula for finding prime numbers by subtracting 1 from powers of 2. (The smallest Mersenne prime is 3 — or 2 to the power of 2, minus 1.) Though far from being the only way to discover primes, the method is slightly easier than others.

As for the usefulness of the discovery, "At present there are few practical uses for these large Mersenne primes, prompting some to ask, 'Why search for these large primes?'" the GIMPS team wrote in the statement. "Those same doubts existed a few decades ago until important cryptography algorithms were developed based on prime numbers."

The discovery has netted Durant a $3,000 cash prize from GIMPS. Further prizes of $150,000 and $250,000 await those who discover the first hundred-million-digit prime and the first billion-digit prime, respectively.

When solutions to the three-body problem are mapped out based on where the three objects start in relation to one another, islands of stability emerge from the chaos, researchers reported in the September issue of the journal Astronomy & Astrophysics. These islands could help scientists detect colliding black holes, the researchers said.

The gravitational interactions between two bodies can be reliably described and mapped using equations. But when you add in a third object, things get wild — the motions of the bodies are unpredictable and often end with one of the bodies being flung out of the system. Even small changes in their starting masses, velocities or positions often lead to drastically different outcomes.

Broadly speaking, researchers use statistics to predict how often any one of the three bodies will be ejected from the system. But when Alessandro Trani, a theoretical physicist at the Niels Bohr Institute in Denmark, and his colleagues ran computer simulations of the three-body problem, their results didn't match the statistical predictions.

Their experiments began with a binary — two objects orbiting each other — and a single object approaching from elsewhere in space. Across more than a million simulations, the team altered the positions of the two bodies in the binary and the angle at which the single object approached. Then, they let the three bodies interact until one was eventually kicked out of the system.

Related: Mathematicians find 12,000 new solutions to 'unsolvable' 3-body problem

In a purely chaotic system, even a small adjustment to the positions or angles of the three bodies could change which of the three objects got ejected. But Trani and his colleagues found several ranges of positions and angles where the same object got kicked out every time. These "isles of regularity" represent gaps in the chaos of the three-body problem.

These non-chaotic zones could complicate how researchers predict three-body interactions in space. The predictions scientists usually apply to these astrophysical interactions rely on statistics, Trani said. But the purely statistical calculations don't work on problems that include regions of both chaos and regularity.

"We need to have a mix of statistical predictions for the chaotic space and a mix of regular mechanical theory or deterministic theory for the regular one, and we also need to know how to mix the outcomes," Trani told Live Science. "That's the most difficult part: identifying where the three-body problem is chaotic and where it's not, without running simulations."

Finding these regions of stability could also help scientists spot and understand gravitational waves, which are released when black holes interact or merge. Interactions among three black holes, which are relatively common inside star clusters, can send two of them careening toward each other. But predictions of these events capture only those resulting from chaotic interactions. There could be more non-chaotic interactions happening among three black holes and, therefore, more opportunities to study gravitational waves, Trani said.

Dark matter makes up the vast majority of the mass of almost every galaxy in the universe, but its exact nature still eludes scientists. It has gravity, but doesn’t interact with light or produce light of its own, so we only have circumstantial evidence of its existence through its gravitational interactions with everything else.

In these circumstances, researchers are desperate to cook up any scenario that might make dark matter more visible. So why not use black holes? It sounds like a ridiculous question: How could black holes, which devour any light that gets too close to them, make dark matter shine? But researchers have put together a scenario that might make it possible. They reported their findings Sept. 20 in the preprint database arXiv. (The findings haven't been peer-reviewed).

The researchers assume that dark matter can, in principle, interact with regular matter (and produce light in the process), but, for some reason, normally doesn't. Perhaps the interaction requires a certain amount of energy that simply isn’t available or is prohibited without a mediator particle doing the work. Black holes could provide the avenue needed to overcome these barriers and get the dark matter to light up.

But not just any black hole will do the trick, only ultra-tiny primordial black holes. These black holes aren’t the leftovers of giant stars but the remnants of the chaotic eras of the extremely early universe, where matter and energy spontaneously compressed to make them. Primordial black holes were first hypothesized by Stephen Hawking, but observations have so far failed to find any. If they do exist, they are extremely uncommon.

Like all black holes, primordial black holes would evaporate Hawking radiation, a strange quantum effect discovered by Stephen Hawking in which virtual particles pop up near a black hole's edge and some are able to escape. The smaller the black hole, the more radiation it emits.so primordial black holes roughly the mass of an asteroid would be emitting plenty of radiation in the present-day universe.

This radiation emitted by black holes isn’t just packets of light, or photons. It’s almost every kind of particle, including dark matter particles. As the primordial black holes decay, they emit dark matter particles that then energize any ambient dark matter particles in their vicinity, triggering a cascade that can briefly release visible light.

The researchers predict that these signals will typically be in the form of gamma ray flashes. They are far too weak for current experiments to detect, but future observatories, like NASA’s proposed All-sky Medium Energy Gamma-ray Observatory eXplorer (AMEGO-X), might have the sensitivity and resolution needed to find these sorts of signals.

In this scenario, the universe would expand to a finite size, but everything would grow so cold that all activity would essentially cease.

Dark energy is the mysterious force responsible for accelerating the expansion of the universe. It was discovered in the 1990s, but despite over two decades of research, it still remains the central mystery of modern cosmology. Over the years, scientists have presented some fascinating research into what it is and how it works.

One idea is known as holographic dark energy. In this scenario, gravity — and space itself — is an illusion. Our universe is really only two-dimensional, and exotic quantum forces on that surface give rise to the appearance of gravity and the structure of 3D space.

A consequence of this theory is a natural accelerated expansion of the universe, which we identify as dark energy.

Related: Our universe is merging with 'baby universes', causing it to expand, new theoretical study suggests

While many researchers have studied holographic dark energy models and ways to test it, a pair of astrophysicists examined the long-term fate of the universe if it is indeed ruled by holographic dark energy. They published their results Sept. 30 to the preprint database arXiv. (It has not been peer-reviewed.)

Dark energy takes up roughly 70% of the energy density of the entire cosmos. As the universe expands, the density of both regular and dark matter drops, while more and more dark energy manifests. To study the ultimate long-term fate of the universe, the researchers ignored matter and focused solely on the evolution of holographic dark energy.

They found that, as expected, holographic dark energy will continue to expand the universe. But, over time, its influence will slowly peter out and slow acceleration. The universe's expansion rate will steadily decrease until the cosmos reaches a nearly static value, essentially locking it to a final size.

But as the universe's expansion slows down, the density of holographic dark energy dwindles along with it. And since the density of matter also shrinks as the universe expands, the universe grinds to a halt. The researchers dub this scenario "the long freeze," in contrast to other commonly known fates of the universe like the "Big Freeze" (the accelerated expansion continues unabated) and the "Big Crunch" (something causes the universe to contract back toward the Big Bang).

The long freeze isn't a rosy scenario. While the universe's expansion will eventually stop, there won’t be any new sources of energy for all the matter inside of it. This means that eventually all the stars will wink out and decay, and all the subatomic particles will drift away from each other in the cold.

Unfortunately, even in their most exotic theories, cosmologists can't come up with a way to give the universe a happy ending.

It is the most famous graffiti in mathematical history, but it looks rather unassuming:

i ² = j ² = k ² = ijk = –1

Yet Hamilton's revelation changed the way mathematicians represent information. And this, in turn, made myriad technical applications simpler — from calculating forces when designing a bridge, an MRI machine or a wind turbine, to programming search engines and orienting a rover on Mars. So, what does this famous graffiti mean?

Rotating objects

The mathematical problem Hamilton was trying to solve was how to represent the relationship between different directions in three-dimensional space. Direction is important in describing forces and velocities, but Hamilton was also interested in 3D rotations.

Mathematicians already knew how to represent the position of an object with coordinates such as x, y and z, but figuring out what happened to these coordinates when you rotated the object required complicated spherical geometry. Hamilton wanted a simpler method.

He was inspired by a remarkable way of representing two-dimensional rotations. The trick was to use what are called "complex numbers", which have a "real" part and an "imaginary" part. The imaginary part is a multiple of the number i, "the square root of minus one", which is defined by the equation i ² = –1.

Related: Imaginary numbers could be needed to describe reality, new studies find

By the early 1800s several mathematicians, including Jean Argand and John Warren, had discovered that a complex number can be represented by a point on a plane. Warren had also shown it was mathematically quite simple to rotate a line through 90° in this new complex plane, like turning a clock hand back from 12.15pm to 12 noon. For this is what happens when you multiply a number by i.

Hamilton was mightily impressed by this connection between complex numbers and geometry, and set about trying to do it in three dimensions. He imagined a 3D complex plane, with a second imaginary axis in the direction of a second imaginary number j, perpendicular to the other two axes.

It took him many arduous months to realize that if he wanted to extend the 2D rotational wizardry of multiplication by i he needed four-dimensional complex numbers, with a third imaginary number, k.

In this 4D mathematical space, the k-axis would be perpendicular to the other three. Not only would k be defined by k ² = –1, its definition also needed k = ij = –ji. (Combining these two equations for k gives ijk = –1.)

Putting all this together gives i ² = j ² = k ² = ijk = –1, the revelation that hit Hamilton like a bolt of lightning at Broome Bridge.

Quaternions and vectors

Hamilton called his 4D numbers "quaternions", and he used them to calculate geometrical rotations in 3D space. This is the kind of rotation used today to move a robot, say, or orient a satellite.

But most of the practical magic comes into it when you consider just the imaginary part of a quaternion. For this is what Hamilton named a "vector".

A vector encodes two kinds of information at once, most famously the magnitude and direction of a spatial quantity such as force, velocity or relative position. For instance, to represent an object's position (x, y, z) relative to the "origin" (the zero point of the position axes), Hamilton visualised an arrow pointing from the origin to the object's location. The arrow represents the "position vector" x i + y j + z k.

This vector's "components" are the numbers x, y and z — the distance the arrow extends along each of the three axes. (Other vectors would have different components, depending on their magnitudes and units.)

Half a century later, the eccentric English telegrapher Oliver Heaviside helped inaugurate modern vector analysis by replacing Hamilton's imaginary framework i, j, k with real unit vectors, i, j, k. But either way, the vector's components stay the same — and therefore the arrow, and the basic rules for multiplying vectors, remain the same, too.

Hamilton defined two ways to multiply vectors together. One produces a number (this is today called the scalar or dot product), and the other produces a vector (known as the vector or cross product). These multiplications crop up today in a multitude of applications, such as the formula for the electromagnetic force that underpins all our electronic devices.

A single mathematical object

Unbeknown to Hamilton, the French mathematician Olinde Rodrigues had come up with a version of these products just three years earlier, in his own work on rotations. But to call Rodrigues' multiplications the products of vectors is hindsight. It is Hamilton who linked the separate components into a single quantity, the vector.

Everyone else, from Isaac Newton to Rodrigues, had no concept of a single mathematical object unifying the components of a position or a force. (Actually, there was one person who had a similar idea: a self-taught German mathematician named Hermann Grassmann, who independently invented a less transparent vectorial system at the same time as Hamilton.)

Hamilton also developed a compact notation to make his equations concise and elegant. He used a Greek letter to denote a quaternion or vector, but today, following Heaviside, it is common to use a boldface Latin letter.

This compact notation changed the way mathematicians represent physical quantities in 3D space.

Take, for example, one of Maxwell's equations relating the electric and magnetic fields:

∇ × E = –∂B/∂t

With just a handful of symbols (we won't get into the physical meanings of ∂/∂t and ∇ ×), this shows how an electric field vector (E) spreads through space in response to changes in a magnetic field vector (B).

Without vector notation, this would be written as three separate equations (one for each component of B and E) — each one a tangle of coordinates, multiplications and subtractions.

The power of perseverance

I chose one of Maxwell's equations as an example because the quirky Scot James Clerk Maxwell was the first major physicist to recognise the power of compact vector symbolism. Unfortunately, Hamilton didn't live to see Maxwell's endorsement. But he never gave up his belief in his new way of representing physical quantities.

Hamilton's perseverance in the face of mainstream rejection really moved me, when I was researching my book on vectors. He hoped that one day — "never mind when" — he might be thanked for his discovery, but this was not vanity. It was excitement at the possible applications he envisaged.

He would be over the moon that vectors are so widely used today, and that they can represent digital as well as physical information. But he'd be especially pleased that in programming rotations, quaternions are still often the best choice — as NASA and computer graphics programmers know.

In recognition of Hamilton's achievements, maths buffs retrace his famous walk every October 16 to celebrate Hamilton Day. But we all use the technological fruits of that unassuming graffiti every single day.

This edited article is republished from The Conversation under a Creative Commons license. Read the original article.

Conservation of energy

Energy conservation — the idea that energy cannot be created or destroyed, only transformed — is one of the most important laws of physics. James Prescott Joule demonstrated this rule, the first law of thermodynamics, when he filled a large container with water and fixed a paddle wheel inside it. The wheel was held in place by an axle with a string around it and then looped over a pulley and attached to a weight, which, when dropped, caused the wheel to spin. By sloshing the water with the wheel, Joule demonstrated that the heat energy gained by the water from the wheel's movement was equal to the potential energy lost by dropping the weight.

Measurement of the electron's charge

As the fundamental carriers of electric charge, electrons carry the smallest amount of electricity possible. But the particles are truly tiny, with a mass 1,838 times smaller than the already-minuscule proton.

So how could you measure the charge on something so small? Physicist Robert Millikan's answer was to drop electrically charged oil drops through the plates of a capacitor and adjust the voltage of the capacitor until the electric field it emitted produced a force on some of the drops that balanced out gravity — thus suspending them in the air. Repeating the experiment for different voltages revealed that, no matter the size of the drops, the total charge it carried was a multiple of a base number. Millikan had found the fundamental charge of the electron.

 "Gold foil experiment" revealing the structure of the atom

Once thought to be indivisible, the atom was slowly divided and split by a series of experiments during the late 19th and early 20th centuries. These included J.J. Thomson's 1897 discovery of the electron and James Chadwick's 1932 identification of the neutron. But perhaps the most famous of these experiments was Hans Geiger and Ernest Marsden's "gold foil experiment." Under the direction of Ernest Rutherford, the students fired positively charged alpha particles at a thin sheet of gold foil. To their surprise, the particles passed through, revealing that atoms consisted of a positively charged nucleus separated by a significant empty space by their orbiting electrons.

Nuclear chain reaction

By the mid-20th century, scientists were aware of the basic structure of the atom and that, according to Einstein, matter and energy were different forms of the same thing. This set the stage for the wartime work of Enrico Fermi, who in 1942 demonstrated that atoms could be split to release enormous quantities of energy.

While working at the University of Chicago with an experimental setup he called an "atomic pile," Fermi demonstrated the first-ever controlled nuclear fission reaction. Fermi fired neutrons at the unstable isotope uranium-235, causing it to split and release more neutrons in a growing chain reaction. The experiment paved the way for the development of nuclear reactors and was used by J. Robert Oppenheimer and the Manhattan Project to build the first atomic bombs.

Wave-particle duality

One of the most famous experiments in physics is also one that illustrates, with disturbing simplicity, the bizarreness of the quantum world. The experiment consisted of two slits, through which electrons would travel to create an interference pattern on a screen, like waves. Scientists were stunned when they placed a detector near the screen and found that its presence caused the electrons to switch their behavior to act instead as particles.

First performed by Thomas Young to demonstrate the wave nature of light, the experiment was later used by physicists in the 20th century to show that all particles, including photons, were both waves and particles at the same time — and they acted more like particles when they were being measured directly.

Splitting of white light into colors

White light is a mixture of all the colors of the rainbow, but before 1672, the composite nature of light was completely unknown. Isaac Newton determined this by using a prism that bent light of different wavelengths, or colors, by different amounts, decomposing white light into its composite colors. The result was one of the most famous experiments in scientific history and a discovery that, alongside other contributions by Newton, gave birth to the modern field of optics.

Discovery of gravity

In perhaps the most widely repeated story in all of science, Newton is said to have chanced upon the theory of gravity while contemplating under the shade of an apple tree. According to the legend, when an apple fell and struck him on the head, he supposedly yelled "Eureka!" as he realized that the same force that brought the apple tumbling to Earth also kept the moon in orbit around our planet and Earth circling the sun. That force, of course, would become known as gravity.

The story is slightly embellished, however. According to Newton's own account, the apple did not strike him on the head, and there's no record of what he said or if he said anything, at the moment of discovery. Nonetheless, the realization led Newton to develop his theory of gravity in 1687, which was updated by Einstein's theory of general relativity 228 years later.

Blackbody radiation

By the turn of the 20th century, many physicists — having advanced theories that explained gravity, mechanics, thermodynamics and the behavior of electromagnetic fields — were confident that they had conquered the vast majority of their field. But one troubling source of doubt remained: Theories predicted the existence of a "blackbody" — an object capable of absorbing and then remitting all incident radiation. The problem was that physicists couldn't find it.

In fact, data from experiments conducted with close approximations of black bodies — a box with a single hole whose inside walls are black — revealed that significantly less energy was emitted from blackbodies than classical theories led scientists to believe, especially at shorter wavelengths. The contradiction between experiment and theory became known as the "ultraviolet catastrophe."

The discovery prompted Max Planck to propose that the energy emitted by blackbodies wasn't continuous but rather split into discrete integer chunks called quanta. His radical proposal catalyzed the development of quantum mechanics, whose bizarre rules are completely unintuitive to observers living in the macroscopic world.

Einstein and the eclipse

Following its publication in 1915, Einstein's groundbreaking theory of general relativity briefly remained just that — a theory. Then, in 1919, astronomer Sir Arthur Eddington devised and completed stunning proof using that year's total solar eclipse.

Key to Einstein's theory was the notion that space — and, therefore, the path that light would follow through it — was warped by powerful gravitational forces. So, as the moon's shadow passed in front of the sun, Eddington recorded the position of nearby stars from his vantage point on the island of Principe in the Gulf of Guinea. By comparing these positions to those he had recorded at night without the sun in the sky, Eddington observed that they had been shifted slightly by the sun's gravity, completing his stunning proof of Einstein's theory.

Higgs boson

In 1964, Peter Higgs suggested that matter gets its mass from a field that permeates all of space, imparting particles with mass through their interactions with a particle known as the Higgs boson.

To search for the boson, thousands of particle physicists planned, constructed and fired up the Large Hadron Collider. In 2012, after trillions upon trillions of collisions in which two protons are smashed together at near light speed, the physicists finally spotted the telltale signature of the boson.

Weighing the world

Although he's perhaps best known for his discovery of hydrogen, 18th-century physicist Henry Cavendish's most ingenious experiment accurately estimated the weight of our entire planet. Using a special piece of equipment known as a torsion balance (two rods with one smaller and one larger pair of lead balls attached to the end), Cavendish measured the minuscule force of gravitational attraction between the masses. Then, by measuring the weight of one of the small balls, he measured the gravitational force between it and Earth, giving him an easy formula for calculating our planet's density and — therefore, its weight — that remains accurate to this day.

Conservation of mass

Much like energy, matter in our universe is finite and cannot be created or destroyed, only rearranged. In 1789, to arrive at this startling conclusion, French chemist Antoine Lavoisier placed a burning candle inside a sealed glass jar. After the candle had burned and melted into a puddle of wax, Lavoisier weighed the jar and its contents, finding that it had not changed

Leaning Tower of Pisa experiment

Greek philosopher Aristotle believed that objects fall at different rates because the force acting upon them was stronger for heavier objects — a claim that went unchallenged for more than a millennium.

Then came the Italian polymath Galileo Galilei, who corrected Aristotle's false claim by showing that two objects with different masses fall at exactly the same rate. Some claim Galileo's famous experiment was conducted by dropping two spheres from the Leaning Tower of Pisa, but others say this part of the story is apocryphal. Nonetheless, the experiment was perhaps most famously demonstrated by Apollo 15 astronaut David Scott, who, while dropping a feather and a hammer on the moon, showed that without air, the two objects fell at the same speed.

Detection of gravitational waves

If gravity warps space-time as Einstein predicted, then the collision of two extremely dense objects, such as neutron stars or black holes, should also create detectable shock waves in space that could reveal physics unseen by light. The problem is that these gravitational waves are tiny, often the size of a few thousandths of a proton or neutron, so detecting them requires an extremely sensitive experiment.

Enter LIGO, the Laser Interferometer Gravitational-Wave Observatory. The L-shaped detector has two 2.5-mile-long (4 km) arms containing two identical laser beams. When a gravitational wave laps at our cosmic shores, the laser in one arm is compressed and the other expands, alerting scientists to the wave's presence. In 2015, LIGO achieved its task, making the first-ever direct detection of gravitational waves and opening up an entirely new window to the cosmos.

Destruction of heliocentrism

The idea that Earth orbits the sun goes back to the fifth century B.C. to Greek philosophers Hicetas and Philolaus. Nonetheless, Claudius Ptolemy's belief that Earth was the center of the universe later took root and dominated scientific thought for more than a millennium.

Then came Nicolaus Copernicus, who proposed that Earth did, in fact, revolve around the sun and not the other way around. Concrete evidence for this was later offered by Galileo, who in 1610 peered through his telescope to observe the planet Venus moving through distinct phases — proof that it, too, orbited the sun. Galileo's discovery did not win him any friends with the Catholic Church, which tried him for heresy for his unorthodox proposal.

Foucault's pendulum

First used by French physicist Jean Bernard Léon Foucault in 1851, the famous pendulum consisted of a brass bob containing sand and suspended by a cable from the ceiling. As it swung back and forth, the angle of the line traced out by the sand changed subtly over time — clear evidence that some unknown rotation was causing it to shift. This rotation was the spinning of Earth on its axis.

Discovery of the electron

In the 19th century, physicists found that by creating a vacuum inside a glass tube and sending electricity through it, they could make the tube give off a fluorescent glow. But exactly what caused this effect, called a cathode ray, was unclear.

Then, in 1897, physicist J.J. Thomson discovered that by applying a magnetic field to the rays inside the tube, he could control the direction in which they traveled. This revelation showed Thomson that the charge within the tube came from tiny particles 1,000 times smaller than hydrogen atoms. The tiny electron had finally been found.

Deflection of an asteroid

In 2022, NASA scientists hit an astronomical "bull's-eye" by intentionally steering the 1,210-pound (550 kilograms), $314 million Double Asteroid Redirection Test (DART) spacecraft into the asteroid Dimorphos just 56 feet (17 meters) from its center. The test was designed to see if a small spacecraft propelled along a planned trajectory could, if given enough lead time, redirect an asteroid from a potentially catastrophic impact with Earth.

DART was a smashing success. The probe's original goal was to change the orbit of Dimorphos around its larger partner — the 2,560-foot-wide (780 m) asteroid Didymos — by at least 73 seconds, but the spacecraft actually altered Dimorphos' orbit by a stunning 32 minutes. NASA hailed the collision as a watershed moment for planetary defense, marking the first time that humans proved capable of diverting Armageddon, and without any assistance from Bruce Willis.

Faraday induction

In 1831, Michael Faraday, the self-taught son of a blacksmith born in rural south England, proposed the law of electromagnetic induction. The law was the result of three experiments by Faraday, the most notable of which involved the movement of a magnet inside a coil made by wrapping a wire around a paper cylinder. As the magnet moved inside the cylinder, it induced an electric current through the coil — proving that electric and magnetic fields were inextricably linked and paving the way for electric generators and devices.

Measurement of the speed of light

Light is the fastest thing in our universe, which makes measuring its speed a unique challenge. In 1676, Danish astronomer Ole Roemer chanced upon the first estimate for light's propagation while studying Io, Jupiter's innermost moon. By timing the eclipses of Io by Jupiter, Roemer was hoping to find the moon's orbital period.

What he noticed instead was that, as Earth's orbit moved closer to Jupiter, the time intervals between successive eclipses became shorter. Roemer's crucial insight was that this was due to a finite speed of light, which he roughly calculated based on Earth's orbit. Other methods later refined the measurement of light's speed, eventually arriving at its current value of 2.98 × 10^8 meters per second (about 186,282 miles per second).

Disproof of the "luminiferous ether"

Most waves, such as sound waves and water waves, require a medium to travel through. In the 19th century, physicists thought the same rule applied to light, too, with electromagnetic waves traveling through a ubiquitous medium dubbed the "luminiferous ether."

Albert Michelson and Edward W. Morley set out to prove this conjecture with a remarkably ingenious hypothesis: As the sun moves through the ether, it should displace some of the strange substance, meaning light should travel detectably faster when it moves with the ether wind than against it. They set up an interferometer experiment that used mirrors to split light beams along two opposing directions before bouncing them back with distant mirrors. If the light beams returned at different times, then the ether was real.

But the light beams inside their interferometer did not vary. Michelson and Morley concluded that their experiment had failed and moved on to other projects. But the result — which had conclusively disproved the ether theory — was later used by Einstein in his theory of special relativity to correctly state that light's speed through a fixed medium does not change, even if its source is moving.

Discovery of radioactivity

In 1897, while working in a converted shed with her husband Pierre, Marie Curie began to investigate the source of a strange new type of radiation emitted from the elements thorium and uranium. Marie Curie discovered that the radiation these elements emitted did not depend on any other factors, such as their temperature or molecular structure, but changed purely based on their quantities. While grinding up an even more radioactive substance known as pitchblende, she also discovered that it consisted of two elements that she dubbed radium and polonium.

Curie's work revealed the nature of radioactivity, a truly random property of atoms that comes from their internal structure. Curie won the Nobel Prize (twice) for her discoveries — making her the first woman to do so — and later trained doctors to use X-rays to image broken bones and bullet wounds. She died of aplastic pernicious anemia, a disease caused by radiation exposure, in 1934.

Expansion of the universe

While using the 100-inch Hooker telescope in California to study the light glimmering from distant galaxies in 1929, Edwin Hubble made a surprising observation: The light from the distant galaxies appeared to be shifted toward the red end of the spectrum — an indication that they were receding from Earth and each other. The farther away a galaxy was, the faster it was moving away.

Hubble's observation became a crucial piece of evidence for the Big Bang theory of our universe. Yet precise measurements for galaxies' recession, known as the Hubble constant, still confound scientists to this day.

Put simply, the universe is indeed expanding, but depending on where cosmologists look, it's doing so at different rates. In the past, the two best experiments to measure the expansion rate were the European Space Agency's Planck satellite and the Hubble Space Telescope. The two observatories, each of which used a different method to measure the expansion rate, arrived at different results. These conflicting measurements have led to what some call a "cosmology crisis" that could reveal new physics or even replace the standard model of cosmology.

Ignition of nuclear fusion

In 2022, scientists at the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory in California used the world's most powerful laser to achieve something physicists have been dreaming about for nearly a century: the ignition of a pellet of fuel by nuclear fusion.

The demonstration marked the first time that the energy going out of the plasma in the nuclear reactor's fiery core exceeded the energy beamed in by the laser, and has been a rallying call for fusion scientists that the distant goal of near-limitless and clean power is, in fact, achievable.

However, scientists have cautioned that the energy from the plasma exceeds only that from the lasers, and not from the energy from the whole reactor. Additionally, the laser-confinement method used by the NIF reactor, built to test thermonuclear explosions for bomb development, will be difficult to scale up.

Measurement of Earth's circumference

By roughly 500 B.C., most ancient Greeks believed the world was round — citing evidence provided by Aristotle and guided by a suggestion from Pythagoras, who believed a sphere was the most aesthetically pleasing shape for our planet.

Then, around 245 B.C., Eratosthenes of Cyrene thought of a way to make the measurement directly. Eratosthenes hired a team of bematists (professional surveyors who measured distances by walking in equal-length steps called stadia) to walk from Syene to Alexandria. They found that the distance between the two cities was roughly 5,000 stadia.

Eratosthenes then visited a well in Syene that had been reported to have an interesting property: At noon on the summer solstice each year, the sun illuminated the well's bottom without casting any shadows. Eratosthenes went to Alexandria during the solstice, stuck a pole in the ground and measured the shadow from it to be about one-fiftieth of a complete circle. Pairing this with his measurement of the distance between the two cities, he determined that Earth's circumference was about 250,000 stadia, or 24,497 miles (39,424 km). Earth is now known to measure 24,901 miles (40,074 km) around the equator, making the ancient Greeks' measurements remarkably accurate.

Discovery of black holes

The acceptance of Einstein's theory of general relativity led to some startling predictions about our universe and the nature of reality. In 1915, Karl Schwarzschild's solutions to Einstein's field equations predicted that it was possible for mass to be compressed into such a small radius that it would collapse into a gravitational singularity from which not even light could escape — a black hole.

Schwarzschild's solution remained speculation until 1971, when Paul Murdin and Louise Webster used NASA's Uhuru X-ray Explorer Satellite to identify a bright X-ray source in the constellation Cygnus that they correctly contended was a black hole.

More conclusive evidence came in 2015, when the LIGO experiment detected gravitational waves from two of the colliding cosmic monsters. Then, in 2019, the Event Horizon Telescope captured the first image of the accretion disk of superheated matter surrounding the supermassive black hole at the center of the galaxy M87.

Discovery of X-rays

While testing whether the radiation produced by cathode rays could escape through glass in 1895, German physicist Wilhelm Conrad Röntgen saw that the radiation could not only do so, but it could also zip through very thick objects, leaving a shadow on a lead screen placed behind them. He quickly realized the medical potential of these rays — later known as X-rays — for imaging skeletons and organs. His observations gave birth to the field of radiology, enabling doctors to safely and noninvasively scan for tumors, broken bones and organ failure.

The Bell test

In 1964, physicist John Stewart Bell proposed a test to prove that quantum entanglement — the weird instantaneous connection between two far-apart particles that Einstein objected to as "spooky action at a distance" — was required by quantum theory.

The test has taken many experimental forms since Bell first proposed it, but the findings remain the same: Despite what our intuition tells us, what happens in one part of the universe can instantaneously affect what happens in another, provided the objects in each region are entangled.

Detection of the quark

In 1968, experiments at the Stanford Linear Accelerator Center found that electrons and their lepton cousins, muons, were scattering from protons in a distinct way that could only be explained by the protons being composed of smaller components. These findings matched predictions by physicist Murray Gell-Mann, who dubbed them "quarks" after a line in James Joyce's "Finnegans Wake."

Archimedes' naked leap from his bathtub

First recorded in the first century B.C. by Roman architect Vitruvius, Archimedes' discovery of buoyancy is one of the most famous stories in science. The prompting for Archimedes' finding came from King Hieron of Syracuse, who suspected that a pure-gold crown a blacksmith made for him actually contained silver. To get an answer, Hieron enlisted Archimedes' help.

The problem stumped Archimedes, but not long after, as the story goes, he filled up a bathtub with water and noticed that the water spilled out as he got in. This caused him to realize that the water displaced by his body was equal to his weight — and because gold weighed more than silver, he had found a method for judging the authenticity of the crown. "Eureka!" ("I've got it!") Archimedes is said to have cried, leaping from his bathtub to announce his discovery to the king.

Deepest and most detailed photo of the universe

In 2022, the James Webb Space Telescope unveiled the deepest and most detailed picture of the universe ever taken. Called "Webb's First Deep Field," the image captures light as it appeared when our universe was just a few hundred million years old, right when galaxies began to form and light from the first stars started flickering.

The image contains an overwhelmingly dense collection of galaxies, the light from which, on its way to us, was warped by the gravitational pull of a galaxy cluster. This process, known as gravitational lensing, brings the fainter light into focus. Despite the dizzying number of galaxies in view, the image represents just a tiny sliver of sky — the speck of sky blocked out by a grain of sand held on the tip of a finger at arm's length.

OSIRIS-REx asteroid-sampling mission

In 2023, NASA's OSIRIS-REx spacecraft came hurtling back through Earth's atmosphere after a years-long journey to Bennu, a "potentially hazardous asteroid" with a 1-in-2,700 chance of smashing cataclysmically into Earth — the highest odds of any identified space object.

The goal of the mission was to see whether the building blocks for life on Earth came from outer space. OSIRIS-REx circled the asteroid for 22 months to search for a landing spot, touching down to collect a 2-ounce (60 grams) sample from Bennu's surface that could contain the extraterrestrial precursors to life on our planet. Scientists have already found many surprising details that have the potential to rewrite the history of our solar system.

To this day, the seemingly simple equation, developed by an 18th-century Presbyterian minister and amateur mathematician, is used in modeling and forecasting to help us predict everything from future weather events, fluctuations in the stock market to the winners of sporting events.

The book "Everything is Predictable," by award-winning science writer Tom Chivers is a captivating tour of this curious theorem and how it impacts modern life, and has been shortlisted for the prestigious 2024 Royal Society Trivedi Science Book Prize. Below is a short excerpt from the book's introduction, which explores to what extent we can predict the future.

Related: 32 sci-fi technology predictions that came true

Can you predict the future? Yes, of course you can.

You can predict with near-certain accuracy that in the next few seconds, you'll take a breath, and let it out again. Your heart will beat, somewhere between one and three times a second. Tomorrow morning, the sun will come up, at a particular time which depends upon your latitude and the time of year but which nonetheless you can find out with great accuracy. All of these events you can predict with confidence.

You can also predict that the train will arrive at a certain time, or that your friend will arrive on time at the restaurant at which you've arranged to meet her. Though, depending on the rail company, or your friend, you might be less confident in that.

And you can predict that the world's population will continue to grow until around the middle of the century, and then start to fall again. You can predict that global average surface temperatures in the year 2030 will be higher than they were in the year 1930.

The future isn't opaque. You can see into it. Some parts are more predictable than others – the Newtonian dance of the planets we can predict out for thousands of years; the Lorenzian chaos of the weather, really only a few days. But you can peer through the murk, after a fashion.

That's not what people normally mean when they say, "I can predict the future." They are referring to something mystical, some psychic or magical vision. We probably can't do that. (You'll read about a scientist in this book who thinks we can, and you’ll also read about why he's almost certainly wrong.) But we don’t need to. All that we do, all the time, is predict the future. We couldn’t function if we couldn’t. We make very basic predictions, like "the air will continue to be breathable," implicitly, with every breath we take. We make more complex predictions, like "The corner shop will have Alpen [a breakfast cereal] when I get there," each time we make a decision. We're not basing them on mystical visions, but on information we have gathered in the past.

The thing with all these predictions is that they are uncertain. The universe may or may not be deterministic; perhaps if we had perfect, God-like knowledge of the position, movement and qualities of every particle in the universe, we could perfectly predict everything, the fall of every sparrow. But we don't. Instead, we have partial information. We can see bits of the universe, imperfectly, through our imperfect senses. We have best guesses for the way those bits move — we know the human-shaped bits tend to seek food and company; we know the rock-shaped bits tend to sit still. We can make messy, imperfect predictions with that information.

Life isn’t chess, a game of perfect information, one that can in theory be "solved." It's poker, a game where you're trying to make the best decisions using the limited information you have. This book is about the equation that lets you do that.

Excerpted from "Everything is predictable: How Bayesian Statistics Explain our World." Copyright © 2024 by Tim Chivers.