## Thales: Ancient Greeks built the cosmos with right triangles

### The ancient Greeks were obsessed with geometry, which may have formed the basis of their philosophical cosmology.

Credit: Michael Dziedzic via Unsplash
• Every triangle inscribed inside a circle on its diameter is a right triangle.
• Upon this discovery, Thales is said to have performed a great ritual sacrifice.
• Might Thales have believed that the entire cosmos was constructed of right triangles?

Thales is credited by the late commentator Proclus, on the authority of Aristotle's student Eudemus, with "discovering" geometrical propositions, some of them more generally and others more practically. Consider some of the diagrams expressing practical examples of right-angled triangles.

From left to right, we have Thales' measurement of (i) the height of a pyramid when its shadow is equal to its height; (ii) the height of a pyramid when its shadow is unequal but proportional to its height; (iii) the distance to a ship at sea from the shoreline; and (iv) the distance to a ship at sea from a tower. Note that, when rotated, they are all the same diagram!

The more general propositions also seem to be relevant to practical geometry:

We have a report about a special accomplishment of Thales. Originating with Diogenes Laertius of the 3rd century BCE on the authority of the mathematician Pamphila, it says that Thales made a splendid ritual sacrifice upon inscribing a right triangle in a circle. Obviously, he thought this was a pretty big deal. More on that a bit later.

The first thing Thales had to know is that the angles of every triangle sum to two right angles. (The angles inside every triangle sum to 180°. Two right angles, each of which is 90°, also sum to 180°.) We have an ancient report that credits Thales' generation of geometers with having grasped this fact in all species of triangles — equilateral, isosceles, and scalene. How might Thales and his geometers have done it? Consider the following diagrams:

By dropping a perpendicular from a vertex to the opposite side in each species of triangle, and then completing the two rectangles formed, one can see immediately that each rectangle (containing four right angles) is halved by the diagonal created by each side of the triangle. Therefore, each half-triangle contains two right angles. And if the two right angles at the base are removed, leaving the three angles of one large triangle, the angles sum to two right angles.

Now, consider how Thales may have proved that every triangle inscribed inside a circle on its diameter must be a right triangle. To show this, he relied on the isosceles triangle proposition and proved that the angle at A [α + β] is right-angled.

Perhaps he did it this way: Based upon the isosceles triangle proposition, Thales knows that segments BD and AD (left diagram) are equal in length because they are both radii of the circle BAC. Thus, their opposite angles — α and α — must be equal. Since every triangle is 180° (that is, contains the equivalent of two right angles) and the angle BDA at the base is a right angle, α + α must also equal one right angle. By itself, α is half of a right angle.

Next, CD and AD are both equal in length since they, too, are both radii of the circle BAC, and so the angles opposite each must also be equal — that is, β equals β. If we acknowledge that the angle at the base ADC is a right angle, and there is the equivalent of two right angles in every triangle, then β + β must equal one right angle. By itself, β is half of a right angle.

Finally, the angle at A is divided into two equal parts, α and β. Because each is half of a right angle, together (α + β) they equal one right angle.

That explains the right angle for an isosceles triangle inscribed inside of a circle. But what about all the varieties of the scalene? More or less, it's the same argument.

Consider triangle ABC (right diagram). It is composed of two triangles ABD and ACD. In ABD, AD must be equal to BD because both are radii of the circle BAC, and so the angles opposite those sides also must be equal. The same argument applies for triangle ADC. Thus, the three angles of triangle ABC are α + β + (α + β). Since we already know that the angles of every triangle sum to 180° (that is, the equivalent of two right angles), then α + β + (α + β) equals two right angles. Thus, α + β must equal one right angle.

Perhaps these lines of proof persuaded Thales and his companions that every triangle inscribed in a circle on its diameter is right. But why the great ritual sacrifice?

The ancient traditions do not give us more insight, and we are left only to speculate. Aristotle claims that Thales posited an underlying unity, water, that alters without changing. Although things look different, water is the substrate of all appearances. Water is merely altered without changing substantially. Had Thales been looking into geometry to try to discover the underlying structure of water, perhaps he followed a similar line of thought as Plato did when he identified the four elements (fire, air, water, and earth) with geometric shapes.

Thales may have identified the right triangle as the fundamental structure of water. Moreover, he now had a way to produce an unlimited number of them for further investigation simply by making a circle, drawing its diameter, and inscribing a triangle inside it.

But there is perhaps another reason for his splendid sacrifice, seen in this metaphysical light. I can imagine one of his compatriots objecting, upon hearing Thales' idea that water was the underlying nature or unity of all things and that the right triangle was its structure. The objection may have gone like this: right triangles may form the basis of every rectilinear figure, but they certainly don't form the basis of the circle. The circle is not constructed out of right triangles, is it? Thus, the right triangle is not the fundamental figure of all appearances.

Thales' reply must have been as astonishing to his compatriots as it is to many of us today. Indeed, the circle too is built out of right triangles! If we plot on the circle's diameter all the possible triangles inscribable inside a circle — starting from one end of the diameter, touching the circle, and then finishing at the other end of the diameter — we produce what modern mathematicians call a "geometrical loci." The circle itself is constructed out of right triangles!

Prof. Robert Hahn has broad interests in the history of ancient and modern astronomy and physics, ancient technologies, the contributions of ancient Egypt and monumental architecture to early Greek philosophy and cosmology, and ancient mathematics and geometry of Egypt and Greece. Every year, he gives "Ancient Legacies" traveling seminars to Greece, Turkey, and Egypt. His latest book is The Metaphysics of the Pythagorean Theorem.

## Will AI replace mathematicians?

### If computers can beat us at chess, maybe they could beat us at math, too.

Credit: Patrick Lux via Getty Images
• Most everyone fears that they will be replaced by robots or AI someday.
• A field like mathematics, which is governed solely by rules that computers thrive on, seems to be ripe for a robot revolution.
• AI may not replace mathematicians but will instead help us ask better questions.

The following is an excerpt adapted from the book Shape. It is reprinted with permission of the author.

Will machines replace us? Since the origin of artificial intelligence (AI), people have worried that computers eventually (or even imminently!) will surpass the human cognitive capacity in every respect.

Artificial intelligence pioneer Oliver Selfridge, in a television interview from the early 1960s, said, "I am convinced that machines can and will think in our lifetime" — though with the proviso, "I don't think my daughter will ever marry a computer." (Apparently, there is no technical advance so abstract that people can't feel sexual anxiety about it.)

## AI Anxiety

Let's make the relevant question more personal: will machines replace me? I'm a mathematician; my profession is often seen from the outside as a very complicated but ultimately purely mechanical game played with fixed rules, like checkers, chess, or Go. These are activities in which machines have already demonstrated superhuman ability.

Some people imagine a world where computers give us all the answers. I dream bigger. I want them to ask good questions.

But for me, math is different: it is a creative pursuit that calls on our intuition as much as our ability to compute. (To be fair, chess players probably feel the same way.) Henri Poincaré, the mathematician who re-envisioned the whole subject of geometry at the beginning of the 20th century, insisted it would be hopeless

"to attempt to replace the mathematician's free initiative by a mechanical process of any kind. In order to obtain a result having any real value, it is not enough to grind out calculations, or to have a machine for putting things in order: it is not order only, but unexpected order, that has a value. A machine can take hold of the bare fact, but the soul of the fact will always escape it."

But machines can make deep changes in mathematical practice without shouldering humans aside. Peter Scholze, winner of a 2018 Fields Medal (sometimes called the "Nobel Prize of math") is deeply involved in an ambitious program at the frontiers of algebra and geometry called "condensed mathematics" — and no, there is no chance that I'm going to try to explain what that is in this space.

## Meet AI, your new research assistant

Credit: Possessed Photography via Unsplash

What I am going to tell you is the result of what Scholze called the "Liquid Tensor Experiment." A community called Lean, started by Leonardo de Moura of Microsoft Research and now open-source and worldwide, has the ambitious goal of developing a computer language with the expressive capacity to capture the entirety of contemporary mathematics. A proposed proof of a new theorem, formalized by translation into this language, could be checked for correctness automatically, rather than staking its reputation on fallible human referees.

Scholze asked last December whether the ideas of condensed mathematics could be formalized in this way. He also wanted to know whether it could express the ideas of a particularly knotty proof that was crucial to the project — a proof that he was pretty sure was right.

When I first heard about Lean, I thought it would probably work well for some easy problems and theorems. I underestimated it. So did Scholze. In a May 2021 blog post, he writes, "[T]he Experiment has verified the entire part of the argument that I was unsure about. I find it absolutely insane that interactive proof assistants are now at the level that within a very reasonable time span they can formally verify difficult original research."

And the contribution of the machine wasn't just to certify that Scholze was right to think his proof was sound; he reports that the work of putting the proof in a form that a machine could read improved his own human understanding of the argument!

The Liquid Tensor Experiment points to a future where machines, rather than replacing human mathematicians, become our indispensable partners. Whether or not they can take hold of the soul of the fact, they can extend our grasp as we reach for the soul.

## Slicing up a knotty problem

That can take the form of "proof assistance," as it did for Scholze, or it can go deeper. In 2018, Lisa Piccirillo, then a PhD student at the University of Texas, solved a long-standing geometry problem about a shape called the Conway knot. She proved the knot was "non-slice" — this is a fact about what the knot looks like from the perspective of four-dimensional beings. (Did you get that? Probably not, but it doesn't matter.) The point is this was a famously difficult problem.

A few years before Piccirillo's breakthrough, a topologist named Mark Hughes at Brigham Young had tried to get a neural network to make good guesses about which knots were slice. He gave it a long list of knots where the answer was known, just as an image-processing neural net would be given a long list of pictures of cats and pictures of non-cats.

Hughes's neural net learned to assign a number to every knot; if the knot were slice, the number was supposed to be 0, while if the knot were non-slice, the net was supposed to return a whole number bigger than 0. In fact, the neural net predicted a value very close to 1 — that is, it predicted the knot was non-slice — for every one of the knots Hughes tested, except for one. That was the Conway knot.

For the Conway knot, Hughes's neural net returned a number very close to 1/2, its way of saying that it was deeply unsure whether to answer 0 or 1. This is fascinating! The neural net correctly identified the knot that posed a really hard and mathematically rich problem (in this case, reproducing an intuition that topologists already had).

Some people imagine a world where computers give us all the answers. I dream bigger. I want them to ask good questions.

Dr. Jordan Ellenberg is a professor of mathematics at the University of Wisconsin and a number theorist whose popular articles about mathematics have appeared in the New York Times, the Wall Street Journal, Wired, and Slate. His most recent book is Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else.

## Hidden philosophy of the Pythagorean theorem

### Pythagoras may have believed that the entire cosmos was constructed out of right triangles.

Credit: Michael Dziedzic via Unsplash
• Ancient Greeks believed that fire, air, water, and earth were the four elements of the universe.
• Plato associated these four elements with 3D geometrical solids.
• Pythagoras may have believed that the right triangle formed the basis of all reality.

In Plato's dialogue, the Timaeus, we are presented with the theory that the cosmos is constructed out of right triangles.

This proposal Timaeus makes after reminding his audience [49Bff] that earlier theories that posited "water" (proposed by Thales), or "air" (proposed by Anaximenes), or "fire" (proposed by Heraclitus) as the original stuff from which the whole cosmos was created ran into an objection: if our world is full of these divergent appearances, how could we identify any one of these candidates as the basic stuff? For if there is fire at the stove, liquid in my cup, breathable invisible air, and temples made of hard stone — and they are all basically only one fundamental stuff — how are we to decide among them which is most basic?

## A cosmos of geometry

However, if the basic underlying unity out of which the cosmos is made turns out to be right triangles, then proposing this underlying structure — i.e., the structure of fire, earth, air, and water — might overcome that objection. Here is what Timaeus proposes:

"In the first place, then, it is of course obvious to anyone, that fire, earth, water, and air are bodies; and all bodies have volume. Volume, moreover, must be bounded by surface, and every surface that is rectilinear is composed of triangles. Now all triangles are derived from two [i.e., scalene and isosceles], each having one right angle and the other angles acute… This we assume as the first beginning of fire and the other bodies, following the account that combines likelihood with necessity…" [Plato. Timaeus 53Cff]

A little later in that dialogue, Timaeus proposes further that from the right triangles, scalene and isosceles, the elements are built — we might call them molecules. If we place on a flat surface equilateral triangles, equilateral rectangles (i.e., squares), equilateral pentagons, and so on, and then determine which combinations "fold-up," Plato shows us the discovery of the five regular solids — sometimes called the Platonic solids.

Three, four, and five equilateral triangles will fold up, and so will three squares and three pentagons.

If the combination of figures around a point sum to four right angles or more, they will not fold up. For the time being, I will leave off the dodecahedron (or combination of three pentagons that makes the "whole" into which the elements fit) to focus on the four elements: tetrahedron (fire), octahedron (air), icosahedron (water), and hexahedron (earth).

## Everything is a right triangle

Now, to elaborate on the argument [53C], I propose to show using diagrams how the right triangle is the fundamental geometrical figure.

All figures can be dissected into triangles. (This is known to contemporary mathematicians as tessellation, or tiling, with triangles.)

Inside every species of triangle — equilateral, isosceles, scalene — there are two right triangles. We can see this by dropping a perpendicular from the vertex to the opposite side.

Inside every right triangle — if you divide from the right angle — we discover two similar right triangles, ad infinitum. Triangles are similar when they are the same shape but different size.

And thus, we arrive at Timaeus' proposal that the right triangle is the fundamental geometrical figure, in its two species, scalene and isosceles, that contain within themselves an endless dissection into similar right triangles.

Now, no one can propose that the cosmos is made out of right triangles without a proof — a compelling line of reasoning — to show that the right triangle is the fundamental geometrical figure. Timaeus comes from Locri, southern Italy, a region where Pythagoras emigrated and Empedocles and Alcmaon lived. The Pythagoreans are a likely source of inspiration in this passage but not the other two. What proof known at this time showed that it was the right triangle? Could it have been the Pythagorean theorem?

## Pythagorean theorem goes beyond squares

We now know that there are more than 400 different proofs of the famous theorem. Does one of them show that the right triangle is the basic geometrical figure? Be sure, it could not be a² + b² = c² because this is algebra, and the Greeks did not have algebra! A more promising source — the proof by similar right triangles — is the proof preserved at VI.31.

Notice that there are no figures at all on the sides of the right triangle. (In the above figure, the right angle is at "A.") What the diagram shows is that inside every right triangle are two similar right triangles, forever divided.

Today, the Pythagorean theorem is taught using squares.

But, the Pythagorean theorem has nothing to do with squares! Squares are only a special case. The theorem holds for all figures similar in shape and proportionately drawn.

So, why the emphasis on squares? Because in the ancient Greek world proportional-scaling was hard to produce exactly and hard to confirm, and the confirmation had to come empirically. But squares eliminate the question of proportional scaling.

## Pythagoras and the philosophy of cosmology

We have an ancient report that upon his proof, Pythagoras made a great ritual sacrifice, perhaps one hundred oxen. What precisely was his discovery that merited such an enormous gesture?

Could this review help us to begin to understand the metaphysical meaning of the hypotenuse theorem — namely, that what was being celebrated was not merely the proof that the area of the square on the hypotenuse of a right triangle was equal to the sum of the areas of the squares on the other two sides, but moreover, was the proof that the fundamental figure out of which the whole cosmos was constructed was the right triangle?

Prof. Robert Hahn has broad interests in the history of ancient and modern astronomy and physics, ancient technologies, the contributions of ancient Egypt and monumental architecture to early Greek philosophy and cosmology, and ancient mathematics and geometry of Egypt and Greece. Every year, he gives "Ancient Legacies" traveling seminars to Greece, Turkey, and Egypt. His latest book is The Metaphysics of the Pythagorean Theorem.

## The foundations of mathematics are unproven

### Philosopher and logician Kurt Gödel upended our understanding of mathematics and truth.

Credit: Peter Macdiarmid via Getty Images
• In 1900, mathematician David Hilbert laid down 23 problems for the mathematics world to solve, the biggest of which was how to prove mathematics itself.
• Far from solving the issue, Kurt Gödel showed just how groundless the axioms of mathematics are.
• Gödel's theorem does not devalue mathematics but reveals that some truths are unprovable.

Everything's a bit crazy at the moment. We're drowning in a sea of lies, half-truths, polarization, debate, argument, and uncertainty. But at least there's math, right? That one sanctuary of truth and certainty. It's the algebraic flotsam we can grip on to, before we're swept away.

Well… look away now if you like it like that, because Kurt Gödel might be about to snatch even that away. His incompleteness theorems shook the foundations of the (math) universe. In fact, he rather did away with those foundations altogether.

## Math problems

In the early 20th century, the famous mathematician David Hilbert laid down 23 problems for the mathematics world to solve. Some of them are particularly esoteric, but the big ones concerned the issues of math's consistency and completeness. Hilbert hated the fact that the whole of mathematics depended on certain "axioms" that were, themselves, not proven. He wanted no loose ends, paradoxes, or unproven items. This was math after all!

Gödel, though, rather quashed all that.

Gödel would have hated what the postmodernists made of his work.

To see how, we have to first know that "axioms" are those statements that we accept as true before we go about doing math. They're like the letters needed to make words. For example, A + B = B + A is an axiom, as are all the functions of arithmetic and so on. Simply put, axioms are the building blocks of mathematics. They're as true for Euclid, drawing squares in ancient Greek dust, as they are for a pained 15-year-old, frowning over some calculus.

The problem is that these axioms are not proven. They're true because they always work, and we observe them as true all of the time. But they're not proven.

## Gödel's challenge

Imagine the whole of mathematics as a huge sack, and inside are all the possible things math can do. It's a mighty big sack, indeed. What Gödel proved is that, first, there exists in this sack a set of things which cannot be proven or disproven, such as axioms. Second, there is no possible way to prove these axioms from within that sack. It's impossible for math, on its own, to prove its own axioms.

Essentially, it's a problem of self-reference. It's an issue seen, too, in Russell's paradox about sets. More famously, the liar paradox imagines a sentence like, "This sentence is false." When you examine it closely, it creates a logical circularity. If the sentence is true, then it's false; but then if it's false, it's true. It's enough to make a robot's brain explode.

Credit: ROBYN BECK via Getty Images

Gödel applied a similar logic to the whole system of mathematics. He took the sentence, "This statement is unproven," and converted it into a number statement about numbers (with a code system known as "Gödel numbering"). He discovered that this proposition cannot be proven within that system.

Going even further than this, Gödel concluded that in every system that's rich enough to allow for arithmetic, there will be a proposition within it that cannot be proven by it's own tools. We need some kind of "meta language" to prove the rules by which a system operates. It's a bit like how we can't see our own eyes or draw around the hand that's holding the pencil.

## How postmodernists weaponized Gödel

Gödel has been misrepresented, even in his lifetime. For instance, certain postmodernist philosophers used him to say, "There is no truth! Even math is groundless!" They wanted to show how everything was meaningless, and truth amounted only to opinion.

But this isn't the point. Gödel only showed that truth does not always need to be proven. This is, of course, no small thing. To pull apart truth and provability, to allow for "unproven truths," seems highly counterintuitive

Gödel, himself, thought there were objective truths. His theory only went to show the limitations of mathematics but not that it was flawed in any way. He would have hated what the postmodernists made of his work.

Jonny Thomson teaches philosophy in Oxford. He runs a popular Instagram account called Mini Philosophy (@philosophyminis). His first book is Mini Philosophy: A Small Book of Big Ideas.

## There is no dark matter. Instead, information has mass, physicist says

### Is information the fifth form of matter?

Photo: Shutterstock
• Researchers have been trying for over 60 years to detect dark matter.
• There are many theories about it, but none are supported by evidence.
• The mass-energy-information equivalence principle combines several theories to offer an alternative to dark matter.

### The  “discovery” of dark matter

We can tell how much matter is in the universe by the motions of the stars. In the1920s, physicists attempting to do so discovered a discrepancy and concluded that there must be more matter in the universe than is detectable. How can this be?

In 1933, Swiss astronomer Fritz Zwicky, while observing the motion of galaxies in the Coma Cluster, began wondering what kept them together. There wasn't enough mass to keep the galaxies from flying apart. Zwicky proposed that some kind of dark matter provided cohesion. But since he had no evidence, his theory was quickly dismissed.

Then, in 1968, astronomer Vera Rubin made a similar discovery. She was studying the Andromeda Galaxy at Kitt Peak Observatory in the mountains of southern Arizona when she came across something that puzzled her. Rubin was examining Andromeda's rotation curve, or the speed at which the stars around the center rotate, and realized that the stars on the outer edges moved at the exact same rate as those at the interior, violating Newton's laws of motion. This meant there was more matter in the galaxy than was detectable. Her punch card readouts are today considered the first evidence of the existence of dark matter.

Many other galaxies were studied throughout the '70s. In each case, the same phenomenon was observed. Today, dark matter is thought to comprise up to 27% of the universe. "Normal" or baryonic matter makes up just 5%. That's the stuff we can detect. Dark energy, which we can't detect either, makes up 68%.

Dark energy is what accounts for the Hubble Constant, or the rate at which the universe is expanding. Dark matter on the other hand, affects how "normal" matter clumps together. It stabilizes galaxy clusters. It also affects the shape of galaxies, their rotation curves, and how stars move within them. Dark matter even affects how galaxies influence one another.

### Leading theories on dark matter

NASA writes: 'This graphic represents a slice of the spider-web-like structure of the universe, called the "cosmic web." These great filaments are made largely of dark matter located in the space between galaxies.'

Credit: NASA, ESA, and E. Hallman (University of Colorado, Boulder)

Since the '70s, astronomers and physicists have been unable to identify any evidence of dark matter. One theory is it's all tied up in space-bound objects called MACHOs (Massive Compact Halo Objects). These include black holes, supermassive black holes, brown dwarfs, and neutron stars.

Another theory is that dark matter is made up of a type of non-baryonic matter, called WIMPS (Weakly Interacting Massive Particles). Baryonic matter is the kind made up of baryons, such as protons and neutrons and everything composed of them, which is anything with an atomic nucleus. Electrons, neutrinos, muons, and tau particles aren't baryons, however, but a class of particles called leptons. Even though the (hypothetical) WIMPS would have ten to a hundred times the mass of a proton, their interactions with normal matter would be weak, making them hard to detect.

Then there are those aforementioned neutrinos. Did you know that giant streams of them pass from the Sun through the Earth each day, without us ever noticing? They're the focus of another theory that says that neutral neutrinos, that only interact with normal matter through gravity, are what dark matter is comprised of. Other candidates include two theoretical particles, the neutral axion and the uncharged photino.

Now, one theoretical physicist posits an even more radical notion. What if dark matter didn't exist at all? Dr. Melvin Vopson of the University of Portsmouth, in the UK, has a hypothesis he calls the mass-energy-information equivalence. It states that information is the fundamental building block of the universe, and it has mass. This accounts for the missing mass within galaxies, thus eliminating the hypothesis of dark matter entirely.

### Information theory

To be clear, the idea that information is an essential building block of the universe isn't new. Classical Information Theory was first posited by Claude Elwood Shannon, the "father of the digital age" in the mid-20th century. The mathematician and engineer, well-known in scientific circles—but not so much outside of them, had a stroke of genius back in 1940. He realized that Boolean algebra coincided perfectly with telephone switching circuits. Soon, he proved that mathematics could be employed to design electrical systems.

Shannon was hired at Bell Labs to figure out how to transfer information over a system of wires. He wrote the bible on using mathematics to set up communication systems, thereby laying the foundation for the digital age. Shannon was also the first to define one unit of information as a bit.

There was perhaps no greater proponent of information theory than another unsung paragon of science, John Archibald Wheeler. Wheeler was part of the Manhattan Project, worked out the "S-Matrix" with Niels Bohr and helped Einstein develop a unified theory of physics. In his later years, he proclaimed, "Everything is information." Then he went about exploring connections between quantum mechanics and information theory.

He also coined the phrase "it from bit" or that every particle in the universe emanates from the information locked inside it. At the Santa Fe Institute in 1989, Wheeler announced that everything, from particles to forces to the fabric of spacetime itself "… derives its function, its meaning, its very existence entirely … from the apparatus-elicited answers to yes-or-no questions, binary choices, bits."

### Part Einstein, part Landauer

Vopson takes this notion one step further. He says that not only is information the essential unit of the universe but also that it is energy and has mass. To support this claim, he unifies and coordinates special relativity with the Landauer Principle. The latter is named after Rolf Landauer. In 1961, he predicted that erasing even one bit of information would release a tiny amount of heat, a figure which he calculated. Landauer said this proves information is more than just a mathematical quantity. This connects information to energy. Through experimental testing over the years, the Landauer Principle has held up.

Vopson says, "He [Landauer] first identified the link between thermodynamics and information by postulating that logical irreversibility of a computational process implies physical irreversibility." This indicates that information is physical, Vopson says, and demonstrates the link between information theory and thermodynamics.

In Vopson's theory, information, once created has "finite and quantifiable mass." It so far applies only to digital systems, but could very well apply to analogue and biological ones too, and even quantum or relativistic-moving systems. "Relativity and quantum mechanics are possible future directions of the mass-energy-information equivalence principle," he says.

In the paper published in the journal AIP Advances, Vopson outlines the mathematical basis for his hypothesis. "I am the first to propose the mechanism and the physics by which information acquires mass," he said, "as well as to formulate this powerful principle and to propose a possible experiment to test it."

### The fifth state of matter

To measure the mass of digital information, you start with an empty data storage device. Next, you measure its total mass with a highly sensitive measuring apparatus. Then, you fill it and determine its mass. Next, you erase one file and evaluate it again. The trouble is, the "ultra-accurate mass measurement" device the paper describes doesn't exist yet. This would be an interferometer, something similar to LIGO. Or perhaps an ultrasensitive weighing machine akin to a Kibble balance.

"Currently, I am in the process of applying for a small grant, with the main objective of designing such an experiment, followed by calculations to check if detection of these small mass changes is even possible," Vopson says. "Assuming the grant is successful and the estimates are positive, then a larger international consortium could be formed to undertake the construction of the instrument." He added, "This is not a workbench laboratory experiment, and it would most likely be a large and costly facility." If eventually proved correct, Vopson will have discovered the fifth form of matter.

So, what's the connection to dark matter? Vopson says, "M.P. Gough published an article in 2008 in which he worked out … the number of bits of information that the visible universe would contain to make up all the missing dark matter. It appears that my estimates of information bit content of the universe are very close to his estimates."