Stephen Wolfram’s hypergraph project aims for a fundamental theory of physics

Simple rules generating complicated networks may be how to build the universe

computer visualization of a hypergraph

A computer visualization depicts a “hypergraph” consisting of relationships between points constructed by repeatedly applying a simple rule for expanding, or “updating,” the network. Stephen Wolfram believes a massively complex hypergraph could represent all of physical space and its contents, a clue to finding the fundamental theory of physics.

Courtesy of Wolfram Research

Thanks to Kevin Bacon, everybody nowadays knows about networks.

There are not only Bacon-like networks of actors, linked by appearing in the same film, but also social networks, neural networks and networks of viral transmission. There are power grid networks, ecological networks and the grandest network of all, the internet. Sometimes it seems like the entire universe must be just one big network.

And maybe it is.

Physicist–computer scientist–entrepreneur Stephen Wolfram believes the universe is a vast, growing network of relationships that constitutes space itself, and everything within it. In this picture, Wolfram sees the basis for the ultimate theory underlying all of physical law.

Wolfram expressed something like this view 18 years ago in a 1,197-page tome entitled A New Kind of Science. But back then his picture was still a little fuzzy. Now he thinks he has found a more sharply focused vision for how to explain reality.

“I’m thrilled to say,” he writes in a summary document released April 14, “that I think we’ve found a path to the fundamental theory of physics.”

At the core of Wolfram’s approach is the notion of a hypergraph. “Graph” in this context is like the diagrammatic representation of a network: lines connecting points. But reality can’t be captured by lines linking points on a flat sheet of paper. Wolfram generates computer visualizations to depict relationships in more complicated “hypergraphs.” (In a hypergraph, the “lines” can connect any number of points, not just one to another.)

Wolfram’s investigations indicate that complex hypergraphs can mimic many features of the universe, including matter and energy, along with reproducing the physical structures and processes described by the theory of relativity and quantum mechanics.

“In our model, everything in the universe — space, matter, whatever — is supposed to be represented by features of our evolving hypergraph,” Wolfram writes.

His key point is that such extremely complex hypergraphs can be produced by applying simple rules to a simple starting point. Suppose you have two “abstract elements” labeled A and B. You have a rule that says every A should be changed to BBB, and every BB should be replaced with A.

Start with A. By the rule, you “update” A to BBB. BBB possesses two BBs. So you update BBB twice: Once, making the first two Bs into an A (making AB), and then making the second two Bs to an A, making BA. So:


is connected to


which is connected to both

AB and BA.

Updates of AB and BA both yield BBBB. But BBBB then makes ABB, BBA and BAB. As you keep on applying the rule, the graph gets more complicated.

These update steps, Wolfram says, correspond to our common notion of time, a sort of ticktock of the cosmic clock. As a rule is repeatedly applied to a set of abstract entities, the resulting connections — the graph of the relationships linking them — correspond to the structure of space. So space (in this picture) is not a mere uniform set of indistinguishable points; rather it is a network of points linked in unfathomably complex patterns that reproduce matter and energy and the relationships collectively known as the laws of physics.

“This is basically how I think space in the universe works,” Wolfram writes. “Underneath, it’s a bunch of discrete, abstract relations between abstract points. But at the scale we’re experiencing it, the pattern of relations it has makes it seem like continuous space of the kind we’re used to.”

It’s sort of like how fish perceive the ocean as a smooth featureless fluid, even though the water is made of discrete tiny molecules.

In a sense, Wolfram believes, everything that exists is basically made from space. “Put another way,” he writes, “it’s the exact same hypergraph that’s giving us the structure of space, and everything that exists in space.”

It almost sounds like theoretical physicists should close up shop and just run some computer simulations using Wolfram’s rules. But as he acknowledges, the job isn’t done yet. So far Wolfram’s project has identified almost 1,000 rules that produce complicated structures that look like a universe. It remains to be seen what rule produces precisely the universe we all actually inhabit.

“Sometime — I hope soon — there might just be a rule … that has all the right properties, and that we’ll slowly discover that, yes, this is it — our universe finally decoded,” Wolfram writes.

In his summary, Wolfram declares that hypergraphs illustrate a principle he calls “causal invariance.” That means that various distinct paths through the hypergraph can sometimes converge. Such convergences allow the cause-and-effect chain of events through time to be preserved.

In a hypergraph “there is not just one path of time; there are many paths, and many ‘histories,’” Wolfram writes. But one supposedly independent path of history can merge with another. “Even when the paths of history that are followed are different, these causal relationships can end up being the same — and that in effect, to an observer embedded in the system, there is still just a single thread of time.”

Thanks to causal invariance, Wolfram’s hypergraphs reproduce many of the consequences of various physical theories, such as Einstein’s special theory of relativity. Traveling rapidly slows down time (as special relativity says) because hypergraph structures corresponding to moving objects make an angle through the hypergraph that extends the distance between updates (or time steps). The speed of light is a maximum velocity, as relativity states, because it represents the maximum rate that information can spread through the hypergraph as it updates. And gravity — described by Einstein’s general theory of relativity — emerges in the relationship between features in the hypergraph that can be interpreted as matter particles. (Particles would be small sets of linked points that persist as the hypergraph updates, something like “little lumps of space” with special properties.)

In an even more complicated extension of these ideas, Wolfram explores how hypergraph properties even correspond to the weird features of quantum mechanics. “In our models, quantum mechanics is not just possible; it’s absolutely inevitable,” Wolfram asserts.

Space as constructed in such hypergraphs can have a very fine structure, like a digital camera sensor with gazillions of megapixels. Wolfram estimates that a hypergraph corresponding to today’s universe might have applied 10500 time steps (incomprehensibly more than the universe’s age in seconds, roughly 1015). So space could be fine-grained enough to contain matter-particle structures much, much smaller than the known particles of physics. In fact, Wolfram suggests, supersmall unknown particles, which he calls oligons, might have been created in abundance shortly after the beginning of the universe. Such oligons, subject only to gravity, could now be hanging out in and around galaxies utterly unnoticed — except for their gravitational impact. Oligons might therefore explain why astronomers infer the existence of vast amounts of invisible “dark matter” in space. (And that could also explain why attempts so far to identify the nature of dark matter have been unsuccessful.)

Similarly, the mysterious “dark energy” that drives the universe to expand at an accelerating rate might just be a natural feature of Wolfram’s hypergraphs. Perhaps dark energy might in essence just be what space itself is made of.

Beyond that, Wolfram believes that his hypergraphs could resolve current disputes about which of many speculative theories are the best bets for explaining fundamental physics. Superstring theory, loop quantum gravity, causal sets and other ideas have all been proposed, and debated, for decades. Wolfram thinks hypergraphs can contain all of them.

“It almost seems like everyone has been right all along,” he writes, “and it just takes adding a new substrate to see how it all fits together.”

Wolfram’s technical paper (and accompanying papers — here and here — by colleague Jonathan Gorard) have been posted on a website promoting his project, and Wolfram is inviting the physics community to participate in pursuing his vision.

“In the end our goal must be to build a bridge that connects our models to existing knowledge about physics,” he writes. “I am extremely optimistic that we are finally on the right track” toward finding the “right” rule for our universe.

That “right rule” would generate a hypergraph with our universe’s precise properties: three (apparent) dimensions of space, the right cosmic expansion rate, the right repertoire of elementary particles with the correct charges and masses, and other features.

But perhaps, Wolfram has realized, seeking one single rule misses a bigger point. Maybe the universe uses all the possible rules. Then all the possible universes are just parts of one really big universe, in which “absolutely everything … can happen — including all events for all possible rules.”

We discern a certain set of physical laws based on the “language” we use to describe and comprehend the world. The elements of this language are tuned to “the kinds of things our senses detect, our measuring devices measure, and our existing physics describes.” The right rule is the one that corresponds to the portion of the hypergraph that we explore from our own particular frame of reference. Life elsewhere might see things differently. “There’s actually an almost infinite diversity of different ways to describe and experience our universe,” Wolfram suggests.

In other words, explaining the physics that applies to our existence might require insight into the mechanisms of a vastly more complex reality, beyond the realm of what we can experience. As Wolfram puts it, “In many ways, we are inevitably skating at the edge of what humans can understand.”

As he acknowledges, much more work will be needed to merge his approach with the successful theories of established physics. And standard physics does have an impressive resume of accomplishments, explaining details about everything from the innards of atoms to the architecture of the universe and the nature of space and time.

Yet mainstream physicists have long suspected that space and time cannot be fundamental concepts. Rather it seems likely that space and time are conventions that must emerge from something deeper. It might be a long shot, but just maybe Wolfram has perceived a path that leads to the depths where reality originates.

Only time — or many more hypergraph updating steps — will tell.

Tom Siegfried is a contributing correspondent. He was editor in chief of Science News from 2007 to 2012 and managing editor from 2014 to 2017.

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