Visualizing Dimensions

Thad Roberts is a theoretical physicist, a philosopher of physics, the inspiration behind the New York Times best seller Sex on the Moon: The Amazing Story Behind the Most Audacious Heist in History, and the author of ‘Einstein’s Intuition: Visualizing Nature in Eleven Dimensions‘.


Does anybody here happen to be interested in other dimensions? All right.

Well, thank you all for your time and your space. Good, I’m glad that one worked here.

All right. Imagine a world whose inhabitants live and die believing only in the existence of two spatial dimensions. A plane. These Flatlanders are going to see some pretty strange things happen; things that are impossible to explain within the constraints of their geometry. For example, imagine that one day, some Flatlander scientists observe this: A set of colorful lights that appear to randomly appear in different locations along the horizon. No matter how hard they try to make sense of these lights, they’ll be unable to come up with a theory that can explain them.

Some of the more clever scientists might come up with a way to probabilistically describe the flashes. For example, for every 4 seconds, there’s 11% chance that a red flash will occur somewhere on the line. But no Flatlander will be able to determine exactly when or where the next red light will be seen.

As a consequence, they start to think that the world contains a sense of indeterminacy, that the reason these lights cannot be explained, is that at the fundamental level nature just doesn’t make sense. Are they right? Does the fact that they were forced to describe these lights probabilistically actually mean that the world is indeterministic?


The lesson we can learn from Flatland is that when we assume only a portion of nature’s full geometry, deterministic events can appear fundamentally indeterministic. However, when we expand our view and gain access to the full geometry of the system, indeterminacy disappears. As you can see, we can now determine exactly when and where the next red light will be seen on this line.

We are here tonight to consider the possibility that we are like the Flatlanders. Because, as it turns out, our world is riddled with mysteries that just don’t seem to fit inside the geometric assumptions we have made. Mysteries like warped space-time, black holes, quantum tunneling, the constants of nature, dark matter, dark energy, et cetera. The list is quite long.

How do we respond to these mysteries? Well, we have two choices: We can either cling to our previous assumptions and invent new equations that exist somehow outside of the metric, as a vague attempt to explain what’s going on, or we can take a bolder step, throw out our old assumptions, and construct a new blueprint for reality. One that already includes those phenomena.

It’s time to take that step. Because we are in the same situation as the Flatlanders. The probabilistic nature of quantum mechanics has our scientists believing that deep down, the world is indeterminant. That the closer we look, the more we will find that nature just doesn’t make sense. Hmm… Perhaps all of these mysteries are actually telling us that there’s more to the picture. That nature has a richer geometry than we have assumed.

Maybe the mysterious phenomena in our world could actually be explained by a richer geometry, with more dimensions. This would mean that we are stuck in our own version of Flatland. And if that’s the case, how do we pop ourselves out? At least conceptually? Well, the first step is to make sure that we know exactly what a dimension is.


A good question to start with is: What is it about X, Y and Z that makes them spatial dimensions? The answer is that a change in position in one dimension does not imply a change in position in the other dimensions. Dimensions are independent descriptors of position. So Z is a dimension because an object can be holding still in X and Y while it’s moving in Z. So, to suggest that there are other spatial dimensions is to say that it must be possible for an object to be holding still in X, Y and Z, yet still moving about in some other spatial sense.

But where might these other dimensions be? To solve that mystery, we need to make a fundamental adjustment to our geometric assumptions about space. We need to assume that space is literally and physically quantized, that it’s made of interactive pieces. If space is quantized, then it cannot be infinitely divided into smaller and smaller increments. Once we get down to a fundamental size, we cannot go any further and still be talking about distances in space.

Let’s consider an analogy. Imagine we have a chunk of pure gold that we mean to cut in half over and over. We can entertain two questions here: How many times can we cut what we have in half? and How many times can we cut what we have in half and still have gold? These are two completely different questions, because once we get down to one atom of gold, we cannot go any further without transcending the definition of gold.

If space is quantized, then the same thing applies. We cannot talk about distances in space that are less than the fundamental unit of space for the same reason we cannot talk about amounts of gold that are less than 1 atom of gold.

Quantizing space brings us to a new geometric picture. One like this, where the collection of these pieces, these quanta, come together to construct the fabric of X, Y and Z. This geometry is eleven-dimensional. So if you’re seeing this, you already got it. It’s not going to be beyond you. We just need to make sense of what’s going on.

Notice that there are three distinct types of volume and all volumes are three-dimensional. Distance between any two points in space becomes equal to the number of quanta that are instantaneously between them. The volume inside each quantum is interspatial, and the volume that the quanta move about in is superspatial.

Notice how having perfect information about X, Y, Z position, only enables us to identify a single quantum of space. Also notice that it’s now possible for an object to be moving about interspatially or superspatially without changing its X, Y, Z position at all. This means that there are 9 independent ways for an object to move about. That makes 9 spatial dimensions. 3 dimensions of X, Y, Z volume, 3 dimensions of superspatial volume, and 3 dimensions of interspatial volume. Then we have time, which can be defined as the whole number of resonations experienced at each quantum. And super-time allows us to describe their motion through super-space.

OK, I know this is a whirlwind, a lot faster than I’d like to do it, because there are so many details we can go into. But there’s a significant advantage to being able to describe space as a medium that can possess density, distortions and ripples. For example, we can now describe Einstein’s curved space-time without dimensionally reducing the picture.


Curvature is a change in the density of these space quanta. The denser the quanta get, the less they can freely resonate so they experience less time. And in the regions of maximum density, and the quanta are all packed completely together, like in black holes, they experience no time. Gravity is simply the result of an object traveling straight through curved space. Going straight through X, Y, Z space means both your left side and your right side travel the same distance, interact with the same number of quanta.

So, when a density gradient exists in space, the straight path is the one that provides an equal spatial experience for all parts of a traveling object.

OK, this is a really big deal. If you’ve ever looked at a graph of Einstein curvature before, space-time curvature, you may have not noticed that one of the dimensions was unlabeled. We assumed we took a plane of our world and anytime there was mass in that plane we’ll stretch it; if there was more mass, we stretch it more, to show how much curvature there is.

But what’s the direction we’re stretching in? We got rid of the Z dimension. We blow over that every single time in our books. Here, we didn’t have to get rid of the Z dimension. We got to show curvature in its full form. And this is a really big deal.

Other mysteries that pop out of this map, like quantum tunneling — Remember our Flatlanders? Well, they’ll see a red light appear somewhere on the horizon and then it’ll disappear, and as far as they’re concerned, it’s gone from the universe. But if a red light appears again somewhere else on the line, they might call it quantum tunneling, the same way when we watch an electron, and then it disappears from the fabric of space and reappears somewhere else, and that somewhere else can actually be beyond the boundary that it’s not supposed to be able to get beyond.

OK? Can you use this picture now to solve that mystery? Can you see how the mysteries of our world can transform into elegant aspects of our new geometric picture? All we have to do to make sense of those mysteries is to change our geometric assumptions, to quantize space.

OK, this picture also has something to say about where the constants of nature come from; like the speed of light, Planck’s constant, the gravitational constant and so on. Since all units of expression, Newtons, Joules, Pascals, et cetera, can be reduced to five combinations of length, mass, time, ampere and temperature, quantizing the fabric of space, means that those five expressions must also come in quantized units. So, this gives us five numbers that stem from our geometric map. Natural consequences of our map, with units of one.

There’s two other numbers in our map. Numbers that reflect the limits of curvature. Pi can be used to represent the minimum state of curvature, or zero curvature, while a number we are calling zhe, can be used to represent the maximum state of curvature. The reason we now have a maximum is because we’ve quantized space. We can’t infinitely continue to go on.


What do these numbers do for us? Well, this long list here is the constants of nature, and if you’ve noticed, even though they’re flying by pretty fast, they’re all made up of the five numbers that come from our geometry and the two numbers that come from the limits of curvature. That’s a really big deal by the way, to me it’s a really big deal. This means that the constants of nature come from the geometry of space; they’re necessary consequences of the model.

This is a lot of fun because there are so many punch lines, it’s hard to know exactly who’s going to get caught where. But, this new map, allows us to explain gravity, in a way that’s totally conceptual now, you get the whole picture in your head, black holes, quantum tunneling, the constants of nature, and in case none of those caught your fancy, or you’ve never heard of any of them before, you’ve definitely just barely heard about dark matter and dark energy. Those too are geometric consequences.

Dark matter, when we look at distant galaxies, and watch the stars that orbit about in those galaxies, the stars out at the edges are moving too fast, they seem to have extra gravity. How do we explain this? Well, we couldn’t, so we say there must be some other matter there, creating more gravity, making those effects. But we can’t see the matter. So we call it dark matter. And we define dark matter as something you can’t see. Which is fine, it’s a good step, it’s a good start, but here in our model we didn’t have to take that kind of a leap. We took a leap, we said space is quantized, but everything else fell out from that.

Here, we’re saying, space is made up of fundamental parts, just the same way we believe air is made out of molecules. If that’s true, then an automatic requirement is you can have changes in density, this is where gravity comes from, but you should also have phase changes.

And what stimulates a phase change? Well, temperature. When something gets cold enough, its geometric arrangement will change, and it will change phase. A change in the density here, at the outer regions of the galaxies, is going to cause a gravitational field, because that’s what gravitational fields are, they’re changes in density.

OK? Totally skipped through all that. And now we’ll go to dark energy, in 15 seconds. When we look out into the cosmos, we see that distant light is red shifted, OK? That it loses some of its energy as it’s traveling to us for billions of years.

Now how do we explain that red shift? Well, currently we say it means the universe is expanding. OK? All of our claims that the universe is expanding come from this, from measurements of how the red shift changes, out of this distance to this distance to that distance. And also we measure the expansion that way.

But there’s another way to explain red shift. Just like there’d be another way to explain how if I had a tuning fork tuned to middle C, and I went in a tunnel and you could hear a B note. Sure, you could say it’s because I’m moving away from you inside the tunnel, but it could also be because the pressure of the atmosphere is decreasing while the sound is traveling to your ear.

Here, that seemed a little far-fetched because atmospheric pressure doesn’t decrease fast, but when we’re talking billions of years of light traveling through space, all we need is the quanta themselves to have a small amount of inelasticity and red shift is imminent.

All right, there’s a lot more to explore in this, because if you’re interested, feel free to check out this website and give all the feedback you can. We’re out of time so let me just say, that this blueprint gives us a mental tool, a tool that can expand the reach of our imagination, and maybe, even respark the romanticism of Einstein’s quest.

Thank you.





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