Into the Unknown, page 22
Could there be different branches of time? I don’t see any reason this would be disallowed; in fact, the many-worlds interpretation of quantum mechanics leads to a branching of the universe into different timelines. In this case, it is interesting to consider why you find your consciousness in this particular branch and not some other. However, I also don’t know how we would know if this topological feature of time were credible. To test this, we might need to find a way to get out of our own timeline, which for the moment seems rather impossible.
Does time have a constant rate? Special and general relativity definitively answer no to this question. But we can take this question to a level deeper—even if you are sitting still in the same place with the same gravitational field, is it possible that the rate of time universally slows down or speeds up? The most annoying part of this question for me is that it isn’t clear what a “rate” of time would be with respect to. For example, if you were watching a toy boat float on a flowing river, you could measure the speed of river flowing with respect to the riverbank and end up with units of something like miles per hour. Since we seem to be trapped in the flow of time with no temporal riverbanks in sight, there is no obvious reference. For all we know, the general flow of time in the universe could be slowing down or speeding up.
If the general flow of time were to change for the entire universe, you might justifiably ask if this would matter (indeed, I have had students ask this in class). In terms of our own physical existence or observable consequences, I am not sure that it does matter in any practical way. However, as someone who cares about the fundamental nature of reality and the limits of our knowledge, I find it deeply unsettling to have so little understanding. Also, I’m just plain curious to know.
In terms of understanding the nature of time, our progress is mixed. Thanks to relativity, we do have a solid sense that time exists in and of itself, which is more than we knew a couple centuries ago. On the other hand, the quantum nature of time opens a line of questions that we didn’t even know we didn’t know a couple centuries ago. To be honest, I’m not even sure I can wrap my head around living in a world in which we understood the true nature of time and were presumably also able to manipulate it. In this case, Temporal Ethics would need to become a standard course in college curriculums.
For the moment, we are left needing to accept time and space as entangled dimensions that—for reasons we don’t understand—seem to manifest themselves differently. Are there other dimensions that could be manifested still differently?
Interstition
Thinking of the dimensions of space as a sort of fabric is already removed from our lived experience, but adding time to this framework brings us face-to-face with the abstract idiosyncrasies of dimensions. These dimensions of space-time are the arena in which everything in our universe seems to play out, yet we understand very little about them.
The role of the dimensionality of our universe is entwined across the topics in this book. We have had to consider whether the dimensions of space and time might have come into existence with the creation of our universe. We have seen them behaving strangely near black holes, and we also have seen that hyperdimensional wormholes are not only not disallowed but may be responsible for the fabric of space-time itself. Even the laws of nature may be dependent on whether there are other dimensions (both big and small).
We seem to be restricted to going about our business in three dimensions of space and a singular dimension of time, but that doesn’t mean there couldn’t be other dimensions to which we don’t have ready access. Of course, probing dimensions that appear to be inaccessible is a particular challenge for scientists, who, as a rule, are fond of empirical inquiry. However, it turns out there are hints of other dimensions in our theories, and as we think more creatively and abstractly, we are coming up with ideas that could help us explore beyond the veil of space-time.
8
Are There Hidden Dimensions?
You may have noticed that the universe we live in appears to have four dimensions—3 of space and 1 of time (which we often write in shorthand as 3 + 1 to indicate there are two types of dimensions). It is easy to take this dimensionality for granted, but this precise arrangement of operative dimensions turns out to be essential for life to emerge in the universe (more on this in the chapter on fine-tuning).
It could be that there are only 3 + 1 dimensions, in which case we might ask why 3 + 1? Could there have been a different number of dimensions? Similarly, we might ask whether there could be different types of dimensions, which are neither spatial nor temporal, but something else entirely. A different number or type of dimensions is challenging to intuit, if for no other reason than we didn’t evolve to do so. Somewhere in our evolution it was important to be able to perceive and construct mental maps of our familiar 3 + 1 dimensions, which enables us to do beneficial things like move around and notice things happening. Given that these 4 familiar dimensions are the extent of the reality that we have ever perceived, it is natural to take these 4 dimensions (and only these 4) as self-evident—one of those axioms that seems so obvious we might go through life not even realizing we take this as given.
Given the limits of our experience and perception, the question of whether other dimensions can or do exist deserves a closer inspection. If other dimensions do exist, we might not perceive them for different reasons, depending on the characteristics of such dimensions. For example, hidden dimensions might be “compactified”—which means curled up on tiny scales that we can’t yet probe, or they might not be coupled to the forces in our 3 + 1 domain.
If the forces we interact with don’t extend beyond the familiar 3 + 1 dimensions, it is hard to fathom a way to probe whether such extra dimensions exist—the experiments and devices we can contrive generally implicitly depend on the known forces and their interactions. A positivist might argue that it doesn’t matter if other dimensions exist if they have no influence on their universe. This position is hard to argue with, but that doesn’t stop me from arguing—in particular, if we assume that either there are no other dimensions or we can’t interact with them, this situation becomes a bit self-fulfilling.
Introduction to Dimensions
Let’s clarify what we mean when we’re talking about dimensions in this context. The essence of what a dimensionality means is the number of coordinates necessary to pinpoint where something is, which in the apparent dimensionality of our universe means in space and time. If something has zero dimensions (which corresponds to a point) then no coordinates are necessary; the thing in zero dimensions has no flexibility in where (or when) it might be, so providing coordinates is superfluous. The underlying concept with dimensions is that they are orthogonal to each other—the properties of each axis are independent of the others.
We can think about going from a lower dimensional object to a higher dimensional object by shifting (formally, we call this “translating”) the lower dimensional object, connecting the vertices, and projecting onto a 2D surface—conveniently like this page of the book.
In principle, you can keep going ad nauseam, but I don’t have the patience to go beyond 4D in this scheme—there are way too many vertices to keep straight on a 2D sheet of paper. Heck, I have trouble even drawing a 4D cube freehand. At least with 4D, you can tell your brain to think about this extra dimension like it might consider time, and it isn’t quite so hard to wrap our heads around—at one moment the 3D cube is in one position, and a moment later it has shifted to the other position.
When we are considering physical objects, it might be challenging to intuit how orthogonal dimensions would behave beyond the familiar 3 + 1. However, you use multidimensional constructs in your daily life, perhaps without even realizing it. So instead of just thinking about shapes and physical objects, we can use an example that is near and dear to my heart: food. Food can be described along many axes that are independent: sweetness, saltiness, spiciness, crunchiness, temperature, caloric value, color, and so on. When deciding what you might be in the mood for, you are mentally moving around in this multidimensional space. You could even make a plot of properties of food along different sets of axes—for example, sweet versus salty, collapsing all the other properties down onto these two dimensions; in making a plot of sweet versus salty, it doesn’t matter how spicy or crunchy the particular food is.
Multidimensional mathematical objects work the same way—generating higher dimensional objects and manipulating them is straightforward. However, visualizing things in higher dimensions requires some mental gymnastics, and it wouldn’t be wise to jump straightaway into doing backflips without warming up (or in my case, backflips wouldn’t be wise to try under any circumstances). So, let’s do a little warm-up exercise, with a simple circle. Your first challenge is to imagine (or draw) a circle and find a point as far away from the boundary of the circle as you can while staying equidistant from the boundary. Hopefully, you’ve landed in the center of the circle.
Here is the next step: go farther. It might take a moment to realize that to go any farther and stay equidistant from the boundary, you need to pop into the third dimension—into or out of the page. Visualizing popping into 3D isn’t so hard for our 3D-wired brains.
But what about higher dimensions? We can do the same exercise as above, but now with a sphere instead. Imagine (or draw) a sphere and find a point as far away from the boundary of the sphere as possible while staying equidistant from the bounding shell, which will lead to a point in the center of the sphere. The next step is to go farther.
The cognitive experience I have when I do this last step feels like my brain knows what it wants to do, it just can’t do it; there is almost a physical sensation of gears getting stuck. I would love to see modern fMRI imagery of people’s brains trying to visualize higher dimensions. I will confess to exposing my kids to videos and imagery of higher dimensional objects at a young age, just in case.
In the meantime, we do have some other tricks to try to get our heads around what higher dimensional objects are like, which generally involves mapping them down to lower dimensions using projections, rotations, slices, and unfolding (and sometimes a combination of these). A necessary limitation with these techniques is that each of them loses information in the translation to lower dimension space, which we try to compensate for by rotating, projecting, and slicing in different ways.
To get a better intuitive sense for how these tricks work, I’ve found a good place to start is by flexing our 3D visualization abilities and envisioning how things would appear to an observer who was limited to 2D perception. Projections are a solid starting point because we regularly encounter them and the mental gymnastics are trivial. Here is an example of a 3D rectangular shell projected down to 2D: if you were only able to see the shadow of this shape, you would need to rotate the rectangle in three orthogonal projections to infer its full 3D dimensionality.
We can also take slices of an object. For example, if a 3D sphere were moving through a 2D surface, it would manifest as a circle that first grew larger and then shrank back before disappearing.
Finally, we can fold and unfold objects between dimensions. I absolutely hate folding things, especially clothes and sheets; if you share my dislike for folding, try thinking of this as a fun craft, like origami. If you lived in 2D, witnessing a cube fold and unfold in 2D would be baffling; it would be moving in and out of dimensions you are unable to perceive as though it were moving through itself.
If we dial this up a notch, we can look at how a 4D cube would manifest in our familiar three dimensions. These hypercubes are also known as “tesseracts,” which is a word and a concept I have held dear for many decades; tesseracts had a profound impact on my imagination as a child when I first read A Wrinkle in Time, in which tesseracts are a major plot device. I have often wondered whether reading this book had an impact on my interests and career trajectory—the power of children’s literature should not be underestimated. To this day, I sometimes assign this book as fun extra credit. Go ahead, you know you want to read it.
We can watch rotating projections of 4D cubes in 3D space (and then mapped onto a 2D surface you are viewing), but sadly, printed media like this book are not (yet?) the best platform for video illustrations (YouTube will not let you down here, so a quick look for rotating hypercubes is worth your while if you have access to the internet at the moment). To my mind, projections of rotating higher dimensional objects look like they are turning themselves inside out.
We can also unfold this hypercube, as with the normal 3D cube. This unfolded hypercube is also known as a “Dali cross,” after Salvador Dali’s use of this form in his art (see his painting Corpus Hypercubus). Looking at this shape and trying to figure out how we might fold it into a 4D cube is perplexing, but our perspective on this is directly analogous to how a 2D being might perceive the folding and unfolding of a 3D cube.
While we are talking about hypercubes, we really must take a quick pause at one other waypoint—5D hypercubes. To spare both you and me, I am not even going to attempt to draw a 3D projection of a 5D cube. But… the two-dimensional projections of 5D cubes reveal some extraordinary behavior.
If you stare at the 2D projection of a 5D hypercube for a bit, you may see hints of 3D cubes appearing to poke in and out of the mosaic. This particular 2D tiling is an example of what are now called “Penrose tiles.” As an undergraduate, the entrance of our math building was tiled with a Penrose mosaic—which I thought was brilliant (and now realize this is not at all unusual for math buildings, which doesn’t make it less brilliant). If you tile a surface with only these two shapes in a set of Penrose tiles to infinity, the resulting pattern can have rotational symmetry, and reflection symmetry, but it will never have translation symmetry. While any discrete pattern within the tiles could be repeated an infinite number of times, when an infinite plan is tiled, this tiling can never be picked up, shifted in one direction, and perfectly line up. Thus, this behavior challenges what we tend to think of as order—nice tidy periodic patterns; while these tilings have order, they are not periodic.
This mosaic is an example of a 5D hypercube crystal structure projected onto 2D for your viewing pleasure.
While Roger Penrose had this tiling named after him, examples of tilings like this were present in Islamic art back to at least the fifteenth century in girih tiles. Given that the Middle East was a hotbed of advanced mathematics at the time,1 I find this temporal coincidence to be intriguing. Some mathematicians have speculated that these tilings were deliberately used in Islamic art to generate patterns that don’t repeat, which would demonstrate an exceptionally sophisticated understanding of these shapes that wasn’t “rediscovered” until the 1970s.2
Beyond merely having aesthetic value, Penrose tiles, like the set above, have some extraordinary properties. For example, the number of possible tilings with these two simple shapes is uncountably infinite, which I realize sounds redundant; in mathematical set theory there are both countable and uncountable infinities.3 By analogy, we could imagine counting from 1 to 10 (heck, you don’t have to imagine), and you would have 10 numbers. But if we included all the possible numbers between 1 and 10 including the nonintegers, suddenly we have an uncountably infinite number of possibilities as we expand the number of decimal points (things like 1.5872 and 6.2841). Related to their uncountably infinite number, the tilings are also self-similar or fractal, which means that patterns can repeat within themselves at arbitrary small scales. As a side effect of these properties, if you lived in the 2D world of Penrose tiles and got lost, you could not determine where you were in the pattern, no matter how much you explored.
The shapes in Penrose tiles themselves are also fascinating; they have the golden ratio (phi or 𝜑 = 1.61803…) baked into them in the ratios of their sides and center lines. We could spend an entire chapter discussing the number phi and why it is fascinating, but for the time being I’ll just mention that, like its better-known sibling pi (𝜋), phi (𝜑) is infinite and nonrepeating, and shows up in the natural world over and over through the ratios of lengths, the related angle, and the Fibonacci sequence. The number phi showing up in these curious tiles, which themselves are projections of 5D cubes, starts to feel like a set-up job.
These tiles create what is termed a “quasicrystal,” which is an object with order, but that lacks translational symmetry. A range of quasicrystals are now known to exist in the “real” world, which include coatings used for nonstick pans; if there is no periodic pattern of molecules coating the pan, it is much harder for food to stick. So, the next time you pull out a nonstick pan to make something delicious, you can think about how fascinating quasicrystals are, and how Penrose tiles are projections of 5D hypercubes.
Hints of Other Dimensions?
Given that we don’t appear to be able to interact directly with other dimensions (if they even exist), you might justifiably be wondering what our motivation might be for considering them at all. To be clear, I would argue that all we really have to go on are some curious behaviors of things in the universe that might be easier to explain if we had extra dimensions laying around. This doesn’t mean that these curious behaviors might not be explainable in some other way we haven’t thought of yet.
