Chapter 1: How to Think About Extra-Dimensions in Physics
Without Having Confirmed Extra-Dimensions in Physics, Our Familiar Dimensions Give Us Clues of What to Look For
This is the introduction of the companion articles to my Unifying Theory of Dimensional Geometry and Interaction series on YouTube. The goal is to go more in-depth and technical on the topics I discussed in my videos. You can view the series in progress by viewing the playlist here: https://www.youtube.com/playlist?list=PLhloW30YwYviMcZ_o6qpaREx2ZQbDL2tr
This article focuses on expanding Chapter 1, which is 08:35–15:11 in my video.
Concept 1: Dimensional Stacking
When we consider our three “traditional” spatial dimensions of x, y, and z, each one is a “stack” of the lower dimension. We start with zero dimensions, which is a point. A line, one dimension, is a series of points. A square, two dimensions, is a series of lines. A cube, three dimensions, is a series of squares. It looks like this:
Logic would then dictate, that if there was a fourth spatial dimension, it should somehow “stack” a cube. However, that is not an easy thing to figure out. Even with a seemingly straight-forward logic, our brains are pretty hardwired to 3D and, to make things worse, we only have 2D (paper) and 3D (modeling) mediums to communicate with. So, taking a guess at a 4D cube or “tesseract” has been a matter of much debate and speculation.
Here is an example of what many think of as a tesseract — It should be noted that some would specify it is more of the “shadow” of a tesseract:
Another way to think about dimensional stacking is with angles. Geometrically, each dimension of space has a center axis that sits 90 degrees perpendicular to the previous dimension. That looks like this:
This way of looking at dimensions is even more difficult when it comes to thinking of a 4th, because from our perspective here in 3D, we don’t have any more room to draw a new axis on a 3D graph. Although, that hasn’t stopped physicists and mathematicians from trying…
While I believe we have been interpreting these patterns (stacking/90 degrees) a little too linearly, I believe they nevertheless are reliable clues to look for while searching extra dimensions.
Hardcore Mode:
The idea of “stacking” is seldom used in these conversations over the more preferred “90 degrees” approach because, while the concept can be drawn in crayon and understood by a child, we have no good mathematical language for it.
A “point” (zero dimensions) has no size, it is “infinitesimally small.” A line, therefore, is finite length made up of non-existent or infinite (depending on how you look at it) points. A line, while having length, has zero height. A square, therefore, “stacked” nothing at all… according to the math. Any time we “collapse” from higher dimensions to lower one or try to “stack” from lower dimensions to higher ones, we get infinities or zeroes.
While calculus, to some degree, has helped us manage these in physics with “limits” (we “skip over” zero, and “brake” before infinity) — it does not allow us to actually describe dimensional transitions inclusively and thoroughly in our understandings. This means that we have to either accept the observable truth at the “crayon level” and include into the meta-knowledge of the physics (which we do with our current three dimensions), or we have to innovate a new form of mathematics to handle it in an “official” way.
Personally, I think we need to change the way we handle “zero,” but that’s a topic for another time….
Concept 2: Distance and Size Change from Different Dimensional Perspectives
Of all the concepts, this one is probably the most familiar. The “tricks” we can do with dimensional perspective are so fun, they have even formed the foundational mechanics for several video games. Right off the bat, the game “FEZ” which came out in 2012 and is still one of the best indie puzzle games of all time, uses dimensional perspective as it’s primary mechanic.
The jist here is that two objects in 3D can be very far apart, but when you force a perspective down to 2D, they can appear to be right next to each other. This can make big objects seem small, and vise-versa. This is popular in vacation photos on Instragram where people make it look like they are pinching the Eiffel Tower or holding up the Leaning Tower of Pisa.
Of course, this is so familiar it may seem like a “big duh!” But it is important to keep in mind as higher dimensions can be in plain sight, but hidden to us through “forced perspective.” In fact, current physics is already riddled with things like “hyperbolic geometry” and “projection.”
For example, is the “singularity” in a black hole an infinitesimally small point in 3D space, or an immeasurable length (hole/hill) into an extra-dimension tangential to our current three? Don’t worry, I’ll get to that question later.
Concept 3: Higher Dimensions Can Only Be Observed in Lower Dimensions Through Relative Motion and Time
One thing we can know with great confidence is that when higher dimensions intersect with lower dimensions, those interactions can only be observed or identified via relative motion or time. For example, if a three-dimensional circular rod passes through a two-dimensional plane, that rod will only be observed as a circle that “appears” and then “disappears” as it passes through. The only evidence of the length of rod is the time the circle appeared. Of course, there is no way to discern between the length and the speed of the rod. Here is a visual example using a 3D sphere passing through a 2D plain:
This presents a dilemma: How can we tell the difference between a lower dimension object that “appears and disappears” and a higher dimensional object “passing through?” The answer to that is energy. Higher dimensional interactions with lower dimensions should still give or take more energy from a system than their lower dimensional doppelgangers.
For example, a flat 2D circle should not be able to have the same weight or force than a higher dimensional rod that just looks like a 2D circle. By observing the mysterious loss or gain of energy in a given system or interaction can help us become aware of when higher dimensions are coming into play.
Have we observed any such phenomenon? Yes, all the time. This is in essence what we have called “gravity.” We see it in the creation and destruction of particles. We see it in the expansion of the Universe. The frustrating part is that academia has decided, so far, to consider throwing out “conservation of energy” rather than pursue higher dimensional study. But, that’s why we’re here…
Concept 4: A Single Object (or Connected Objects) in a Higher Dimension Can Be Perceived as Separate Objects on Lower Dimensions.
This my favorite higher dimensional concept. A complex object, being a strange shape or several objects connected into a single group or system, can appear as separate and distinct objects in a lower dimension. The only way to know if these objects are united in a higher dimension is if interacting with one of them instantly effects the other.
For example, consider this extra dimensional donut (or torus if you are nerdy):
This idea alone is a much better assumption/interpretation of what we observe with quantum entanglement. If entangled particles are interacting in a higher dimension, then there is no violation of locality, causality, or the “speed of light.” But again, academia would currently rather burn all our laws of physics than consider this as a possibility.
Ironically, the “multiverse” or “many worlds” are currently considered a more reasonable explanation (certainly more popular in science journalism). It boggles the mind.
Concept 5: Extra Dimensions as Objects vs Space
OK, this one can also seem very “no duh!” But I have to talk about it as it is a serious problem in how we interpret higher dimensional geometry. Higher dimensions can be experienced as either objects OR the containing space around the object.
When trying to understand the effects and influences of higher dimensions, it is absolutely critical to discern if the thing we are calculating is an object, the path of an object (I neglected this option in the video), or the containing space of the object.
For example the “curve of space” in General Relativity comes from interpreting the influences of gravity on an object’s vector as the space itself, rather than the influence of the containing space on the path of the object. Another attempt at higher dimensional geometry is “manifolds.” Which, depending on the example, are either higher dimensions “projected” into lower dimensions, or an attempt to “bend” extra dimensions into shapes we can comprehend in three dimensions.
As you can see in the above image, it’s hard to tell now what we are looking at. Is this a 6D object or 6D space? I believe this is best understood as the calculated path (or potential paths) of an object in 6D space. However, the real question is: Do the mathematicians making it know what it is? The problem is that most of the math we have developed is designed to calculate the path of an object, not the containing space. But that’s something I will dive more into later.
Either way, this simple yet critical concept that is often lost in conversation, communication, and interpretation of the math.
Concept 6: Limited vs Infinite Extra Dimensions
This is a more difficult concept and, unlike some of the previous ones, can feel counter-intuitive. The description is simple: A dimension can be “infinite” or limited, and a limited dimension can still be everywhere at once. But getting your head around it can be a challenge.
The easier part is that a dimension can be “infinite” or limited means that we have to consider the possibility that the containing space of the Universe may not be equal on all sides. As far as we can tell, length, width, and height (or x,y,z), seem to go on forever. However, there is strong evidence that we may have some limited dimensions that do impose some restrictions.
A mostly uncontroversial example of this would be the “Plank Length” — although that is not universally recognized as a dimension of space. The harder part is the “everywhere at once.” If a dimension is limited, wouldn’t we run into it? The answer is no, but that is because a dimension of space or “side of the Universe” would be a part of the containing space, not an object. In other words, it’s not like a room with limited square feet. It’s simply a limit imposed on a particular direction or angle.
For example, if I had a fourth dimensional box (or “tesseract”) and the fourth dimension was limited, then my box could be any length in three of the dimensions, but can’t go past a certain length in the fourth. However, I can move that box anywhere in space, and the restriction would remain the same. The “limit” is everywhere relative to the other dimensions.
In my video, I call this an “Ant Farm Dimension.” As an ant farm is a mostly two-dimensional environment (length, height) with a very limited third dimension (width). No matter where they go in their two-dimensional world, the third dimension would always be there limiting movement in a particular direction.
If the two dimensions went on forever, the ants would still think they had an “infinite” universe, but the third dimension would seem more like a strange physics restriction than “space.” They might call it “rotation” as it is only wide enough to allow them to turn around. Of course, this mental experiment also implies more Pixar-like intelligent ants.
Concept 7: Energy Distribution Through Extra Dimensions
In my video, this was my worst explained concept. The idea here is that for every dimension to add to a system, you have exponentially more degrees of motion to express the energy of that system. This means that something which seems like a large expression of energy in a single dimension, would look like a small of expression of energy over many dimensions.
The most recognizable analog to this is the idea of force divided by area. If you push against someone with an open palm, you are not going to do much damage. But, with only a small application of force behind the tip of a needle, you can easily pierce the skin. That’s because the force of pressure you can generate from your arm goes from being spread out across your hand, to focused down into the tiny point of a needle.
So, if were to imagine an expression of energy that applies evenly in every direction, like a bomb exploding in the vacuum of space, we can use the following thought experiment: I believe this effect is what we are seeing in the case of nuclear bombs. I am proposing that atoms, and sub-atomic particles, are examples of huge amounts of energy being expressed over many more dimensions.
The massive explosions we get from a nuclear reaction are all the energy in the atom being pulled out of “higher” dimensions and compressed into our more familiar “lower” dimensions. Every nuclear reaction, fusion or fission, requires a particle to “give up mass.” The energy we get from the reaction is from the mass surrendered by the particle. The implication I am making is that mass itself, and what determines if a particle has mass or not, is based on the dimensions in which energy is expressed.
We already know mass and energy are interchangeable, so now we just need to understand what the “line” is crossed that turns one into the other. Of course, these are all just ideas at the moment, my goal as I continue on is to prove them. Or, at the very least, make a strong case for them.
Caveat: What if Extra Dimensions Aren’t “Linear”
With all these concepts and examples, there is one major potential problem: they all assume a linear/consistent scaling to extra dimensions.
What if there are extra dimensions that exist at angles we haven’t even been able to imagine yet? What if the logic we are applying to extra dimensions is a little too dependent on our understanding of our current three spatial dimensions?
I present all of these a “concepts” because they are far from “rules.” They are clues to point us in a direction, but we still have to open to discovery.
You Don’t Have to Take My Word For It:
Here are some other great videos that cover extra-dimensional concepts that I believe are well done and useful for education:
Previous Article: Introduction: A Unifying Theory of Dimensional Geometry and Interaction
Next Chapter: Misconceptions About Gravity (will link here when ready)
