Actually, the real title of this was going to be “Defining the 4th Dimension and Beyond, Using Spacial Geometry.” While that sounds incredibly wimpy to the mathematicians out there, it would surely scare away most average readers.
But, this was envisioned by a child.
Even though it makes use of Pascal’s triangle, tesseracts, and other mathematical concepts, it mostly uses reasoning and basic knowledge, as it came out of a child’s mind. Be assured, it is accessible to everyone.
Be assured also, it has a valid scientific model of the fourth dimension, along with several cosmological implications that are derived from this model.
When I was in junior high school (dinosaur era), I was reading science fiction a lot—something that my teachers didn’t feel was all that useful. And that sci-fi stuff got me thinking—something else my teachers didn’t feel was all that useful either. LOL You would think they’d have appreciated that I was doing some real pondering about the universe, but they wanted me to be doing more useful thinking… like memorizing dates in history or practicing looking up log tables.
Much as they tried to entice me with their subordinate clauses and subjunctive verbs, my mind kept drifting back to the concepts of time and space, which fascinated me. I was always really good about being able to wrap my brain around concepts and visualize them, so I started trying to imagine what a four-dimensional world would look like…and how “time” factored in (as that was what scientists were theorizing were the properties of the fourth dimension).
Well, I did it. And when I was done, I had an actual model of the fourth dimension—along with practical guidelines for defining other higher dimensions as well.
I now know that one element of my overall concept was already developed many years before I did mine—it was obscure mathematics, so no one knew much about it at the time. However, my model went considerably beyond that.
Most importantly, my particular model may yet help science by providing a base algorithm. I don’t believe there is anything that has been able to clearly define the fourth dimension, or reliably predict higher dimensions…or been presented in this way. My model offers the ability to truly visualize a spatial fourth dimension from several perspectives. Additionally, it does point toward deriving possible insights into cosmological mysteries.
To make sure you understand the concepts in the specific way that I developed and used them, I will explain how I got there, so you won’t be giving me the same funny looks my teachers did. Be assured, it’s still amazingly simple.
First, you will need to know what a “dimension” is.
The Definition of a “Dimension”
NOTE: If you already would know a parallel line and a flatlander if you saw one, you can skip to Part 3.
We live in and experience a world we see as “three-dimensional.” Traditionally, that has been defined very much the same way you think of when you define a box—length, width, height (three dimensions). Although, the way my model works out, that isn’t the ideal way to view it when defining multiple dimensions, but we’ll get to that later. Just keep in mind that in our three-dimensional world, the “volume” we live in has length-width-height, and we call that: space.
Space, itself, is made up of both one-dimensional and two-dimensional elements.
A one-dimensional object is a line. Just a line. If we use the box above as an example, either what we called the “length” or the “width” edges of it could be considered a one-dimensional object. A one-dimensional object is any single line—having no physical depth, width, height, or anything else associated with it—only length. It is just a very, very thin line (mathematical lines are infinitely thin).
A two-dimensional object only has length and width, and is just the area on a single “plane” (like a tabletop). For instance, each side (face) of that box is a plane—a two-dimensional-like surface that defines/outlines the space of that box. If the sides of that box had no mass, they would be truly two-dimensional planes.
The size of a plane that would define OUR whole space would be a flat plane that would extend outward, forever straight, all the way to the end of our universe…a super-sized side of a box. Of course, scientists really do have a sense of humor, so they have now confused the issue by informing us that space may actually be CURVED. We won’t go into that one at this point since it doesn’t affect our current discussion, but I know there are some of you out there that love those kind of paradoxes.
Now, when trying to think in universe-sized “spaces,” it gets to be a bit mind-boggling. In order to define something beyond our own “space,” we have to define what space is, so we know where it ends—or at least, where it is NOT. So, how big is space—do we know of something that is outside of our space?
Well, what we commonly call “outer space”, by cosmological definition, is still really OUR own, single, 3-D space. Within the context of mathematical definitions of lines, planes, and spaces, they are considered infinite. A pure “line” would be assumed to extend to infinity.
The length-width-height where we exist is a single “space”—until we get to a wall of some kind. Therefore, our solar system, our own Milky Way galaxy, and all the other galaxies, all the dust and matter…all the way as far as the eye can see… In fact, the whole universe is the 3-D space in which we exist—it’s OUR SPACE. Space can be pretty big. LOL
If it can take up the whole universe, then you’re probably wondering how there can be any place that isn’t our space. Where would the fourth dimension be hiding? Where do we have to go to get outside our “space”?
Current cosmology has evidence that our universe may be finite—meaning there is a limit…a boundary…a definable edge of some kind to our universe. The way they explain it, there was a “big bang” that exploded, which started our universe (another point we will discuss later). The matter/energy from that big bang *IS* our universe—it defines our universe. And ever since then, the matter/energy has been expanding outward from the center of that big bang…which means our universe is expanding…which in turn, means our “space” is expanding. This general concept, as surreal as it seems, is probably accurate. And except for the little detail that our space, therefore, isn’t precisely definable (as it is constantly growing), there could very well be an “out there” that isn’t part of our own universe. That means our three-dimensional space would exist within something that goes beyond three dimensions.
What would the fourth dimension be? It would logically have to be something that would include our space—in the same way that our space includes any lines and planes of the first and second dimensions.
The accepted theory within the scientific community is that “time” is (or is part of) the fourth dimension. So, going back to that box we started with… It exists in that space. But if we were to take a picture of the box there now, and then someone were to remove the box, and then we took a picture of that same space after it was gone, then we would see how that space would have been altered by time. That is the general principle of how the fourth dimension factors into our universe, although there are several variations.
Some current theories embrace parallel universes, as well as alternate universes. But would that imply there are other big bangs and “universes”? Possibly. It could also mean a single event that spawned multiple timeframes. One concept of a “parallel universe” assumes a starting point (e.g. the big bang), but then at every juncture where there is a possibility to change the course of an event via “choice,” two paths/threads are started—one where the path was unaltered, and another where a choice was made, which altered it in some way. For every choice/change, there would be a different timeline created for that alternate universe—spawning an infinite number of universes.
Hopefully, I have all of you following along now, as those definitions are crucial to understanding where this is going.
Visualizing Dimensions 1, 2, and 3
It would help a bit if we could try to visualize what it might be like to exist in another dimension. Seeing things from other perspectives is hard enough in a familiar world, but we’re trying to visit worlds where we couldn’t even exist in our physical state. This is a good brain exercise—as even the futuristic scientists from our favorite sci-fi stories would have trouble replicating a two-dimensional being.
Here is a common example that allows us to visualize fairly easily how a two-dimensional world would see a three-dimensional object. This visualization was made famous in a novella called Flatland, by Edwin A. Abbot…
Existence in the second dimension would be like living on a flat plane, where there would be no up-down at all—only sideways. Imagine standing on a very, very flat piece of land. The initial trouble with imagining this, is that we couldn’t actually “stand” in a two-dimensional world—we’d all be pancakes. A true two-dimensional state would be so “squished down” to be truly, ultimately flat, such that it’s hard to explain how anything could really even be there. But for the sake of this exercise, we have presence and vision on that two-dimensional plane.
Now, if a 3-D box were to “visit” our plane, what would we see? Let’s think about this. The box is comprised of six sides—each one a plane. Depending on the box’s position, a couple of those planes might be exactly parallel to our own plane, while the others would be generally perpendicular to our plane. That means it would appear totally different to us—based on the orientation of the box when we have our close encounter.
We wouldn’t be able to see up or down, so the only parts we would see are at our level, where the box intersects our plane.
Imagine that the box is set down into our 2-D world, so the bottom side of the box (a plane) lined up perfectly with our plane—like when someone puts a box down onto a table. Then, it would exist in our 2-D world as a whole plane. It would create a tile-like square where it sat—and only that square—displacing everything that existed there at the place it intersected.
If a 3-D alien came along and plopped down a box around you, all you would see of that box would be the lines where your plane intersected the sides of the box. It would form an outline of a square around you (a two-dimensional wall), and you’d be trapped. Eek!
On the other hand, if you were outside the area where the box “touched down,” you would only see the intersection of the box plane with your 2-D world. That would be a line.
That is how the box alien would look for those of us viewing it in two-dimensions. If we “two-dimensionals” were then asked what a box (aka cube) looked like, we would be convinced it looked like just a line or a line-drawn square.
As you can see… it’s all in your point of view (or as Einstein might have said, “It’s all relative”).
Of course, living in a one-dimensional world would be even worse. Travel would only be possible along a single, infinitely straight line. It would be like living in a pipe—where you can’t even turn around. Infinitely boring
Now that you have had some practice “seeing” things from one of the dimensions as you exist in another, you can now view some of my different perspectives for visualizing the 4th dimension.
Hypothesis and Basis for My Model
Standard Geometry tells us about the properties/rules of points, lines planes and spaces…
- A point has no form, as it is infinitely small.
- A line has only length, is comprised of at least 2 points, and is infinitely long.
- A plane has two dimensions (length and width, x,y only), a flat surface area with no thickness that extends infinitely, and its area is comprised of points and lines.
- A space has three dimensions—length, width, and height (x,y,z), and its area is comprised of points, lines, and planes.
We also know that…
- A minimum of 2 points are needed for a line to exist.
- A minimum of 2 lines (parallel or intersecting), or 1 line+1 point, are needed for a plane to exist.
- A minimum of 2 planes (intersecting or parallel) are needed for a space to exist.
So, if we had a minimum of 2 spaces, what would now exist? (And following that logic, I wondered what a “parallel” or “intersecting” space would look like.)
This is how I tried to visualize the 4th dimension. Since the dimensions already were defined by numbers (and to make it easier on myself so I wouldn’t have to re-program my brain when thinking this through), I used the existing numbers. However, a point didn’t have any number, but since it was infinitesimally small, and since a zero would work for the sequence, I assigned the “dimension” 0 to a point.
I did feel good when I later found out that the science field had also decided to adopt that convention. And, it makes this part of the explanation easier.
Using the dimensional numbers to represent the elements that define each dimension, we have…
I kept trying to visualize what “Time” would look like to us if we weren’t trapped in a three-dimensional viewpoint. I also kept trying to visualize what we would look like to someone in a “time” dimension. But that was too hard without any hints. I needed some hints.
I got a little farther along when I tried to visualize going from one space to another. In order to do that, you would have to go through *something.* Time? Space-time?
To help visualize that part, I went back and thought about pretending I was that two-dimensional (flat) being, living on a single plane. That plane would, then, be my entire universe—any other plane would be a different universe. In order to get from one plane to another, the “something” that I would have to traverse would be a space. That is because, to get off that plane, the only direction I could go is “up” or “down”—aka 3-D. Trouble is, I would not have any knowledge about “up” or “down” because my perspective is limited within my plane-world. Likewise, in my 3-D world, it’s hard to imagine a fourth dimensional direction I would need to traverse to get to a “different” space.
I considered how I would get between any of the other dimensions…
Going between two Points — you would have to travel along a Line.
Going between parallel Lines — you would have to travel on a Plane.
Going between parallel Planes — you would have to travel through Space.
Going between [parallel?] Spaces — you would have to travel across ????
In the process of that, it occurred to me that a space could also be defined as a whole bunch of planes stacked on top of one-another. Now, I was getting somewhere…i was starting to think “outside the box.” LOL
So, what is it I would need to traverse to go between spaces? It would be something that contained a bunch of spaces? That *something* would, of course, be the 4th dimension, but how could I define it? Like the planes all stacked up to create a space…what if I were to stack a whole bunch of spaces—what would it create? What would it look like?
That brought back a very scary image in my mind. When I was a kid, my grandmother used to drag me to the bank before going shopping, where I would have to wait in sheer boredom while she visited with the teller. The bank had one of those heavy green glass tabletops that was just about kid-height. I remember putting my eyes right up to the edge and looking in. What I saw was both the most fascinating and the scariest image I have ever seen. Inside a huge, dark abyss, were an infinite number of reflections of the glass plane and the hardware holding the glass. It was bottomless and topless, with the eerie planes stretching out forever. I imagined our spaces stacked like that, and tried to visualize exactly what a whole bunch of spaces might form. A table sitting in a 4-D room? A honeycombed hive of 4-dimensional bees?
For one thing, I just couldn’t figure out how to get a relation to time out of that. I kept visualizing multiple universes all clumped together, but there wasn’t a unique time element that was obvious. Sure, you would take time to traverse them, but it would take time to traverse along our two-dimensional plane. Time appears to be an inherent property of all existence, but it could just be the highest level of the first four dimensions. The scientists were fairly sure time was it, but it still didn’t feel to me as though time was the main structure for a 4th dimension.
What if there was a space-time dimension that sort of bridged the third dimension with a fifth (time) dimension? Would that work? Maybe.
The Minimal/Triangular Space Model
I thought it might be easier if I could pare down space to a minimal size. If I could better define what a “space” actually is. So, what is a space?
(I know…”it’s something to a void”…*ew, bad joke*)
Seriously, space is a three-dimensional area. How do you define three dimensions? One way, of course, is with x, y, and z coordinates But I needed something much more basic—something that would be the minimal, base structure of space, and preferably of each dimension.
I thought about the least number of elements needed to create each dimension.
What would be the least possible number of points needed to define each dimensional object…
Aha! I was getting closer to something concrete. I could interpolate that the fourth dimension would be defined with a minimum of 5 points.
Now, I only had to figure out where that fifth point needed to go. Sure, it wasn’t going to be simple, but I only had to worry about placing one point, rather than wrapping my mind around a whole stack of spaces.
Where was that point supposed to go? I kept looking at my box and my stack of spaces. It wasn’t obvious, but I was going to figure it out…it was only one stupid point…how hard could that be?
When the answer didn’t come instantly, I dabbled a bit with looking at it from other similar perspectives. For instance, what would a least number of points *look* like? Maybe that would help me visualize it better.
I sketched a very simple representation for myself that basically looked like this…
What suddenly hit me was that I was drawing TRIANGLES. And since I was reducing this down to least elements, the concept of an equilateral triangle dawned on me. What if this new dimension was easiest to define if EVERYTHING was equal? In other words, what if, just like in an equilateral triangle, the spacing of the points were equal, the length of the lines were equal, the size of the planes were equal, etc.?
This was so simple now, it *had* to be solvable. I was sure I had something. I kept looking at the three-dimensional equilateral triangle. NOW, where do I put that fifth point?
I remember staring at it numerous times during school the next day. Where do I put a point that is equally distant from all other points in an equilateral tetrahedron? If I put it outside the tetrahedron, some of the lines and planes would be too long. If I put it on the surface of the it, some of the lines and planes would be too short. If I put it inside the tetrahedron (like in the lower right drawing in the diagram above), ALL of the lines and planes would be too short. There didn’t seem to be any place left—but I was sure there had to be.
But then something clicked. The only place that was *equally* too short for all points and planes was the dead-center. What if that center could somehow burrow in deeper than just our space—poke through to…yup…the FOURTH DIMENSION?
Aha!!! Okay, now I could plop that point in there—a virtual equilateral tetrahedron in the fourth dimension—then go back and fill in the rest of my model.
What would be the least number of elements for any of these definitions?
Least number for each one…
I really had something going. I could easily interpolate that the “magic” number of spaces for this next dimension would be “5.” I even played a little by expanding my sequence in order to see if going much farther out would give me any additional hints or help at all. Just for fun, I also played with going to “negative” dimensions–wondering if that might apply on a sub-atomic scale.
Okay, so we have a minimum number of spaces (5) needed to create a 4th dimension, and my model with the inward-point that poked out into another universe fit that number series. Wahoo! I would try to draw the fourth dimension.
What would it actually look like? I was getting an image of the extended space bulging back out on the “other” side…that is…way inside. Anti-matter and parallel worlds were popular sci-fi concepts then, and I had a mental picture of an “anti-space” projection on the other side of that pinhole. I drew it (it reminded me of a crystalline structure).
Given that I now had a three-dimensional representation of it, I tried to rationalize what the fourth dimension would really be that would encompass our real-world universe. It somehow involved poking a hole through our existing universe.
A new concept, called a “black hole,” was making news at the time. I realized that the black holes, which crushed everything down to a very small point—crushing so hard, it could be poking a hole right through to another “space” in another universe, which would mean that both spaces together would comprise a fourth dimensional existence. That was it—that was my real-world visualization. The fourth dimension would, therefore, include whatever was on the other side of that black hole.
So, was time the fourth dimension? Umm, that didn’t seem quite right, based on what I was visualizing. I thought it was more likely a time-space, as the “event horizon” would imply that. It could also be something more exotic. But this was close enough. I was so excited, I had to share it with an adult. I had a good enough model with enough supporting evidence, that I felt confident to tell someone “in authority.”
White Holes—Tying It All Together With Some Tantalizing Theories
I was not in the class of the best math teacher at my school, but I needed the best. So, I went to see her and tell her what I had found and presented my “time-space” rationale and drawings. However, without really investigating what I had, she told me I was wrong, and reiterated the wisdom of that era. I realized I had hit a brick wall, but I also lost confidence in pursuing it.
I was totally bummed. Based on how it all fit together and seemed so intuitive, I was still pretty sure I was right. And yet, I didn’t know who else to go to, and it didn’t seem worth trying to convince anyone else anyway.
Fast-forward to around 1980. I was watching a “Cosmos” episode, and Carl Sagan pulls out a tesseract. I had never heard of that before. Whoa!!!! That’s it! I was right! That’s my model! I was elated…also very upset that I hadn’t been taken seriously.
But…but…that’s a CUBE. It should be a TRIANGLE! If the model is to work, it should be represented with triangles, shouldn’t it? But back then, there wasn’t any World Wide Web, so there was no place to look it up and see what was known without spending a lot of time in a research library. Not something I could justify spending time doing if it wasn’t going to add much to people’s knowledge. The scientists seemed to have been well onto it, so even if they didn’t have the rest, it wouldn’t be too much longer before they got there. Once again, I let it go—but this time, it was with the realization that I really had been on the right track.
Shortly after that, I met my partner, whose background is astronomy and physics. We discussed it a couple of times, which helped to fill in some gaps for me about the known universe. During one of those discussions, some things hit me…
* Are photons a real-world example of dimension 0?
* And What if our “big bang” in this space/universe of ours, is actually a black hole that imploded in another universe and is now leaking in from another space/universe—a WHITE hole here?!
* What if all of the black holes we see here have little spaces poking out into other universes—becoming ‘white holes’ and spawning new spaces? (Note that my 4-D triangle model creates spaces in the shape of an inverted triangle within the 4th dimension—a bubble universe).
* And could there be any other spaces ‘intruding’ on the edge of our space?
Although well-versed in astronomy, my partner isn’t a cosmologist, and we didn’t have easy access to any. So, once again, back onto the proverbial shelf this project went.
Fast-forward to around 1990. My partner was doing some work at NASA, and when talking with some researchers about the Hubble Telescope and black holes, I was now confident enough to present the model to them. They didn’t think anything much was being done with dimensions and tesseracts at the time, and seemed enthusiastic about my concepts, but I don’t think they ever followed through with it, as they never got back to me either way. Yet it was worth a shot.
On to the year 2000. I met someone who is incredibly strong in both science and engineering. I showed my model to him. He basically dismissed it because he had seen it before, “Oh, that’s just Pascal’s triangle.” Not knowing how Pascal’s triangle could have related to the fourth dimension, I was confused that he didn’t wonder about that himself and look more carefully. But, now I was encouraged that my model was based on something that the mathematical community accepted as a legitimate foundation, and had a historical background as well. At that time, there wasn’t much online about Pascal’s triangle, but between the library and the online info, I read up on Pascal and the “Khayyam triangle” (the early version of it). There wasn’t much about how it related to the fourth dimension.
40+ Years — I Hope Someone Will Find This Valuable
Unfortunately, that was an intense time in my life, and my personal situation had to take precedence. Therefore, spending time searching down someone who might possibly be interested in my little model was not a rational use of my resources. I kept telling myself that “one of these days,” even without the help of an expert, I will get some of this info up on the Web so it could fill in the gaps for those who could make use of it. I had a bunch of other vital info to get up on the Web “one of these days” too, so, I was adamant it would eventually happen.
Now it’s 2010. Still didn’t find anyone to assist me privately, but I’m ready to put my findings up. I wrote out my little “dissertation.” While doing so, I thought of another analogy to that two-dimensional world. If a three-dimensional cube would be perceived as a square that is confining the area within a two-dimensional world, maybe what we see as the edge of our universe is similar. The boundary of our universe could be our 3-D view of the fourth dimension. And I could see where that could relate to time, as time governs the rate of our universe’s expansion.
I checked online to see what was current about the tesseract, and found there is a bunch more work on it, but not a lot about equilateral triangles. I then looked up the Khayyam triangle. I see there is now a Pascal’s Tetrahedron and a lot of other related work. Hmm, I FINALLY was able to get this all written out for public disclosure, and a bunch of my “revelations” are already in articles on Wiki. *sigh*
Did I do this in vain? I don’t think so. I looked carefully, and so far, no one has connected all the dots (pun intended) that bring all these elements together in one place. And even though there is plenty of math that goes far beyond what I could ever contribute, I offer the basic theory, and a visualization of the concepts in a way that hasn’t been done before. Most importantly, this provides a foundation that contains not only the definitions and rules for working with dimensions—but the ability to predict structure and the way to visualize it—on which others can now build. With the help of this model, what we observe about our universe (on both macro and micro levels) may be more easily understood.
I hope you all get something out of this—it would make all my work and frustration worthwhile.
And if you have a kid come up to you with a new idea, please, please—listen and take the time to think it through and consider if it might have merit. At the very least, encourage the child to keep going on it and keep thinking. Thank you!