Dimensionality

The first topic I want to lay out is an understanding of dimensionality. We perceive a four dimensional universe: three spacial dimensions and one dimension of time. For the moment, I want to set time aside because, as dimensions go it is weird and prickly and will, I am sure, get its own post or 50.

The question I'd like to explore here is how a higher dimension phenomenon would appear to an observer perceiving a lower dimension universe. Because it is difficult to envision more than 3 spacial dimensions, I would like to begin by considering the questions "If a three dimensional being interacted with a two dimensional universe, how would the residents of the 2D universe perceive this interaction?"

To begin thinking about this, I imagine a two dimensional universe, something like an infinitely thin sheet of paper. Though the surface of this universe could be any shape, let's imagine it is flat and is actually the size and shape of a piece of notebook paper.

What is life like for the residents of this universe? Well, they can only move forwards/backwards or left/right. There is no such thing as going up or down. The residents of this universe could be any shape, but let's assume they are sentient triangles. Growth in this universe would not be growing taller and in volume like we do, but rather growing in perimeter and area. If anyone has read Flatland by Abbott, you are familiar with this set up.

Objects in this flat universe would be other shapes: squares, pentagons, etc. Unless there was a gap in the perimeter of one of these objects, the residents of the 2D universe would never be able to see the interior of any of these objects. Houses in this 2D universe might look like an architect's blueprint...certainly would lower the cost of new builds. An interesting question might be what blueprints drawn up by the 2D residents would be like...just smaller versions of the actual 2D house?

Ok, so hopefully we have a good idea of what the 2D universe is like. Now the question is, how do the residents of this universe perceive an encounter with a 3D being? Specifically, let's imagine that I take my hand and shove my fingers through the sheet of paper that is their universe. How would they perceive my fingers?

Well, my fingers would appear to be 5 seemingly separate circles that appear in their universe. One of the 2D-ers might be looking at the cross-section of my thumb while another might be looking at the cross-section of my pinky finger. Because they can't "look up" to see the rest of my hand, these 5 mysterious circles would seem utterly unrelated to each other.

Of course, being 3D, I would be able to look down on their notebook paper universe and watch them poking and prodding at these objects. Interactions with these circles could be quite strange indeed. If they poked my thumb, I might twitch my pinky. To them the thumb "circle" and the pinky "circle" might seem be located very far apart (depending on how small the 2D residents are) so to see one circle react to a poke given to a separate circle far away would certainly seem mystifying, perhaps even "spooky action from a distance."

So the question is, how could the 2D beings ever deduce that my finger circles are actually 5 separate manifestations of a larger, higher dimensional structure? Even if I wanted to solve the mystery for the 2D beings, how could I tell them? As a 3D being, the most I could ever show in their 2D universe is a cross-section of myself which would make speaking to them difficult, not to mention that they could only perceive the pieces of the soundwaves that my voice made that happened to lie in their plane. I suppose that if I were really motivated, I could learn their language and try to write them a note. Of course, to execute that, I would have to be able to use a 2D writing utensil to write either a 1 or 2D message. Which, as a 3D being would prove quite difficult.

That lays out the basic metaphor. Let's change it a little. Instead of imagining my 5 fingers intersecting with the 2D universe, let's imagine a hand with a nigh infinite number of tiny fingers piercing the 2D universe. In fact, the entire 2D universe and everything in it is made up of one of these tiny fingers. I am now imagining the 80's novelty "pin impression" item where you could press your face or hand into a bed of hanging pins and see the image arise on the other side. But in this 2D universe, the residents are so much bigger than the pins that the surface of their universe seems smooth to them and they don't see that themselves and every other object is actually made out of these tiny pin fingers.

Of course, eventually if they developed powerful magnification techniques, they might figure it out. Anyway, the pin fingers that happened to fall inside of a 2D being would appear to belong to that particular being, those outside to another object or being. In reality however all of the pin fingers are attached to the same hand and source in 3D.

This is not to say that the 3D pin fingers have any will or volition of their own per se. The only thing we are saying is that the substance of the 2D universe is actually made up of tiny cross-sections of these 3D pin fingers. The pin fingers might be totally inert.

BUT, the fact that the pin fingers are all attached to the same hand in the 3D universe could have bizarre implications for the 2D beings, just like poking my thumb and seeing my pinky twitch would have seemed odd. If all the pin fingers are connected in 3D, then certainly information could be passed from the pin fingers in one being to the pin fingers in another seemingly separate being very quickly, even if the recipient were far away. This would certainly seem inexplicable and weird to the 2D beings.

So this brings me to one of my primary hypotheses: the universe as we perceive it is composed of building blocks that are of a higher dimension than the ones we perceive. We are like the 2D beings that perceive the pin fingers (assuming we detect them) and other objects/forces as separate entities when in fact the substance of the universe as we perceive it is actually a 3D (4 if you include time) cross-section of a potentially linked higher dimensional space. If this is the case, then it raises a lot of questions about how we might be connected, if at all. Perhaps more importantly, it begs the question of what type of super-structure our universe is a "cross-section" of?