One of the things that’s weird about n-D space (where n is 1 or greater) is that time is required regardless of whatever value “n” takes on, in order to allow motion. It’s almost as if time is a liquid form of the next dimension up (n+2). And, there’s a relationship between “n” and “n+1” that I haven’t rambled on about yet.

The last few blog entries have focused on the idea of adding one more dimension to whatever set-up we were looking at, at the time. But we can go the other way by taking cross sections. We take cross sections all the time, either with CAT scans in order to build up a full 3D image of the human body, core samples of the earth and trees, and when we saw through wood. In fact, a cross section of a tree is almost like looking at frozen time. The thickness and spacing between rings tells us how much rain, sun and nutrients the tree received each year, and we can go back and review that as either a line (core sample) or a plane (cutting across the tree).

However, this raises a serious question – how do we decide on where along the 3D space represented by the tree (the height) do we take our 1D or 2D sample? Obviously, being closer to the ground is going to give us different data than if we go up into the branches. It’s still a cross section, and it’s still the same tree, but the results are going to be wildly different.

Here’s another example.

Say you have a flat plane, and two people standing in the middle of nothingness in this plane. Person A and person B. In the left diagram, there’s nothing for A and B to stand on so they just remain where they are and there’s no growth. In the second diagram from the left, A and B both get to stand on little line segments. This gives them some wiggle room to move along, but they still can’t travel far enough to meet up. In the third diagram, B’s line segment gets longer, and he (or she) decides to travel along it to the opposite end. Then, in the last diagram, the long segment gets disrupted and turns into 2 smaller segments, with the result that B is now farther away and has no way of getting back to where he had been before. Let’s pretend that all four diagrams are cross sections of the same 2-D object, which is perpendicular to the plane A and B are on, and that by moving the object up and down, A and B are both subjected to varying “realities”; i.e. – their ability to travel, and the line segments that they see will change. The choice of “height” for the object controls the cross section we get. But, if we could somehow “layer” all of the cross sections together we’d be able to tell what that object looks like in 3-D space; or, what it “is”.

Let’s start over, this time with a cross section of a truly 3-D object as seen within our 2-D plane. It’s the same object, but with a width as well as a length. In the left diagram, we get a circle and A and B are standing opposite each other. On this circle, one or the other of them could decide to walk over to join the other person. In the second diagram, the ring turns into a wall, and neither person can look past it to see the other one. To get to the other side, they either have to walk around the outer perimeter; OR, if they’re unlucky, B had been on the wrong side of the ring when it started getting thicker and is now on the inner perimeter with absolutely no way to reach A because the wall is in the way. In the third diagram, B’s side of the wall has sprouted a projection of some type that pushes B farther away from A. Then, in the fourth diagram the projection breaks off from the wall, leaving B stranded. Again, it’s the same object in all 4 drawings, just with different “heights” in determining where to take the cross section. If we layered them all together over time, we’d be able to take our 2-D world and “see” what it looks like in 3-D space.

In fact, it’s a coffee mug. If the 2-D cross section is taken with z at about 0, what A and B will see is a solid disk, and they can travel along the circumference to join up. But, if we limit them to a 1-D cross section of the base of the mug, they just see a single long line segment that they can walk along.

Which brings me to the “frame of reference”. The assumption would be that the coffee mug is sitting flat on the table and we’re just traveling “straight up and down” along the z axis for these cross sections. But, what if that’s not true? What if the mug is sitting at an angle? One possible cross section is what we’d get in the right drawing – a kind of C-shaped wall that we could walk around.

And then we can get into the entire concept of which direction is “x”, which is “y” and which is “z”? If we have a donut (center drawing), laying it flat could give us the cross section on the left. But, standing it up and down could give us the drawing on the right, if “z” is only a few centimeters up. All of these objects still look the same if we layer the 2-D cross sections together to get a 3-D object, it’s just that they’re rotations of the same thing.

If we stand the donut on edge and vary “z”, we can get something of an oval, two separated circles, or two ovals almost, but not quite, touching.

Why go through all this? Because if each higher dimension is a “fluid representation of space” with respect to time, we have something we can worth with to project “n” space to “n+1”. That is, a point plus time, if layered together over a long enough period of time is going to give us our 1-D line segment. That segment plus time, when layered will give us a 2-D ring. That ring + time, when layered will give us a 3-D coffee cup…

In order to make the jump to the next level, we need to observe something that varies with time. Just moving the mug on the table isn’t going to be enough to say “here’s where we’re taking the cross section”. But, there are examples of “frozen time” in 2-D, such as with tree core samples. What would give us “frozen time” in three dimensions? Because, this really would be what would allow us to make hyperspace jumps across the galaxy. Just as with A and B standing on a cross section of the coffee mug, the value of z controls their ability to easily “cover great distances” to meet up. If B is stranded on the handle of the mug with z half way up the mug, too bad. But, move z closer to 0 and B can not only reach the main body of the mug, but the perimeter distance may be smaller when person B walks over to person A. A hyperspace jump could become feasible if: 1) We know what shape the universe is in 4-D space; 2) We know how it’s rotated (the reference frame); 3) We can figure out how to manipulate “z” (AKA: D4).

(Hint: Gravity wells.)