Wednesday, 1 June 2011

Like fingering the third dimension

Good. I've got your attention with some double entendre. Now read this damn post.

For today's glorious, confusing, slightly maths-themed post I have some further discussion on dimensions and how to understand them, specifically how it's possible for them to interact with each other (I suggest you start with this post). We'll go through things slowly so you can understand them (and making the diagrams has helped myself understand them), and soon enough I'll be able to show you what it would look like if a fourth-dimensional person stuck their finger into our 3D world.

You're looking a little plane
Before we think about fingering a 3D world (cough) we're going to see what it would be like if you stuck your appendage into a 2D world (cough). And yes, potential innuendo will be a core part of this post.


So here's our first diagram. Ignoring your cries of 'what's a 2D plane? How can it exist?' we'll just assume that it does. A 2D plane is a 2D space, it's not necessarily finite like the square you see here, but it does exist in a three-dimensional space, which is why we can make it interact with 3D objects. So we have our 2D 'world', K, and we have a regular tube, a cylinder, named L. Following that handy arrow, we're going to take L and stick it through the plane K. Think of a 2D plane as a piece of paper, but it's more of a space than an object. That's why we can stick things through it - it's super handy.

So we now move L along the arrow axis, perpendicular to the 2D plane (remember it has two dimensions - up/down and left/right - x and y), and thus it will intersect it.


There we go, it's intersecting it. The blue circle is the shape of L that is intersecting the 2D plane - because a cylinder is effectively a circular prism, if we record the shape of L on the plane we can see that it is a cross-section of the shape. Imagine it as a slice of the 3D object, like a cucumber.

So we can conclude that if you stuck your cylindrical finger through a 2D world, those inside the 2D world (ignore the details) would see it as a roughly circular shape, and - because it's a cross section - they would see all the insides of your finger, such as your bone in the middle. It's quite a handy X-ray technique, apart from the whole '2D world's can't exist' thang, which isn't wholly true.

Frustum in
Now we're going to try the 3D-into-2D interaction with a shape that does not have a consistent cross-section. Because the tube is a circular prism, no matter how far into the 2D plane you stick it, its 'image' on the plane will always be the same, circular. If you move it around in the x- and y- direction, you'll only move that image around the plane (ie. up/down and left/right).

Now we're going to taper our cylinder L slightly, so one circular face is smaller than the other. We have what's known mathematically as a frustum -  a cone with its top sliced off and fed to rapid deer. N.B. the deer bit might not be in all textbooks.


Here it is, out little frustum, endearingly named J. It's going in the same direction and into the same plane as L was, only it's a frustum. The face nearest plane K is smaller than the opposite face.


Now we've moved forwards a little and boy, are things getting crazy! Frustum J has moved parrallel to the arrow and has moved to intersect plane K, but close to the small circular face. We'll call this stage a.


Keep that frustum moving, son, and now we come to stage b. J is further into the plane and thus is intersecting it at a point where its cross-section (still a circle) is larger than it was at stage a.


Compare the two stages (a is before b, remember) and we can see that the circular cross section gets larger the further the frustum is into the 2D plane. That means if you were in that 2D world and someone stuck a frustum in like above, you'd see it as a circle getting bigger. Stick the frustum in with the larger face first and the 2D people would see it as a diminishing circle. Clever, eh?

Cube your enthusiasm
Now we're gonna get serious. Because it's time to stick your hyperobject in a 3D world and perhaps jiggle it about a bit (cough).

Let's explain. Instead of a 2D world, or 'plane', we're going to use a 3D world - which is really just a 3D space. I don't like saying 'world' because it makes it sound like it's a finite space with tiny trees and tinier dogs, like a little cube of '3D stuff' whereas it's actually just a space. That plane in the previous section could have gone on forever, objects J and L are seperate from the plane in the third dimension (outwards), regardless of how large the plane is in the x- and y- directions.


So, as I displayed the 2D space as a square in a greater 3D space (notice it was in perspective), and I'm going to show the 3D space as a cube in a greater 4D space that we can't really show. Into the 3D space M we're going to poke a four-dimensional hyperobject, N. N is the 4D equivalent of a cylinder - a hypercylinder, I suppose.


Remember I showed the 'image' of the 3D object on the 2D plane in blue? Well now I'm showing the 'image' of 4D object N on 3D space M in blue. Just as the 2D plane showed its image as a cross section in the same amount of dimensions as itself (a circle is a 2D shape), we'd assume the 3D world would show its image in three dimensions. And since our hyperobject is a cylinder of sorts, the image would be a circle + one dimension. AKA a sphere!

From this we can discern that the cross-section of a hypercylinder is a sphere, which makes sense because the 3D cylinder has a 2D cross section, therefore an nth dimensional object has a (n-1)th dimensional cross-section when intersected with an (n-1)nth dimensional space.

The second diagram above shows what would happen if that hyperobject, N, had a varying cross-section. Just like the frustum having a small circle front face and a large circle back face, N could be a 'hyperfrustum' and thus have a small sphere front cell (4D cell ~ 3D face ~ 2D side ~ 1D point) and a large sphere back cell. That means if a 4D frustum is poked into a 3D world, you'd see it as an enlarging sphere.

And that's what if would be like if a four-dimensional person stuck their four-dimensional finger into a three-dimensional space. You would also be able to see the inside of their finger - though whether they'd have fingers or whether they'd exist at all is a totally different matter.

But there's more: Slice me some, Joe
Wait up. Before we leave this post I have two further things to show you re: dimensions intersecting. Let's go back to our original 2D plane and the 3D cylinder that's intersecting it. Remember I took the cylinder away and showed you, in blue, the 'image' it made on the 2D plane? Well here's a similar diagram, but there's three:


I've showed the 'image' on the 2D plane when the cylinder intersects it at different points. Stage a is when the cylinder's very edge is touching the 2D plane and, as before, a circle image (a slice, a cross section) is created on it. Move the cylinder through the plane and we get to stage b, then to stage c. The images are the same, but remember something vital: they are simply 2D images so only have 2 dimensions, x and y. To the person in the 2D world, they are not linked! They are separate shapes! The only thing linking them together to the 2D observer is the fact that they are in the same 2D place, and after each other in time. Mathematically, their only linking factor is that they are linked in the third dimension.



Now we shift it up a dimension and go back to our 4D hyperobject sliding through a 3D space. It creates a 3D 'image' (or projection) in the 3D space, a sphere. At points a, b and c the hyperobject is intersecting the 3D space at different points along its length, but because it is a 4D 'prism', these slices are the same - always the same size of sphere. These spheres have values in the x, y and z directions but apart from that are not linked, as the circles were before. That means that us, 3D observers, cannot see the axis along which they become part of the same object.

So, potentially, objects we see in everyday life could be projections of the same four-dimensional object intersecting our 3D world, and could actually be linked together in higher dimensions. For all we know, the entire Earth or galaxy could be projections of one 4D object, and thus we are all linked.


And finally, before we go, remember we can also show different dimensions in graph format, keeping with the Cartesian standard that I nattered on about in the last post. A cylinder is shown above - in the x and y axes it's simply a circle, but then add in another graph comparing the y axis to a third, the z (3D) axis and we can see that it is more than simply a 2D object. It's a 3D object!


Similarly, the above diagram shows a hypercylinder. In our three dimensions, it's simply a sphere - but show its fourth value, the w axis (4thD) and we can see it has 'depth' in that dimension also, making it a four-dimensional object. N.B. if the line was flat in the second graph, then it would have no 4D depth - it may have a place on the axes (w=2) but it does not have any 'length' along it.


And there you have it, folks. Now if anyone asks you what it would be like if a 4D being fingered a three-dimensional space, you can tell them! Or just direct them to this blog! Or if they don't ask, direct them to this blog anyway! Then go get dinner with them, because people like that need dinner.*

Look at it this way: at least you now know the effects of sticking your frustum into mysterious spaces. (That one's for all of you who were only interested in this post for its innuendo)

~John

*Also, people need love, people need loving, people need the trst of a frail old man. Just sayin'.

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