The fourth dimension.

Well, I didn't really time it, I expected some dramatic moment to happen with the music

there but it didn't happen.

So alright, this is A Coding Challenge, I am so crazily excited about this.

If this actually works I think that, I don't know what's going to happen, stuff will just

like, smoke, and brain matter will start just like leaking out of my nostrils.

I don't even know, so wait 'til the end of this video to see.

Now what I'm starting with here looks like a plain old spinning cube and actually that's

what it is.

What I want to do is not just make a plain old spinning cube in three dimensions, I want

to make a hypercube spinning in the...

Fourth dimension.

So I want to make something that looks like this, right?

So this is a hypercube, otherwise known as a tesseract.

And I'm going to, before I start coding any of this I'm going to explain to you what that

is exactly.

So let me come over here, so I, I think my favorite dimension is one dimension.

No, zero, let's start with zero dimensions.

A point is a shape that exists in 0D, that's a zero.

Now the line which is bound by two points is something that exists in one dimensional

space.

This is one dimensional space.

A rectangle, a square, a plane which is bound by one, two, three, four lines exists in two

dimensional space.

Now notice the pattern here.

This is bound by two lines, this is bound by four.

I'm sorry, this is bound by two points, this is bound by four lines.

Now if I create two planes and then connect all the edges I have what is commonly referred

to as a cube.

This is something that exists in three dimensions, are the number of dimensions that we live

in on this planet where we live and it is bound by six planes.

So you can see a pattern here.

I was going to say you double it but this is not bound by anything I guess so this sort

of makes sense.

This is bound by zero things, this is the beginning.

Now what does it mean, then, to have four dimensions?

Well I'm not really going to be able to draw this too easily but we could make the case,

right, if I'm following this pattern, that four dimensions is bound by eight cubes.

And this is mathematically true, this is accurate.

The problem is I'm going to sit here and I'm going to visualize this in my head.

Are you doing this with me?

I can't, I just cannot do it at all.

Human beings did not evolve to understand the world in an dimensions higher than three.

Anyway, I could keep going on about this since this is a mathematical truth, why isn't there

some way that we could somehow unlock is and see this on a computer screen in some way?

And in fact, there is, so this is how.

How do we even see this?

How do we even see this 3D shape on a 2D computer screen?

Well the way we do that is by writing code like this one that I've done right here.

Now the truth of the matter is, in processing, which the programming language environment

I'm using right now, I could just say box and I would get that because I'm in the P3D

renderer.

And the P3D renderer knows how to take the mathematics of a 3D shape and project it into

a 2D canvas to create the illusion of the third dimension.

And it totally makes sense to our brains 'cause we're so used to 3D.

We can imagine it, we can see the 2D version of it and imagine it in three dimensions.

And I did, you don't have to have watched that video to watch this one but I did a whole

previous Coding Challenge where I made exactly this put only using the P2D renderer.

So I did the math of taking 3D points and projecting them into 2D, so if I can do that,

why can't I create the math for a 4D shape, do the projection into 3D and render it with

the P3D renderer?

The truth of the matter is I could then project it into 2D and render it with the P2D renderer

as well, but I'm lazy, I'm just going to do one projection.

And if this is true there's no reason why I couldn't create a 5D shape and a 6D shape

and a 7D shape.

And I project 7D into 6D into 5D into 4D and then into 3D and visualize it.

So this is what I'm going to attempt to do in this video.

It is going to require matrices.

There are other ways to do it without it or create the visual illusion of it but I'm going

to be using matrices and the matrices are going to be used for a couple things.

I need a projection matrix, this is a matrix that takes a 4D point and turns it into a

3D point.

And I'm going to also need a rotation matrix.

I don't need the rotation matrix but the rotation matrix is what's going to make this funbecause

I can start to rotate around weird axis to see crazy things happen.

So this is kind of like optional but this is what's going to make the visualizations,

so I'm going to do the most basic wire frame version of this and hopefully you are then

going to make beautiful, interesting, weird things about this.

If somebody watching this, but who can go to the highest dimension possible in processing?

I would like to see that as a challenge.

Now if you want to learn more or see a condensed version of the explanation of what the tester

act is and how this works I would highly recommend, for me, I'm making this coding challenge because

of this particular video.

From, how do you pronounce that?

Anybody know how you say this?

Okay, I can't figure out how to pronounce it, I'm going to guess, LeiosOS.

But this is a wonderful, wonderful YouTube channel with many excellent explanatory videos

and this one about understanding 4D, the tesseract, is excellent.

So you could pause this, go watch that, come back later, all of the above.

Okay, alright so now I think we're good, we're good, we're good, and I minimize the browser

and here I am.

Okay, alright, now what do I need?

The first thing I need to do, if I'm going to make something 4D, to do something in 3D

I make heavy use of this idea of a PVector which is a data structure that holds an X,

Y, and a Z.

I need something that can hold an X, Y, Z, and W for that fourth dimension so I'm going

to make a new tab and I'm going to call it P4Vector and I'm going to say class P4Vector

and I'm going to say it has an X, Y, Z, and a W and it needs a constructor.

And it can get a X, a Y, a Z, and a W. And then I'm going to say this.x, isn't it fun

when you have to use the this.

in java 'cause it's like so rare?

Every once in a while you do.

Z, W, X, Y, Z, W, so now this is my P4Vector.

Alright, P4Vector, there we go, and now I am going to make this cube, this very cube,

all the points of the cube and actually one thing I want to do just not, I'll leave it

right now, I'm going to make it in 4D.

Zero, zero, the fourth dimension And then while I'm adding the fourth dimension I got

to change these P4Vector.

Then I've got to change this P4Vector, P4Vector.

And now this stuff I'm going to worry about later.

And let's see, we're going to watch, it's going to be four dimensions now, everybody.

Here it comes, four dimensions.

Look away This is the same exact cube, right?

It's the same, is that four dimensions?

Oh I'm so confused.

Well how are you even going to see this fourth dimension?

First of all, all the points are on zero.

So what you can think about it is it's flat.

I took a four dimensional shape and flattened it, flattened it, you know, we say flat 'cause

usually we're taking a three dimensional shape and flattening it into 2D so a cube if you

flatten it would just be a square.

So I took this hypercube and flattened it and I just have the cube so I need to make

the hypercube.

You know what, I'm going to, against my better judgment I'm just going to extend this array

to have 16 points.

Right, a hypercube is made up of 16 vertices because it's basically two cubes with all

the points connected.

So I am going to, I was going to make like a cube object and a hypercube as two cubes

but I'm going to just keep going.

So I'm going to say eight, nine, 10, 11, 12, 13, 14, 15.

And then I'm actually going to put this in 4D.

So all of these zeroes, right, they should all be 100.

The rest of them at negative 100.

I know I'm doing this in a highly manual way but first of all I find it kind of relaxing

sometimes just to sit here and copy paste the same thing over and over again, you should

try it, it's very soothing.

And then also I can always refactor it later but just so I really know that it's working,

okay.

I've got 16 four dimensional points,, I am going to now draw them again.

Okay, so it still just looks like those points.

What's missing here?

Well I'm not actually doing the projection so now it's time for me, like, I'm just taking

the 4D point, ignoring that v.w part and then doing a projection.

So really what I need to do right now is think about how I do the projection so I'm going

to create a projection matrix for a 4D point.

So this is the point X,Y,W,Z.

The projection matrix to take a 4D point and turn it into, oh wait, X, Y, Z, W. And project

it into something I can draw in 3D, X,Y,Z.

The idea is I want to look at the shadow of the 4D object in 3D, just like I might look

at the shadow of a 3D object in 2D so to do that I need a matrix that has four columns

and three rows so the typical way to do that would be like this.

One, zero, zero, zero, zero, one, zero, zero.

Zero, zero, one, zero, right, I think I got everything.

The idea here, if I actually use this matrix and perform matrix multiplication which I'll

link to two videos where I go through that in more detail, this actually gives me literally

this.

It's like, just chopping off the W. So let's actually put this in our code and see what

happens.

So I'm going to create a matrix.

Now one thing I didn't mention is I'm working with a set of helper functions and I worked

these functions out into separate video if you're interested but it's not, too much,

it's way too much to be in this video.

That has actually the matrix multiplication math in it, as well as some helper stuff to

do matrix multiplication with a matrix and a vector.

I'm going to have to change this to P4Vector, I'm just realizing that, but that's no problem.

Okay so what I first want to do is create that projection matrix.

So I'm just going to do it right here as a local variable and a little bit later it'll

sort of make sense why.

So I'm going to say, float projection equals, so I need to make that matrix which is an

array or arrays.

So there are three rows, 0, 1, 0, 0, and then 0, 0, 1, 0.

This is the equivalent of orthographic projection, if you've heard that term before.

Okay so now we've done that, then what I'm going to say is, P4Vector, actually P Vector,

I don't need it to be a P4Vector, I'm taking the 3D point.

P Vector equals matrix multiplication, the projection times V, okay?

So now it's telling me it doesn't know how to do matrix multiplication with a projection,

matrix, and V which is a P4Vector, which is strange because if I go in here it has a function

to matrix multiply a two dimensional array and a P Vector.

But I need this to be a P4Vector, okay.

Oh, vecToMatrix, doesn't know how to make a P4Vector, so I'm just going to change these

functions to deal with 4D.

M3 equals VW, so again if you want to see on a separate video where I wrote all this

code and now I'm just adjusting it to add this fourth dimension.

And then matrix to vec is going to be, you know what, I'm going to always be doing this,

this is fine, this stays the same, so that stays the same 'cause I still want, what I'm

going to get is a 3D vector inside a matrix and I want to turn it into a P Vector so let's

see.

Nobody's complaining at me and now I want to look at the projected points, projected,

projected, projected.

Here we go, oh what's wrong here?

Vec to matrix... array out of bounds exception.

I have an error somewhere, where is this called?

Oh this has to be four, this has to be a four.

So I forget that I'm in java and I really have to specify types and lengths of things

so that has to be a four.

Is there anywhere else where I need to do that?

We'll find out soon enough.

Okay, let's try this one more time, we're entering the fourth dimension.

Whoa, okay, it's still the same Why?

There are eight points in four dimensions.

Well guess what, if I took a cube and I showed it to you in orthographic projection facing

the camera basically it would look like just four points.

The four points on the back of the cube would be sitting right exactly behind the four points

at the front and you wouldn't see them as different.