Name: 編碼挑戰#113：4D超立方體（又名 "魔術方塊"）。 (Coding Challenge #113: 4D Hypercube (aka "Tesseract"))
Uploaded: 2021-01-14T10:25:36.000Z
Duration: 42 min 44 s
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Welcome to A Coding Challenge, where I will enter...

程度副詞

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

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

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

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

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

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

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.

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

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.

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?

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

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 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 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.

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

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?

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

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

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,

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

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編碼挑戰#113：4D超立方體（又名 "魔術方塊"）。 (Coding Challenge #113: 4D Hypercube (aka "Tesseract"))

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