4D Cube Explained-part 2

Hello again, this is the promised follow up, in which we dive deeper into the secrets of the 4D cube!

In the last chapter, we covered the basic ideas of a 4D cube as an introduction. In this chapter, we would explain further into the idea of 4D geometry, as well as answering the questions proposed in the last chapter.

https://scienceangles.wordpress.com/2019/01/04/4d-cube-explained-part-1/

So, we have derived what a 4D cube look like, and to view it from a 3D perspective, we have two ways of representing that image:

Graph 1
Graph 2

                                                                                  

Wait a sec, those two look quite different, are they both correct? What’s the difference?

As we can see, in Graph 1, the two 3D cubes that are supposed to be opposite each other are of different sizes. However, in Graph 2, the two 3D cubes opposite each other are the same sizes. In the previous chapter, we explained how to intercept Graph 2, but what about Graph 1? How do we understand Graph 1?

Let’s go back to the relationship between 3D and 2D. What is the 3D equivalent of Graph 1?

We get a picture like this:

Two opposite sides of a 3D cube are two squares. One is bigger, one is smaller. They are connected at all vertices, to create the picture above. Where have I seen this picture before? Think back on the museums you’ve visited, the Renaissance paintings you’ve seen. Think about the study of perspective. Or better yet, think about the room you are in. Go to one end of this room, place your back against the wall, and look across the room at the other end. You get a picture kind of like the one below.

Is it not exactly what we drew just then? When we place ourselves inside the 3D cube, Given a variable distance, we can see the two sides opposite each other at the same time, only of different sizes.

Artists know this. They use this skill to create realistic feeling of a 3D world on a 2D plane. Take a look at this painting, School of Athens. In this famous painting, the people far from us seem smaller, whereas the people near us seem bigger. This creates the feeling of 3D world on the 2D panel.

School Of Athens by Raphael

So, if we have a 4D cube that looks like Graph 1 it means that we are placing ourselves inside a 4D space, the smaller cube farther from us and therefore looking smaller, the bigger cube nearer us, looking bigger. I’m using the term ”farther” “nearer”, but these adjectives are the adjective of a new variable, a variable that does not exist in our dimension, the 4th variable in space. What might that be? It could be anything, colour, temperature…anything will do. But for our convenience let’s consider the 4th dimension as time, which is the easiest thing to understand in our 3D world, but the one thing that we cannot control. Ok, so now our variable is time, and the cubes are just two cubes through different time lapse. We are standing at a starting position “Now”, and the smaller cube we see in the distance is the exact same cube, labelled ”a year from Now”. That cube is considerably farther from us, so it seems smaller.

hat is how we interpret the difference between Graph 1 and Graph 2. Also, we gained a brand new understanding of a cube in 4th dimension. It is a cube, consisting of several 3D cubes that are spread across a time span, but also tangible at the same time.

Check out these photographs from photographer Steven Wilkins, showing the same landscape at different times on the same panel. In these pictures, the same thing at different times coexists. These pictures amazingly share the same idea as the structure of a 4D cube.

https://www.stephenwilkes.com/fine-art/day-to-night/52fa9a11-c5e8-4048-8071-0b560af4b6c2

So through these two articles, I hope I have given you a better understanding of a 4D world, and I would be very glad if you would be more fascinated in the mysteries of the higher dimensional space. There are endless magic to that topic and still a lot to explore!

4D Cube Explained-part 1

When you look at the Dali painting, Corpus Hypercubus, what comes to your mind? Weird, bizarre structure? Eerie feeling? What is the secret behind the cubes arranged together in the shape of a cross?

The answer may shock you: It is a hidden picture of a 4D cube.

Huh, I heard you sneer. We don’t even know what a 4D cube look like, how can one draw one on a 2D plane, and how can you understand it?

Well, the best way to imagine a higher dimension is to think of the relationship between our dimension and a lower dimension, and then picture that relationship onto a higher dimension and our dimension. This is what we’ll do a lot in the understanding of the fourth dimension. Take a 3D cube first. We all know what a cube looks like, right? It looks like this:


The cube sonsists of 6 sides which are 6 identical 2D squares, with 2 squares opposite each other. And similarly, “cube” in 2D space, a square, consists of 4 identical sides of 1 dimension. Therefore we can imagine that a 4D cube should have been consisted of 3D cubes with two 3D cubes facing  each other on opposite sides. Grab a piece of paper, and draw down 2 3D cubes slightly apart, then we have the two opposite sides of the 4D cube. Now we connect the vertices together, we get a very curious picture below:

Now this is what a 4D cube look like from a 3D perspective. It consists of 8 cubes (count it yourself!). The cubes are a little deformed because they are connected in the 4D space, and we cannot draw that connection out in the Euclid Geometry. In the picture below, the orange part marked out by the marker is another cube that is created by connecting two 2D sides of the opposite cubes.

Now that we’ve pictured what a 4D cube looks like in out 3D vision, let’s try to change the cube into a 3D model that could very well represent the cube in 4D. Let’s get back to the relationship between 3D and 2D: what happens when we cut a 3D cube up into a 2D picture? We take a paper cube and cut it up along the edges. We get the picture on the left. If we print it out on a piece of paper and then cut it out, we can fold it up into a 3D cube.

Now, if we cut the 4D cube up, we get a 3D model. In order to visualize it better, I drew a different version of a 4D cube. The difference between this 4D cube and the 4D cube I drew before would be explained in Part 2.

Anyways if we take this cube, and cut it up along the edges, we can get a model in 3D that can be folded up into a 4D shape. What does this model look like? Wait for it…

Did you get it right? And, hey, that really look kind of similar! That’s the cross in Dali’s painting!

Secret revealed! The surreal artist used a 3D model painted on the 2D canvas to demonstrate a 4D cube. He is telling us that God and the divine powers are a form of energy in higher dimensions that we cannot perceive. We can only glimpse their power and glory from a fraction of 3D perspective. Dalí’s inspiration for Corpus Hypercubus came from his change in artistic style during the 1940s and 1950s. Around that time, his interest in traditional surrealism diminished and he became fascinated with nuclear science. Sparked by science, his imagination takes him to explore concepts of a higher dimension, and thus born this interesting picture.

So in Part 1, we explained the shape and form of a 4D cube, imagined in a 3D perspective, and hopefully gave you some understanding of 4D geometry. But we have this weird and mythical picture in front of us, does it actually mean anything? How can we understand the nature of the fourth dimension through a cube? Is there any other way of picturing a 4D cube? All this and more, we will explain in the next part!

The next part will be posted on the 15th of January. Before then, follow us!!

-Ian