Second chapter : The four-dimensional space

I./ What is a dimensional space?

In mathematics, a dimension of a mathematical space, is defined as the number of coordinates needed to specify any point within it.

The one-dimensional space: is defined by a line.

On this one dimensional space, the point can only go right or left.

The two-dimensional space: can be defined by a plane. Here, we will define the plane with two perpendicular lines.

In this two-dimensional space, the point can go right or left, and up or down.

The three-dimensional space: can be represented by our environment. Indeed, we live in a three-dimensional space. So a point in our dimensional space, can go up or down, left or right and back and front!

So, how can we describe a four dimensional-space? How can we imagine a fourth direction, which is independent of the first three? II./ How to describe a three-dimensional solid to a two-dimensional creature

Thanks to this part we will get one step closer to answer the question in the last part.

Edwin A.Abbott is the first one to think about it.

In his book, Flatland, he tells the story of a two-dimensional square, which lives in a plane called Flatland, which encounters a three-dimensional sphere!

First of all, we can imagine that the square cannot see in three dimension. So it will not see the sphere. But it will see something! It will see the cross-section of the sphere.

The video in the link bellow will explain it: https://www.youtube.com/watch?v=HL2Z8NL4IQ0

So can we guess what it will be if each solid will come to Flatland?

Here we can see: https://www.youtube.com/watch?v=DkvdMkR8V8g

This is one way to describe a three-dimensional solid to a two-dimensional creature.

III./ Few representations of four-dimensional space

a./ Cross section of a four-dimensional space solid

One way to imagine a four-dimensional space solid is to imagine it would cross our universe, like the sphere did in Flatland.

When we tried to imagine a representation of a three dimensional solid in a plane, we thought about the cross sections of the solid. The cross section of a sphere is a circle, the cross section of a cube could be a square, and so on.

So we can imagine that the cross section of a four-dimensional solid in our universe could be a three- dimensional solid.

Let’s watch this video that explains well the situation:

https://www.youtube.com/watch?v=IbV0UoXXcOY&list=PLNefH6S6myiOfykOcgIc2sYrpr1Zk5Mhi&i ndex=2

b./ Oblique projection

As we can see in this video, there are other ways to imagine the fourth dimension.

One of those is the oblique projection. The oblique projection is a projection that permits us to draw a solid in a plane.

We will try it with the tesseract (four-dimensional space cube).

c./ Using colors

The last possibility we will study is using colors. Watch this video to understand:

https://www.youtube.com/watch?v=IbV0UoXXcOY&list=PLNefH6S6myiOfykOcgIc2sYrpr1Zk5Mhi&i ndex=2