A dot can not represent the actual object, the two-dimensional plane can not show the full picture of the object, the three-dimensional structure completes the description of the world, so how should the 4-dimensional space be represented?

This problem was not initially elaborated until the **early 20th century**, but more often through mathematical language to express the state of 4-dimensional space.

This academic problem later perfected Einstein’s theory of relativity, but as Einstein’s mentor, **the German** mathematician **Hermann Minkowski’s** theoretical analysis of high-dimensional spaces gave him a pivotal position in the mathematical community.

But before that, no one knew what a **4-dimensional space** looked like, but strictly speaking, it was the same now.

But with mathematical descriptions and model understanding, **we can now deduce the projection of 4-dimensional space in the 3-dimensional world from the 3-dimensional world**, just as we draw on paper.

However, to prove the existence of the 4-dimensional, **Minkowski** has spent a lot of effort.

Without much discussion here on special relativity, let’s look directly at how the **German** mathematical wizard proved 4-dimensional space.

# Minkowski space-time

Minkowski space-time needs to be applied to the Lorentz transform, taking into account the problem of appropriate time and length contraction, and the main solution tool is the “**Minkowski diagram**“.

From a mathematical structure point of view , Minkowski ‘s metrics and derivatives also have **group theory** , and in terms of space-time manifolds as a result of the hypothesis of special relativity , space-time intervals represent invariance , because curved space-time is local Lorentz.

Both **lorentz transforms** and **special relativity** propose the concept of **absolute space-time**, and the observation of facts depends on the observer’s system of reference, so Minkowski’s expression of space-time in mathematics is equally space-time invariance.

But due to the invariance of the intervals, the classification of any vector is the same in all reference frames related to the Lorentz transform.

So Minkowski’s spatial event will have a variety of different vector sets to represent the light cone of that event.

The direction of time and the spatial changes cause Minkowski space-time to have different sets in four sets.

In the geometry of space-time, Minkowski space has a very important distinction in terms of time.

**In 3D space**, Minkowski space-time has an additional dimension whose coordinates **Xº are derived from time**, so that the distance differentiation satisfies the formula.

This is also what we later said, in 4-dimensional space there will be a time reference.

But what needs to be understood here is that the existence of time is not time as we generally understand it.

Usually the time we use is absolute time in space, but Minkowski space-time in special relativity can be expressed as the invariance of any inertial reference frame observing space-time intervals.

**That is, the 4D distance between any two events, this rotational symmetry of the existence of Minkowski space-time expresses changes in 4-dimensional space.**

In contrast, time in four-dimensional space acts as an additional axis and is orthogonal to the other three axes.

From the geometric structure of mathematics , Minkowski space-time retains orthogonality about curves in hyperbolic rotation , while Euclidean diagrams maintain orthogonality by rotation.

This is the hyperbolic orthogonality of Minkowski’s space-time, which was later used in special relativity to define simultaneous events.

Through various mathematical representations, Minkowski proved the representation of 4-dimensional space, although this is not the same as the general physical space-time representation, but the application of relativity verifies the correctness of Minkowski’s space-time.

# What should a 4-dimensional space look like?

Due to the addition of additional degrees of freedom, geometry in **four-dimensional space** is **more complex** than geometry in three-dimensional space.

In the three-dimensional world, a circle can be squeezed into a cylinder, while in the **4-dimensional world, several different cylindrical objects appear**.

The best proof pattern is the Klein bottle, where curves can form knots in three dimensions, but surfaces cannot unless they intersect themselves.

However, in 4D space, the variation of the curve can be easily untangled by moving in the fourth direction, and the 2D surface can be formed in 4D space.

So what should a 4-dimensional space look like for humans? And what will it look like for people who enter the 4-dimensional space?

Through imagination and dimensional analogy, the most common way we use is to express the high-dimensional world through projection, but after entering the 4-dimensional space, everything will become different.

In the three-dimensional world, we can easily imagine physical images of different 3 dimensions in our minds, and to understand **the 4-dimensional** we can apply the changes in **Minkowski’s space-time**.

However, on four dimensions, each axis will have a cube, so the four dimensions are multiplied by 2 faces, and every 8 faces form a surface.

**Due to the increase in dimension and the change of motion, the structure in the 4-dimensional space will change in various forms according to the angle of the observer**, and from the perspective of human vision, no one knows exactly what the real form of the object in this space is.

If you still can’t understand very well what’s going on in 4-dimensional space, then look at the following diagram.

Since the world in which humans live is a **3-dimensional structure**, we cannot truly understand 4-dimensional space, but only from **mathematical graphics**, and even so, there will still be many incomprehensible structures.

If you are fantasizing about entering 4-dimensional space, the reality of the situation is likely to be very complicated, because people who enter 4-dimensional space will die quickly.

# 3D creatures in 4-dimensional space

From the Klein bottle we can see that the things of the 3-dimensional world do not exist in 4-dimensional space. After entering the **4-dimensional space**, the atomic structure of all matter will become different, and the atomic orbitals will contain more electrons.

So in this dimension, some metallic elements become gases, such as magnesium.

Similarly, our bodies undergo very strange changes, where most of the elements we rely on in the body fail due to changes in spatial dimensions, where functions that function properly in 3-dimensional space fail.

Theoretically, **people will be decomposed in 4-dimensional space**, and assuming that people can live at this time, then we may see various body fragments moving in 4-dimensional space.

In fact, living things in 3-dimensional space have no meaning in 4-dimensional space.

As a very simple example, a painter can draw a very real portrait or animal on a piece of paper, but the 2-dimensional structure cannot show their internal organs.

Therefore, there is only a “surface” in the 2-dimensional world, and there is no concept of “inside” and “outside” in the 2-dimensional world.

**So, if objects in 2 dimensions can somehow enter the 3D world, they will also collapse because there is no three-dimensional support**.

A similar concept is an example of a human being, because after entering the 4 dimension, we do not have 4 dimensions of hands, feet, and bodies, then changes from any direction can destroy the human body.

**In the face of the complex changes in space, Minkowski space-time gave a reasonable explanation** and expressed it in mathematical language, which was very great in the 20th century.

It is also Minkowski’s research that makes people realize that time and space are a space-time continuum and are coupled together in 4 dimensions.

However, such a genius and master **of mathematics** could not escape the pain, and when Minkowski was 44 years old, he had to face death due to the onset of **appendicitis**.

Due to the lack of medical care at that time, surgery could not solve the problem of appendicitis, and he died in 1909.

However, his student **Albert Einstein** did a good job of bringing Minkowski space-time into his theory, which is also his great point, and he finally got the theory of relativity.

Perhaps reality is like 4-dimensional space, and we never know what the next change in direction will look like.

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