Starting to Visualize the Fourth Dimension

Of course, there is no direct way to "see" things in four dimensions. Consequently, we need to use some backdoor approach to acquire a visual feel for manifolds or other topological spaces that live in four dimensions. For over 100 years, geometers and topologists have created many techniques that allow one to study geometrical objects without beeing able to "see" them. After all, seeing is often deceiving. However, seeing can be fun and sometimes helps bolster one's intuition when trying to make generalizations of geometrical concepts to higher dimensions.
One way to visualize objects in four dimensions is to view level slices. Now before I jump to using three dimensions to view four dimensional objects, I should explain what I am going to do using two dimensions to visualize surfaces in three dimensions.

Starting with three dimensions

Let's consider the surface given by what I'm going to call the double Tchebycheff. It's not exactly what I've shown to the right. Take what I've shown to the right and turn it upside down.
Define f(x) = 8x⁴-8x²+1 and set F(x,y) = f(x)+f(y). Then the surface shown to the right is the upside down of z=F(x,y) over a suitable domain. (I have turned to the surface upside down because it's easier to see the interesting parts this way.)

A way to see the above surface by restricting oneself to only two dimensions is to view a collection of level curves or slices. A level curve is the curve you obtain when you intersect the surface with a plane, usually of the form z = c for some constant c. I have done this with the double Tchebycheff and displayed in to the right.
It's quite easy to pick out the maxima, minima and saddles of the double Tchebycheff in the above picture but how about in this one?
Another approach one can take, especially with a computer, is to use time frames as well as color. In the graphic below, each frame shows a successive slice.

Moving on to four dimensions

Using the same function f as defined above, define G(x,y,z) = f(x)+f(y)+f(z). By analogy with the "double" case, let's call this function the triple Tchebycheff. We would like to try to visualize the surface in four dimensions defined by w = G(x,y,z). To this end, we take horizontal w slices and animate them as we did in the last example.

We can easily observe the surface of the double Tchebycheff and see how the level curves piece together to make up the surface. As we try to understand how these level surfaces piece together to form the surface of the triple Tchebycheff, we are beginning to visualize a three dimensional surface lying in four dimensions.

Of course, there is no direct way to "see" things in four dimensions. Consequently, we need to use some backdoor approach to acquire a visual feel for manifolds or other topological spaces that live in four dimensions. For over 100 years, geometers and topologists have created many techniques that allow one to study geometrical objects without beeing able to "see" them. After all, seeing is often deceiving. However, seeing can be fun and sometimes helps bolster one's intuition when trying to make generalizations of geometrical concepts to higher dimensions. One way to visualize objects in four dimensions is to view level slices. Now before I jump to using three dimensions to view four dimensional objects, I should explain what I am going to do using two dimensions to visualize surfaces in three dimensions.
Starting with three dimensions
Let's consider the surface given by what I'm going to call the double Tchebycheff. It's not exactly what I've shown to the right. Take what I've shown to the right and turn it upside down. Define f(x) = 8x⁴-8x²+1 and set F(x,y) = f(x)+f(y). Then the surface shown to the right is the upside down of z=F(x,y) over a suitable domain. (I have turned to the surface upside down because it's easier to see the interesting parts this way.)
A way to see the above surface by restricting oneself to only two dimensions is to view a collection of level curves or slices. A level curve is the curve you obtain when you intersect the surface with a plane, usually of the form z = c for some constant c. I have done this with the double Tchebycheff and displayed in to the right. It's quite easy to pick out the maxima, minima and saddles of the double Tchebycheff in the above picture but how about in this one? Another approach one can take, especially with a computer, is to use time frames as well as color. In the graphic below, each frame shows a successive slice.

Moving on to four dimensions
Using the same function f as defined above, define G(x,y,z) = f(x)+f(y)+f(z). By analogy with the "double" case, let's call this function the triple Tchebycheff. We would like to try to visualize the surface in four dimensions defined by w = G(x,y,z). To this end, we take horizontal w slices and animate them as we did in the last example.

We can easily observe the surface of the double Tchebycheff and see how the level curves piece together to make up the surface. As we try to understand how these level surfaces piece together to form the surface of the triple Tchebycheff, we are beginning to visualize a three dimensional surface lying in four dimensions.

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