How to visualize(inside ones brain) the Four-dimensional_space

Question

Can the fourth dimension https://en.wikipedia.org/wiki/Four-dimensional_space be visualized intuitively by the humans.

Does the professional mathematicians can do this ? If so what are the things to be learned to acquire this ability.

Answer 1

Professional mathematicians can sometimes visualize simple 4 dimensional shapes and movements to a limited extent, but nowhere near the way we can visualize 3 dimensional space. To visualize four dimensional objects myself, I generally draw (or imagine) 2 planes, each with 2 axes, side by side. Any object I draw in one plane, I also mark points accordingly in the other. This helps in locating points and vectors, but not curves and shapes such as hyperspheres. Also, ability to visualize even a 3 dimensional space is a quality that some people are born with better than others.

Answer 2

You can get pretty far by visualizing $(d+1)$-dimensional things as movies of $d$-dimensional things moving in time. These kinds of visualizations are closely related to Morse theory. For example, it's a bit tricky to visualize the $3$-dimensional sphere $S^3$, but it's not at all tricky to visualize the following movie:

• Initially, there is nothing.
• Then there is a point.
• The point expands to a $2$-dimensional sphere $S^2$.
• The sphere gets bigger.
• The sphere gets smaller.
• The sphere contracts to a point.
• Then there is nothing again.

Similarly, to visualize $\mathbb{R}^4$ you can visualize a movie where you're looking at $\mathbb{R}^3$ and nothing happens. That's a bit boring, but it gets more interesting if you put stuff into $\mathbb{R}^4$. For example, you can try to visualize a knotted surface in $\mathbb{R}^4$ by visualizing a movie where you're looking at $\mathbb{R}^3$ and some knots and links appear and disappear...