Visualizing the 4th dimension.

In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects?
I searched that we can at least see their 3-D cross-sections. A Tesseract hypercube would be a good example.
Can we conclude that a 3-D cube is a shadow of a 4-D tesseract?

But, how can a shadow be 3-D? Was the screen used for casting shadow also 3-D; or else, what way is it different from basic physics of shadows we learnt?

edit: The responses are pretty good for 4th dimensional analysis, but can we generalize this projection idea for n dimensions, i.e. all n dimensional objects will have n-1 dimensional projections?
This makes me think about higher dimensions discussed in string theory.
What other areas of Mathematics will be helpful?

Answer

The animations below accompany an introductory talk on high-dimensional geometry.

Mathematically, the second was made by putting a “light source” at a point (0, 0, 0, h) (with h > 0) and sending each point (x, y, z, w) (with w < h) to \frac{h}{h - w}(x, y, z).

Shadow of a rotating cube
Shadow of a rotating hypercube

Attribution
Source : Link , Question Author : pooja somani , Answer Author : Andrew D. Hwang

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