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?


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

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

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