Projecting an $n$-dimensional cube in $n-1$ dimensions

Question

When projecting an $n$-dimensional cube in $n-1 $ space, how many faces do not retain perpendicularity?

Answer 1

The answer varies. The "look down into box" projection of a cube onto a plane preserves perpendicularity for two faces. Projecting along a long diagonal preserves perpendicularity for none.

Each of these extreme cases generalizes to higher dimensions.

For projections of the $4$-cube, see the wikipedia page on the tesseract (https://en.wikipedia.org/wiki/Tesseract) and other google images. When you see these on your screen, they are plane projections - $2$ dimensional pictures of projections to $3$-space.

When thinking about projections you have to be careful about the definition. In linear algebra the projections are those with a viewpoint "at infinity". There's no perspective in the drawings. Then the only thing that matters for your question is which edges of the cube are seen end on, as just a point. (That's essentially @SergeBallestra 's comment.) If you project down one dimension you can see at most one edge direction that way. If you project from $n$-space to ($n-k$)-space you can see up to $k$ edge directions end on.

For the more general projections from any viewpoint, with "vanishing points" for parallel lines, you have to study projective geometry. Artists began using it in the renaissance.