Question Platonic Solids and their shadows

[Max Pain]

While watching stuff about 4th dimension on youtube I had an idea:
If you have, for example, an triangle and project it (perspectivic) on a 2-dimensional surface you can make any triangle out of it. For example you project a regular triangle and can get a rectangular one as the "shadow" (of a point-lightsource).

I'm pretty sure this works for triangles, but I'm not so sure that it does work for every geometric thing, also a 3-dimensional one (if not then everything that follows is rubbish).

So a 3-dimensional thing (like a platonic solid) casts a 3-dimensional shadow on a 3-dimensional surface (analogue as above in 2 dimensions).

This reminded me of Platons "Theory Of Forms", where everything in our imperfect world is just a shadow from an perfect form in an ideal world. Here the five Platonic Solids came into my mind, and that their regularity gets lost if you see only their shadows.

Maybe Plato had a similar idea when he formulated his idea. However, this involves he knew of the 4th dimension. I did an excessive research on the net and couldn't find anything about this particular thing.

It would be interesting to know if this idea is plausible (or complete nonsense caused by watching too much videos on higher dimensions)?

 

[kwan3217]

I don't think you can make any shape with a three dimensional shadow of a four dimensional object. For one thing, the shadow will have at most the number of corners and edges the 4D shape has. But you can do what you are talking about, project an arbitrary 4D shape into 3D. When you do, the regularity may be lost, just like the shadow of an equilateral triangle may be scalene.

Wikipedia has articles on the 4D platonic solids, of which there are six, five which are roughly analogous to the 3D platonic solids, and one (the 24-cell) which has no correspondence. Those articles have many images attached to them, see if they are the idea you are thinking of. Also, I don't know if you have seen Dimensions, a 2-hour video on how to imagine and project 4D ideas onto a 3D world or a 2D screen, among other things. It was put together by mathematicians, so while it may not be complete, the things they do show are accurate. It starts out slow, and you may think "I already know this 2D and 3D stuff, get on with the good 4D stuff!" but pay close attention. The fourth dimension is not that far away, and many of the ideas we use in 3D and 2D apply to 4D, if you are careful.

 

[Max Pain]

Thanks for the link on the video, it looks very interesting.

Actually my thought was to project an 3-d shape on a 3-d "surface". Then the regularity of the body also can get lost.
Due to Plato's "Theory of Form" everything in our imperfect world is a shadow from an ideal Form in an ideal world. Also everything consists of the four elements, which are made up by the Platonic Solids. It just came into my mind that there is an connection between the shadow thing and the Platonic solids thing.
It would be just interesting to know if the old Greeks already knew of the 4th dimension.
Nevertheless the idea looks now much more far fetched, then in the moment I had it.

 

[cjp]

I think such nD-to-nD projection in an (n+1)D space (e.g. 3D-to-3D in 4D space) mathematically comes down to a multiplication of the generalized coordinates(*) of each corner point with a (n+1)*(n+1) matrix.

I don't know exactly what is possible and what isn't, but the first thing I see is that you can't change the number of corner points. At least you can't increase it; you may be able to decrease it by letting certain corner points fall exactly between other ones, so that they are effectively eliminated. This is also what happens when the shadow of a triangle becomes a line segment.

(*) For a point (x,y,z), the generalized coordinates are (x,y,z,1). So, you basically add an extra dimension to each vector. This has certain advantages for describing certain transformations with matrices.