wich rotation is it?

[batmanmg]

when images of a 4 hypercube are shown at different angles... or even being rotated... is this rotation 3d or 4d rotation?

[thigle]

4 hypercube you mean hypercube i suppose.

consider this: you have an ordinary cube executed in wireframe. a light far enough to seem like parallel rays source (like the sun), casts shadow of the cube onto flat 2-space. rotating the cube in 3-space makes the 2d shadow change. in one extreme (viewed along the diagonal, vertex-on) you see hexagon with diagonals as the shadow, in other extreme (face-on) a square. between are various intermediary shadow/shapes.

now analogically for the 4-cube aka hypercube: it dwells within 4space just as 3cube dwells naturally within 3space. its 'shadow' is 3d. now the shape it has is due to the position of the 4cube in 4space and the way it is projected into 3space. parallel projection gives rhombic dodecahedron in one extreme, cube in the other. in between are all the various 3shadow-shapes that seem like 8 cubes connected with various deformations (though they are all 3cubes in 4space).

so to sum it up:
when images of a 4 hypercube are shown at different angles... or even being rotated... is this rotation 3d or 4d rotation?
when unchanging shape just rotates in 3space, that is viewer rotating in 3space around the shadow of the hypercube. if the shape is changing (lenghts and angles) then it means that the hypercube is being rotated in 4space.

its relatively simple, just think about 3cube to 2d projection and then analogicaly about 4cube to 3d.

[Rkyeun]

To represent a 3D object like a cube in the 2D space of your computer screen, we have to fudge it a little. We do this by adding fake perspective, which is where you get images like... two squares connected by rhombuses. It can be a little confusing, but if we rotate that image in a 3D way the 2D shape changes and you see a little better the shape we're trying to show you.

To represent a 4D object like a hypercube in the 3D space of our world, we also have to do a little fudging. More fake perspective, more confusion, but as we rotate it in a 4D way you start to see what shape we're talking about, even if at first it might be hard to wrap your mind around. At every 4D rotation we have a 3D fudged shape to show you. The problem is we have to fudge that shape too in order to make it show up on your 2D screen. So we take that shape and rotate it in several 3D ways to make sure you see it, which is a little confusing, and then we rotate the original object in a 4D way, which is a lot confusing, and more often than not the result is totally incomprehensible.

Sorry.