Just as you cannot have a real cube in a two dimensional space – only a shadow, or projection, or drawing of one – so you can only have a shadow or projection on higher dimensional forms into 3 dimensional space. And just as you can rotate a wireframe cube and see some interesting changes in the shadow that it projects onto a two dimensional surface, so you can rotate a higher dimensional object and observe the interesting changes in its two dimensional (or three dimensional) shadow.
The best interactive hyper-dimensional solids (polytopes) animations that I have yet seen online are at a site called DogFeathers. You can choose what four dimensional shape to rotate, and other parameters such as the speed of rotation, the way it is coloured, etc etc.
You can also choose to watch a stereo projection so that you can see it in 3D. You can do this using red green or red blue glasses, or if you know how to cross your eyes and merge the images then there is a setting for this as well. For instance here is a still image taken from the cross eye stereo animation
NOTE: You will need to have Java installed for your web browser to display these properly… which most machines already have. If you see a still gif image instead of the animations and their control panel then you will need to install Java from here
I intend to write a full page about hyperdimensional forms soon, and also a page about stereographic projections and the perception of 3D space… but you will have to come back for that 😉
Okay why am I still writing and why are you still reading – just go there and check it out – they really are incredible ! 🙂