3D - 2

Can a 4D shape be represented in a 3D volume?

A 3 -dimesional shape can be repesented on a 2-dimensional plane eg. a cube drawn on a piece of paper (although I guess it would mean nothing to a creature confined to a 2D world). My question is can the same be done from 3D to 4D? Can a 4D shape be represented using a 3D volume? I appreciate that we (being 3D creatures) probably wouldn't be able to comprehend the result, but I just want to know if it can be done?

Public Comments

1. yes, and that is what we call a generalization of a projection. There are many different kinds of projections; the example you gave from 3D to 2D is one such; but the most common is what we call a stereographic projection. It would be better to look it up on the net or some math book, but the keywords you should be using are projections and stereographic projection. One simple example is the polychoron. If in 2D, we learned of polygons (triangles, squares, etc) and in 3D we have polyhedra (tetrahedra, cube, pyramid, etc), then in 4D we have polychorons (the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, 600-cell and many others). In a lot of books, the way to view these objects is to use the schlegel diagrams, either drawn in 2D or 3D. The schlegel diagrams are another example of projections from 4D down to 2D/3D. In general, you can project any object from higher dimension to 2D or 3D must some distortion would definitely occur.