I have 3 fun representations of hyperdimensional 'objects' such as 'Hypercubes'. The 'Hyper' part of 'Hypercube' means it is a representation beyond the 3rd dimension. You will find hypercubes (and spheres, pyramids, cones, etc) representing 4d and up (unlimited). The first is a 'Tesseract', or a four dimensional cube (aka 'hypercube'). It has 8 cells (3d cubes), 24 faces, 32 edges and 16 vertices. The 4th dimension is a temporal dimension (what we think of as time). The tesseract is to the cube as the cube is to the square. formally, the tesseract can be described as a regular convex 4-polytope whose boundary consists of eight cubical cells. A tesseract is in principle obtained by combining two cubes, only in 4d they are 1. My guess is that the hypercube does not actually have more faces exactly, it's more that in the 4th dimension the same faces occupy more than one location at the same time, which translates here in 3d to more faces 24 faces arranges into 8 'cells'. http://www.youtube.com/watch?v=5xN4DxdiFrs To better understand it, in this video you can see them 'Unwrapping a tesseract ' http://www.youtube.com/watch?v=BVo2igbFSPE The second is a 5th Dimensional Hypercube...well we can only see the shadow of one by using our imagination. We see the cube shifting along the fourth dimension (time) and the 5th dimension is the space that makes the transformation possible. At least that's how I understand it. http://www.youtube.com/watch?v=lFvUaFuv5Uw (As described by the creator of the animation it is a "5d cube. the 2-dimensional projection of the 3-dimensional shadow of the 4-dimensional shadow of the 5-dimensional analog of a simple wire-frame cube rotated in various directions") The third is "How to Turn a Sphere Inside Out (without cutting it!)" http://www.youtube.com/watch?v=R_w4HYXuo9M I saved the best for last. Thge fourth are Hypercubes, starting from dimension 0 up to dimension 6 http://www.youtube.com/watch?v=-x4P65EKjt0 or 'Tour 6D' http://www.youtube.com/watch?v=Bn7HDBj9ZQQ There are 4D and 5D Rubik's cube type game software for people to solve. magiccube 4d mc 4d. melinda greene don hatch 2002. Introduction to understanding the Rubik's 3x3x3x3 (4D Rubk's Cube) http://www.youtube.com/watch?v=Aonf34s0Bqg Doesn't have to be all cubes. Here is a Hypersphere http://www.youtube.com/watch?v=BqfwPQvb7KA How to Draw a Hypercube (Tesseract) with 8 Squares http://www.youtube.com/watch?v=e5OgCgj1UoE 'Hypercuube' on wiki: http://en.wikipedia.org/wiki/Hypercube 'Tesseract' on wiki: http://en.wikipedia.org/wiki/Tesseract 'Five-dimensional space' on wiki: http://en.wikipedia.org/wiki/Five-dimensional_space I may follow up with other stuff