Cubes

Posted January 2004.

1. Introduction

Headlines have been made recently because cosmologists have been proposing the notion that physical space might involve 10 or more dimensions. This need for using higher dimensional space grows out of the relatively recent development of string theory, a subject on the border between physics and mathematics. Just as we were getting more comfortable with how to think about 4-dimensional space-time, we are faced with a new challenge. Conceptualizing higher dimensional spaces or objects in them does not come naturally to most people. Perhaps this explains why the study of higher dimensional spaces is surprisingly recent. Higher dimensional spaces are intriguing to mathematicians because many of the properties of objects that one takes for granted in two and three dimensions are not shared by the analogous objects in higher dimensions. It becomes fascinating to compare and contrast what happens in two and three dimensions with what happens in n dimensions. One way of constructing a bridge to higher dimensional space is to take something familiar in 3-space, such as a cube (or sphere), construct the analogue of the cube in higher dimensions, sometimes called hypercubes, and see what properties these higher dimensional cubes have.

Joseph Malkevitch
York College (CUNY)

Email: malkevitch@york.cuny.edu

1. Introduction
2. Some history
3. The 3-dimensional cube
4. Combinatorial perspectives on cubes
5. A recursive way of constructing cubes
6. Cube puzzles
7. Symmetries of the cube
8. The Sharir-Ziegler cube
9. References