Ed Scheinerman is a professor in the Applied Mathematics & Statistics department of the Whiting School of Engineering at Johns Hopkins University where he also serves as a vice dean. Scheinerman studied mathematics as an undergraduate at Brown and as a graduate student at Princeton. His research focus is graph theory—the mathematics that models networks (roads, social, neurons, computers, and so forth). He has twice been honored with the Paul Halmos-Lester Ford award for mathematical writing. He is the author of a variety of books including The Mathematics Lover’s Companion which is also written for a general audience.
Solution to the hypercube problem
A four-dimensional hypercube has 32 edges. Here’s why. Start with a simple line segment. It has only one edge. We transform it into a square by copying it to a parallel line segment; its endpoints trace out two new edges, giving the four edges of a square: two copies of the original line segment (shown solid in the figure) plus two that are traced out by the endpoints of the line segment (shown dashed).
Next, we transform a square into a cube. We move the square to a parallel square to form a cube. As before, we get two copies of the square (original and parallel) giving 8 edges plus 4 more traced out by the corners of the square. The cube has 12 edges
Now comes the leap into the fourth dimension. Just as before, we move the cube parallel to itself making two copies. That gives 12+12=24 edges. And, just as before, the 8 corners of the cube trace out an additional 8 new edges, for a grand total of 24+8=32 edges.
Edward R. Scheinerman, A Guide to Infinity: Ten Mathematical Journeys, Yale University Press, 192 pages, 6 x 9 inches, 70 b-w illustrations, ISBN: 9780300284799
We don't have paywalls. We don't sell your data. Please help to keep this running!