UM, HOW MANY DIMENSIONS WAS

UM, HOW MANY DIMENSIONS WAS THAT?….Today in The Corner, John Derbyshire asks, “How do you fit the Empire State Building into a 1-inch cube?” The answer, although not very practical, goes like this:

The fact, by the way, that there is a figure equivalent to the cube ? i.e. a sort of super-cube ? in any number of dimensions, provides the answer to the question in my subject line. A 1-inch square (i.e. every one of its sides is just 1 inch long) has a longest diagonal whose length is the square root of 2. A 1-inch cube (every edge 1 inch long) has a longest diagonal whose length is the square root of 3. A 1-inch 4-dimensional hypercube has a longest diagonal whose length is the square root of 4. And so on. This is a general rule: A 1-inch n-dimensional super-duper-hypercube has a longest diagonal whose length is the square root of n. The Empire State Building is around 15,000 inches high, and that is the square root of 225,000,000. So if you construct a 1-inch cube-equivalent in a space of 225 million dimensions, the ESB will fit into it very nicely.

If Derb would just stick with math he’d be pretty readable.