Big Ben In N Dimensions.

[PatrickPowers]

Everybody knows about Big Ben, the London clock with four faces. How many faces need N dimensional Big Ben have?

It turns out with N>2 the answer is always four. Let's say there are two clock faces are in the e12 plane, facing in opposite directions, and a similar two in the e34 plane. Then if one is a reasonable distance away from the clocks (you can't see from inside or if very close to Big Ben) then one will have an angle of at least 45 degrees with at least one of the faces. Adding more dimensions doesn't change the situation. Any length in the remaining N-4 dimensions can only serve to increase the maximum angle with the two planes. The greater of the two angles will always be 45 degrees or more.

Example: Observer is at e1. They cannot see the e12 face but are perpendicular to the e34 face.

Observer is at e1+e3. They have an angle of 45 degrees with each of the two planes. Any decrease in either angle causes an increase of the other angle. So there is always an angle of at least 45 degrees. Adding more dimensions may increase and never decreases these angles. All the new dimensions are perpendicular to both clocks so anyone solely in some combination of these new dimensions sees them both full on.