"Say, That Looks Big Enough to Fly a B-25 through!"
I'm reprinting this from the comments section on james_nicoll's blog, where, in a discussion of planes hitting tall buildings, ethelmay writes, intriguingly: Tangent: I once met a guy who had flown a B-25 UNDER the Eiffel Tower. He and his wife were friends of my father and stepmother.
The spaces at the base of the Eiffel Tower appear to be defined by semicircular arcs. Their diameter is 74.24 meters, for a radius of 37.12 m.
A North American B-25 has a wingspan of-- well, some online sources say 66 feet. Some say 67 feet. Some say 68. I found one that says 118 inches, but it turns out to be describing a 1:7 scale model. Let's take 68 feet, or 20.7 m, for a half-span of 10.35 m.
Presuming your transgressive* friend is skilled enough to fly his bomber down the centerline of the archway (as projected onto the ground; a plumbline dropped from the highest point of the arch would touch this), and presuming the wingspan is the dominant constraint (e.g. the twin tailfins are not tall enough to intersect the arch if the wingtips clear it), what is the maximum altitude at which the wings safely pass through?
This height is the length of a vertical side of a right triangle, whose horizontal side is a half-wingspan in length with a vertex on the arch, and whose hypotenuse is one radius in length with one end at this vertex and the other on the centerline at the ground.
Pythagoras teaches us that the maximum height is therefore
SQRT(37.12^2 - 10.352) = 35.6 meters, or 115 feet.
This answer is a bit too simple. I have treated the semicircle of the arch as if it were in a vertical plane. As anyone can see, the arch is actually tilted, so viewed from the side the tower's base appears to be a trapezoid. So height of the semicircle, and the maximum safe height for the bomber, are actually lower by a factor depending on the angle of the trapezoid's sides. Improving this result is left as an exercise for any student able to determine this angle.
I have also neglected the calculation of minimum height, which might involve gathering data on the umbrellas over vendors' pushcarts and such.
I am sure your friend, as a methodical and safety-minded aviator, took all these things into account in making his own calculations.
* Yet completely awesome.