# The math behind Gravity’s Rainbow (Part3): Double Integrals

A part of Gravity’s Rainbow plays in the mines of Kohnstein that were repurposed for production of the V1 and V2 rockets. Prisoners of the Nazis concentration camp Mittelbau-Dora worked there under inhumane conditions. More people died manufacturing the V2 than were killed by its deployment.

One of this section’s themes is the architecture of the place and how it relates to destruction and death. We listen in to the conversation between the master architect and one of his apprentices: To get a feel for the resemblance, take a look at the mine’s layout and the sign for a double integral:  Alright. The text then goes on with a reflection on the “meaning of the shape of the tunnels”:   Ok, since there was no GPS back in World War 2 the rocket needed to measure acceleration to know about the distance it travelled. Let’s do some calculations:

We will assume that the rocket always has constant acceleration that is mostly directed upwards (y), but also has some components that push forward (z) and left (x):

acceleration(t) = a(t) = (x,y,z) m/s^2 = (-1, 5, 2) m/s^2

We can now calculate the double integral of this, to get the position of the rocket at time t:

position(t) = ∫∫ a(t) dt = ∫∫ (-1, 6, 4) m/s^2 dt = ∫ (-1t, 6t, 4t) m/s dt = (-1t^2, 3t^2, 2t^2) m

Let’s say we want the rocket to reach a height of 50km = 50.000m. We can then calculate how many seconds the rocket will fly:

3t^2= 50.000 <=> t = 100 sqrt(5/3) <=> t = ~129s

So “Brennschluss” is reached after about 129 seconds.

There are another two meanings of the double S shape that Pynchon talks about. One is “the shape of lovers curled asleep” which is less interesting mathematically. The other one is this:  He poses the question “And what is the specific shape whose centre of gravity is the Brennschluss Point?” and says that there is only one shape that satisfies this criterion.

I really have no idea what specific shape he is alluding to here. Maybe it’s just his way of toying with the reader.  But please let me know if you have a more satisfying solution!