Since we know the height is 128 stories, and one story in a typical building equals 3.3 meters, then I can find the number of hex pods that will stack up in the inverted V. But that leaves a question: How many layers of hex pods would you need for all 1.14 billion pods?

OK, let’s get to this. I will start with the side area of the V. I’m going to assume it’s a right triangle and the vertical side is 128 x 3.3 = 422.4 meters tall, with a vertex of 33 degrees. Now I just need to find the total side area of the rocket V and divide by the area of each hex pod to find the number that fit in one vertical layer. After that, I can find the number of layers needed for all those people.

Since you might want to use your own estimated values, I put it all in Python. Here are my results. (Yes, you can click the pencil and change the values to make yourself happy.)

But here you see a problem: That 128-story rocket would have to have a width of 38 kilometers. That’s 23.6 miles! The robots sure build weird-looking fat rockets. Oh well, I guess they know best. They are the machines, after all.

How Much Energy Would It Take to Launch Them?

Now you’ve got this giant rocket full of people and hex pods. You want to send it into space, never to return. How much energy would that take?

The robots obviously aren’t going to use rockets with chemical fuel—that’s so barbaric only a human would do it. But either way, I can calculate the energy needed. There are two important parts of this calculation: What is the mass of each V-rocket, and how fast does it need to go?

First, let’s take the mass. I’ll use a rough estimate that most of the mass is due to all those humans. (It’s possible that the pods are just some type of force field with no mass, and that the structure of the rocket itself is super efficient and has negligible mass compared to the people.) So, if there are 1.14 billion people per rocket and their average mass is 75 kilograms, that’s a total mass of 8.55 x 10^{10} kilograms.

If the robots want these humans to never come back, they would need to launch the rocket with a velocity equal to the escape velocity. Here is my more detailed explanation of escape velocity—but essentially it’s the minimum speed needed to reach an infinite distance from the Earth. Using the mass and radius of the Earth, this is a velocity of 1.118 x 10^{4} m/s.