# I want to build my own pyramid tomb, and I want it to be big. Instead of the slope being 45°, the slope can be 89.995°. How big can this be?

Short answer: The biggest pyramid possible is close to the height of Mt. Everest, but if you assume a loose definition of “pyramid” it could reach a quarter of the distance to the moon.

Long answer: This question was asked by Ivan. Ivan would make a great pharaoh. They were all about building bigger and better than the last guy. You know, showing off. Well, Ivan wants to show off and end the one-upsmanship once and for all, by building the biggest pyramid possible.

How big can a pyramid be? Well, a pyramid is basically just a man-made mountain, so how big can a mountain be?

I’ve previously approximated the tallest mountain possible when I calculated the potato radius. With some basic physics considerations you can guess that it will be close to the size of Mt. Everest. The reason is that, as the mountain gets taller, the pressure on the base gets bigger. Basically, gravity is pulling all that weight down, and eventually the pressure will exceed the strength of the base so the mountain will crumble.

This argument (for approximating the height of the tallest possible mountain) doesn’t require any information about the slope of the mountain, only the height of the peak. Whether the mountain is a rounded cone, or a bit more angular like a pyramid, or even just a straight-up cube, the ceiling height can be determined by the strength of the base rock you’re using. Geometry doesn’t matter, to first order.

In reality, it will be more complicated. Mountains actually make the continental plates sag, sinking a bit into the mantle, and then there is a bouyant force that keeps it all afloat, just like a person lying on an inflatable bed in a pool. The vocab word here is “isostasy” and it has a great Wikipedia article (I apologize to the geologists reading this, you must hate me right now).

Anyway, if material costs aren’t a concern, then the tallest 45° pyramid is the same as the tallest 89.995°. Even a 90° pyramid, which might better be called a cube, is possible. If we use a rock like granite, with a strength of around 200 MPa and a density of 2.7 g/cc, then the tallest possible pyramid is basically just the same as the tallest possible mountain, which again, is something like Everest.

But suppose we get even more flexible in our definition of pyramid. We just want a tomb with the tallest possible structure on top. It turns out, if we let the slope be greater than 90°, then we can build a tower up to 100,000 km up from the earth.

Sounds insane, right? If we were to build such a structure, it would extend a quarter of the way to the moon. As a consequence it’s no longer supported by the base; it’s supported from the top. We would need to put a massive weight at the top of this structure, so that as the earth rotates every day it will be swung around in a circle, sort of like a toy on a string getting spun in a circle. The tension from this weight keeps the structure upright, supported from above.

This eliminates the problem of gravity crushing our base, but makes a new problem. The structure must get wider as it goes up, until it reaches geostationary orbit, and then it must narrow again towards the massive space-weight.

There is a huge problem though. How do we get a material strong enough for this? If you do the math for the tapering required to make this work, a steel structure would have a taper ratio (the ratio of the width at geostationary orbit to the width at the base) of 1033 – not possible. Carbon nanotubes, on the other hand, have a taper ratio of 1.6, which is a bit more feasible, so your tomb should be built from the lightest, strongest material mankind can manufacture [1].

And unlike the Egyptians, your tomb serves a practical purpose. Because it extends most of the way to the moon, we can climb up it, and use it to launch cargo into orbit for really cheap. Have you guessed the punchline yet? Ivan, the tomb you want to build is a space elevator.

asked by Ivan V.

cover image credit: Wikimedia Commons

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