Gravity does not care about the ground
What If the Earth Were a Cube?
Gravity pulls towards the centre, so on a cube the middle of each face is the bottom of a valley.
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Step 1 of 5
Down is not the direction your feet point.
Gravity pulls everything towards the centre of the planet. On a ball those two directions are always the same. On a cube they are only the same at one spot on each face — the exact middle.
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Step 2 of 5
So the eight corners are mountains.
A cube's corner is about seventy per cent further from the centre than the middle of a face. On Earth-sized numbers that is a climb of thousands of kilometres — far, far taller than Everest.
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Step 3 of 5
The air and water would run into the middles.
Water does not care which face it is on. It only cares which way is down. Water always flows downhill, and downhill means towards a face centre. Every ocean would collect into six round seas, one per face, with the atmosphere pooled on top of them.
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Step 4 of 5
The corners would stick out above the air.
With the atmosphere pooled in the middles, the edges and corners would be in vacuum — airless, silent, with black sky and unfiltered sunlight, like standing on the Moon.
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Step 5 of 5
It would not stay a cube for long.
This is why planets are round in the first place. Anything big enough has gravity strong enough to crush its own rock, and rock under enough pressure flows. The corners would slump until the whole thing settled into a ball.
Small things stay lumpy for exactly this reason — asteroids are not round because their gravity is too weak to crush them.
The short version
A corner is seventy per cent further from the centre than a face's middle — a mountain thousands of kilometres high, above the air, with all the water pooled into six round seas. Rock that deep flows, so it would not stay a cube.
Try it yourself
Ask which is higher: the top of Everest, or a beach on the other side of the world. Measured from Earth's centre, some beaches near the equator win.