## Playing with 4D Planetary Scales

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### Playing with 4D Planetary Scales

Lets assume that, to maintain life, a 4D planet needs active geology just like Earth, which means retaining heat to drive tectonics and volcanism. If primordial heat is the same between a 4D planet and Earth (very debatable, but I'll use it as a first approximation because I don't know of anything better), and assuming that thermal conductivities and emissivity and so on are all equal, then we want the the surface-to-volume ratio of Earth to match the surface-volume-to-bulk ratio of the 4D planet. That ratio will always be equal to the radius of the planet scaled by the ratio of dimension-specific coefficients. Going from 3D to 4D, it turns out that the radius must increase by a factor of 4/3rds to maintain the surface-to-volume ratio.

In order for figure out how much material that corresponds to, we'll measure radius in units of 2 angstroms (i.e., one average atomic diameter). Calculating the volumes of Earth and the 4D planet in terms of cubic atomic volumes and quartic atomic bulks, and then dividing, we find that this 4D world with equal heat retention capacity as Earth must contain 1.18515e17 times as many atoms! Earth's mass in kilograms in about 6e24kg. Multiplying it out and assuming atomic masses are comparable between universes, this 4D planet would have a mass of about 7e+41kg--which is somewhere between 1/3 and 1/4 the mass of our entire galaxy! Meanwhile, its surface comes out to 2e28 cubic kilometers--equal to the volume of a sphere 23 AUs in diameter! (That's a couple of AUs larger than Saturn's orbit.)

So, uh... dang. That is a lot of space, and all crunched down in four dimensions so its all within reasonable Earthy travel distances....

Of course, if you fiddle with physics in other ways, planets could retain heat at smaller sizes. Or maybe life doesn't need geological recycling, and then they don't need to retain heat at all. But hey--the volume of a solar system to play with all accessible on the surface of a planet? That's pretty neat.
loganrk
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