#fourth dimension

IT'S REAL!!!

IT'S HAPPENING!!!

Minecraft Dungeons 2 is coming out THIS YEAR!!!!!!!!

We're going to the darkest depths of the Overworld and into another dimension!

We know very little currently, but more information will come in the next couple months!

She had a hard time understanding why I didn't understand what she was talking about with respect to the mechanisms for controlling a spacecraft's double rotation. My travel guide is at least more well versed with explaining things by dimensional analogy.

But I did learn some things about their space program. As previously mentioned, planets can orbit in any plane, in any 3-volume, in the entire planetary system. This makes proper interplanetary space travel (aside from certain flybys) incredibly difficult.

But it gets worse. Let's look at Tsiolkovsky's Rocket Equation:

Δv = ln(m1/m2) * Ve

Δv = Delta v, the change in velocity a rocket is capable of executing. ln = the natural logarithm function. m1 = the loaded total mass of the rocket. m2 = the empty mass of the rocket. m1/m2 = the mass ratio. Ve = the effective exhaust velocity of the propellant.

The first and most important disadvantage is that the mass ratio is lower in four dimensions. The square cube law in 3 dimensions says that as a container with a given thickness of wall scales up in linear dimensions, the area (and mass of the wall) scales as the square of the linear dimension, while the volume (the interior of the container) scales as the cube of the linear dimension. Effectively, the mass ratio scales as the (3/2=1.5) power of the linear size of the tank.

In four dimensions though, the mass ratio scales as the (4/3=1.25) power of the linear size of the tank.

Uh-oh. An exponent of 1.25 grows more slowly than an exponent of 1.5.

Red is y=x^(3/2), blue is y = x^(4/3)

For each kilogram of fuel that can be stored in a tank, more weight must be put aside for the tank wall. Even though the tank can store vastly, vastly more atoms of fuel, the wall has to be much, much more complete.

This affects the mass ratio. That's the part of the Tsiolkovsky rocket equation that is inside of a logarithm. 4D rockets are exponentially less efficient than 3D rockets.

But there is a saving grace, at least for orbits around the 4D home planet.

4D planets are pretty small. There's a truly enormous amount of mass in the 4D universe but it's been spread out, since there are no accretion disks/balls. Also, surface gravity grows much faster--as the square of radius rather than linearly with radius.

The reason gravity grows linearly with radius (for a constant density) is that the mass grows with the cube of radius (because it grows linearly with volume) but the force of gravity falls off with the inverse square of the radius. The terms cancel out to a linear relationship.

If 4D space had inverse cubic gravity, this relationship would also hold, but then stable orbits would be effectively impossible--there would only be inward or outward spirals. The 4D universe I am writing from has inverse square gravity, like our own. So the mass of a planet grows with the tesseract of radius, while the force of gravity still grows with the inverse square of the radius. So surface gravity grows with the square of radius.

4D atoms are also a bit smaller than 3D atoms, although the actual mass within each atom happens to be smaller too.

A planet with the diameter of Earth in the 4D universe would have a mass of something like 1,000 solar masses.

The gravitational constant of the 4D universe is smaller than that of our own.

All told the diameter of this planet is about 608,000 meters, or about 1/20th the diameter of the Earth, it has a far far greater mass than the Earth, and a surface gravity of about 0.8 times Earth gravity.

This is actually really good! Anyone who has played Kerbal Space Program with rescale mods knows that the delta-v requirements are proportional to the square root of the rescale factor, and since orbits are ultimately still keplerian here in the 4D universe, that means that delta-v costs for orbital maneuvers are in the ballpark of (1/20)^(1/2)=0.22 times that of orbital maneuvers on Earth.

If you're wondering why the atmosphere doesn't escape--the kinetic energy and thus temperature of a gas scales at a higher rate with velocity, so the gas is moving slower for a given temperature.

I did begin to wonder if rocket exhaust efficiency here were at a disadvantage too, for the same reason. After all, if a rocket's hot exhaust gases produces a lower effective exhaust velocity, then the Ve part of the Tsiolkovsky equation is reduced, linearly reducing delta-v. That would affect both chemical and nuclear thermal rockets. But my contact at the 4D space program assured me that compared to us, 4D rocket chemists have some much more potent stuff available to them, and that the higher resulting temperatures avoid melting or vaporizing the engine because the 4D analogues of metals have much higher melting points.

Delta-v to orbit the 4D planet is about 2.2 kilometers per second, which is very modest compared to Earth's 10 km/s. Their orbital rockets end up being smaller in linear dimensions than ours.

Unfortunately the relative velocities between planets are measured in 10s to 100s of km/s, making them impractical to reach.

But it's not all doom and gloom. The planet has four moons. Two of them nearly share an orbital plane, a third nearly shares an orbital volume, and the fourth does not share a plane or volume with the other three but does nearly share an orbital plane with the orbit of the planet around its star.

There are three main active space stations, one for each relevant orbital plane. They are filled with rocket propellant for refueling ships traveling between low orbit and the moons. The one that serves both the outer moon and stellar orbits is also closest to one of the planes of the planet's rotation, so it is the most heavily built up, taking the form of an artificial gravity ring.

I asked if it was possible to have an artificial gravity double-rotating duocylinder and she said yes, but it'd be more expensive than a simple rotating ring for not much benefit.

Then I had an idea.

Earth-built rockets are better than 4D rockets. Suppose we reprojected a spacecraft capable of traveling to the other planets? We could use a high framerate 2D video camera and record footage of the planet as it flew by, then 4D scientists would be able to take that data and display it as a 3D image.

The rocket scientist chuckled. "I'll keep that idea in mind," she said.

And why wouldn't NASA or ESA or the CNSA want to embark on such a mission? The 4D people have tetrachromatic vision in a different range of wavelengths than humans have, so even though we've developed software to convert transmitted 3D imagery into 2D+Time footage, any pictures we have of their universe are in false color.

Such a collaboration would amount to a win-win!