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4D planetary mechanics

If one wishes to construct a four-dimensional universe with similar^{[dubious–discuss]} mechanics to ours, planets are a necessary addition. Unfortunately, however, not only do they have a few problems, but they are also quite difficult to understand. This page explains how 4D planets work and explains away said issues.

Clarification

To clarify, "four-dimensional" here means a space with four mutually perpendicular axes, all being isotropic (cannot be distinguished). This is about the same definition for 3D and below. Obviously, this type of space does not exist in our universe, and no one has ever claimed that there is. After all, if it did in fact exist this page wouldn't be necessary. And no, Einstein has never said that 4D is time, and rather he said that spacetime can be described as a four-dimensional system. If you still insist 4D space is illogical then you are completely wrong and are willfully ignorant of over a century of professionally done established geometry.

With that out the way, let's begin.

Background

Four-dimensional planets would be physically "similar" to ours. In any dimension, they would be large clumps of matter, not enough to be a star, but not small enough to be considered an asteroid. Just like in our universe, large enough clumps of matter would, due to gravitational forces, shape the mass into a (hyper)spherical shape. In 4D, this shape is called a glome (mathematically, 3-sphere), which has a three-dimensional ground. In general, surfaces in any dimension are of the lower dimension. Surfaces in 2D are 1D, surfaces in 3D are 2D, surfaces in 4D are 3D, et cetera.

While their sole existence in a 4D universe isn't problematic, there are several issues with their formation. For one, extrapolating from 3D physics makes gravity inversely proportional to the cube of the distance ($g\propto r^{-3}$) instead of to the square of the distance ($g\propto r^{-2}$) as it is in our physics. This makes orbits completely impossible with any exponent ≤−3, as they will eventually pass through the orbiting body. Let's just assume everything follows the same inverse square law. Sure, it creates an even stronger version of Olber's paradox but with gravity instead of starlight, however it at least makes orbits possible.

Another major problem is that flat systems such as galaxies and stellar systems are only possible due to one property of odd dimensions: there is one free non-rotational axis. Any rotation in 3D can be described by some plane, which is 2D. This leaves a singular axis from which material can collapse into the plane of rotation.

In even dimensions however, there is no such free non-rotational axis. In 4D, there are two such rotational planes (such rotation being called a double rotation), meaning that material can never collapse into a 3D plane of rotation. This is corroborated by simulations of 4D dust clouds ran by Tessimal (simulation link), which show that spinning clouds of dust in 4D never flatten into anything and remain fully 4D clouds.

To explain away this problem, we'll assume that double rotating clouds will "flatten" into bounded zero-radius dihyperbolae, a shape produced by the equation $x^2+y^2-m^2{z}^2-m^2{w}^2=0\cap x^2+y^2+m^2{z}^2+m^2{w}^2=R^2$ , where $m$ represents the ratio between the double rotations and $R$ represents the system radius (Desmos 3D graph). It is particularly interesting as its radial sections are Clifford tori which bodies orbit on.

Double rotations

Double rotations are a confusing thing. As they are a novel phaenomenon in dimensions above 3D, they are quite hard for "4D beginners" to understand. A (hopefully) simple way I've found to understand them is like so: in 2D, objects rotate around a point; in 3D, objects rotate around an axis, so it follows that in 4D, objects would rotate around a plane, which itself can be rotated.

For those of you who like to dabble with non-Euclidean spaces, they can be visualized quite elegantly in S³: a plane of rotation can be imagined as every point within a plane that intersects a glome, which yields a great circle. It turns out that the points furthest away from some great circle form another great circle. It also turns out that rotating everything in the direction of one great circle doesn't affect the other. As rotation in the direction of either of these great circles are independent, it is possible to rotate them both independently.

Interestingly, double rotations are chiral: their mirror images are distinguishable, albeit only in 4D. Looking in the direction of one of the rotations, the other could be rotating either clockwise, called a dexter double rotation, or counterclockwise, called a sinister double rotation.

For some terminology, rotations of only one plane are called simple and double rotations where revolution rates are equal are called isoclinic.

Rings & cardinal directions

Diagram of the ring system and cardinal directions.

Since planets are objects,^{[citation needed]} they can also double rotate. And just as 3D planets with no rotation at all are practically nonexistent, 4D planets with simple rotation would also be practically nonexistent.

As outlined in the previous section, the planes of double rotations can be imagined as two opposite great circles ("rings") in S³, the geometry of a glome. Despite being called "circles", great circles do not actually appear to curve from the ground, just like Earth's equator which is also a great circle. As planets would generally be aligned to their star, one of these rings (specifically the one most aligned with the ecliptic plane) would be by far the most sunny. Let's call it the tropical ring, its opposite being the polar ring, both lying 90° from each other. A planet whose tropical ring spins faster than its polar ring is called tachyotropical and one whose polar ring spins faster than the tropical ring is called tachyopolar.

Distance from the tropical ring is called artitude. Despite being more commonly called latitude, it is completely different: artitude spans 90° and is unsigned (can only be positive), while latitude spans 180° and can be positive or negative. Planes of constant artitude form Clifford tori, and at 45° the shape is a regular^{(is this its actual name?)} Clifford torus. The other two glomeral axes are longitudinal, both looping in 360°. Distance along the rotation of the tropical ring is called longitude and distance along the rotation of the polar ring is called colongitude. Planes of constant longitude or colongitude form hemispheres.

Alright, let's get to naming the cardinal directions. The direction toward the polar ring is called kirn, its opposite being thern, although they are more commonly called north and south, same reason why I don't like latitude for artitude. These names are derived respectively from the Greek words for snow and warmth, χιών και θερμός (khiṓn kai thermós). For longitude, the direction toward the rotation of the tropical ring is called east, its opposite being west. For colongitude, the direction toward the rotation of the polar ring is called anst, its opposite being dyst. These names are derived respectively from the Greek words for east and west, ἀνατολή και δυσμή (anatolḗ kai dysmḗ), with the -st suffix indicating longitudinalness. Both longitude and colongitude loop in circles, so they can be expressed in the same way longitudes are normally written. Just as east and west have singularities at Earth's poles, east and west also have singularities in the polar ring, and anst and dyst also have singularities in the tropical ring.

Personal directions

Personal directions mapped in a 3D field of vision.

As a side tangent, on Earth it is possible to tell cardinal directions from which cardinal you're facing: facing north, east would be to your right, south to your back and west to your left. To understand how these work in 4D, we'll need to get familiar with personal directions. In 3D, there are six directions: up, down, front, back, left and right. These last two form the basis for the multidimensional family of orthogonal ground directions, being the horizonal directions orthogonal to front/back. In 4D, orthogonal ground directions form a circle instead of two points, and splitting this circle into four perpendicular sections creates the four orthogonal ground directions. Looking from up in clockwise order, they are: on, right, gain and left.

4D compass, showing both dexter and sinister variants. Of course, 4D beings would use only one of these for their specific planet's type.

Getting back to what was originally being discussed, that way of telling is quite more complex due to there being more ways to orient yourself relative to the planet. Due to this, it is way easier to simply memorize how the compass looks like. An example of what that way of telling would be on a dexter planet is as follows: facing kirn with anst being to your on, thern would be to your back, dyst to your gain, east to your right and west to your left, east and west switching in sinister planets. Here's a slightly more advanced example in a sinister planet: if you were facing west with kirn being to your left, east would be to your back, thern to your right, anst to your on and dyst to your gain, anst and dyst switching in dexter planets.

Small additional notes

Interestingly, hemispheres are completely different in a 4D planet. While a naïve hemisphere could be built cutting a glome in half by a 3D plane, forming a shape called the semiglome, a much more useful application of a hemisphere relating to double rotations can be built by cutting a glome with a zero-radius dihyperbola. This results in a shape bounded by a solid Clifford torus and a strange ring pyramid-like structure, called a rupiglome (Desmos 3D graph), which is a wordplay on cliff (Latin: rūpēs) from Clifford torus.

Despite what the double rotation might make it seem, time systems wouldn't be all that difficult. After all, it's quite likely that planets would be generally aligned with the ecliptic, and keeping track of the polar rotation wouldn't be all that useful for keeping track of time as it only affects the position of the star along an east/west-facing circle of fixed distance from the horizon.

Due to the centrifugal effects of Earth's rotation, the equator is slightly farther from the center, making Earth an oblate spheroid. In 4D, the same effect occurs with two axes, making planets that rotate fast enough bioblate glomeroids. If the planet rotates isoclinically, there is no oblation and will simply expand while remaining glomeral. In general, for any dimension the rotational planes of planets will expand proportional to the plane's rotational velocity.

Coordinate system

Navigation in 4D worlds would be just as prevalent as in 3D worlds. For that purpose, glomeral coordinates must be devised for the most natural result. A naïve set of coordinates can be made extrapolating usual 3D coordinates, using longitude $\lambda$, latitude $\phi$ and an axis joining two apices on either end of the glome $\chi$. The formulae for converting between this set and spatial coordinates is as follows: $\begin{array}{l}x=\cos \lambda \cos \phi \cos \chi \\ y=\sin \lambda \cos \phi \cos \chi \\ z=\sin \phi \cos \chi \\ w=\sin \chi \end{array}$ However, if you've understood the last section correctly, it should be quite clear that this is not the most natural set of glomeral coordinates (albeit it is still valid). After all, double rotations don't have pointlike apices, they have circular apices. These more natural coordinates that make use of these properties are called Hopf coordinates, using artitude $\chi$, longitude $\lambda$ and colongitude $\mu$. The formulae for converting between Hopf coordinates and spatial coordinates is as follows: $\begin{array}{l}x=\cos \lambda \cos \chi \\ y=\sin \lambda \cos \chi \\ z=\cos \mu \sin \chi \\ w=\sin \mu \sin \chi \end{array}$ Despite seeming complicated, in actual usage Hopf coordinates would be written about as simply as we write normal coordinates, exempli gratia, 30° K 80° E 110° D indicates a point of 30° artitude, 80° longitude and −110° colongitude.

This coordinate set does, however, make some formulae more complex. For example, the surface speed on a 3D planet is given by $v=r\omega \cos \phi$ , $r$ being the planet's radius, $\omega$ the rotational velocity and $\phi$ the latitude. On a 4D planet however, this becomes $v=r\sqrt{{\omega }_1^2{\cos }^2\chi +{\omega }_2^2{\sin }^2\chi }$ , $r$ being the planet's radius, ${\omega }_1$ the tropical rotational velocity, ${\omega }_2$ the polar rotational velocity and $\chi$ the artitude.

In higher dimensions

Five dimensions

Not a whole lot is mechnically different for five-dimensional planets, which are shaped like phennia (singular phennion), 4-spheres with a 4D surface. As it's still only possible to double rotate in 5D, the two-ring system remains for the equator, being extruded with latitude, with the same two cardinal directions, north and south. Due to this, a "normal" 5D hemisphere is back to being a phennion cut by a 4D plane, called a hemiphennion. Interestingly, the ecliptic would likely still be aligned with the tropical ring, resulting in the polar ring, which is in the equator to be colder. Even when they're not aligned, this phaenomenon happens with a whole (hollow) great sphere for any planet.

As double rotations in odd dimensions are not chiral, there is no sinister/dexter for 5D planets, as the image of the planet could be seen spinning either sinister or dexter depending if you're observing from the north or south apex, respectively. This is similar to how the Earth can be seen spinning either clockwise or counterclockwise from the north or south pole, respectively.

Due to there being a free non-rotational axis, the whole cloud problem is solved, and planets can orbit in a glomeral plane. The orbits crashing into the Sun with gravity being inversely proportional to the quartic of distance problem is still not solved, though.

The formulae for converting phennical coordinates to spacial coordinates, also given latitude $\phi$, are: $\begin{array}{l}x=\cos \lambda \cos \chi \cos \phi \\ y=\sin \lambda \cos \chi \cos \phi \\ z=\cos \mu \sin \chi \cos \phi \\ w=\sin \mu \sin \chi \cos \phi \\ v=\sin \phi \end{array}$

Six dimensions

Things get much more interesting in six-dimensional space. As there are now three rotational planes, triple rotations become possible, and therefore planets begin to have three rings, one tropical and two polar. Just like how double rotations are chiral in 4D, triple rotations are also chiral in 6D, and in fact in any even dimension 2n, n-rotations are chiral. Planets are shaped like 5-spheres (haven't given them a proper name) with a 5D surface. I believe 6D planets have three longitudinal and two artitudinal axes, although this could be wrong as the high dimensionality makes this difficult to think about. In general, an nD planet should have ⌊n/2⌋ longitudes, ⌊n/2-1⌋ artitudes and one latitude only in odd dimensions. The formulae for converting 5-spherical coordinates to spacial coordinates, given third longitude $\nu$ and second artitude (coartitude?) $\psi$, are: $\begin{array}{l}x=\cos \lambda \cos \chi \\ y=\sin \lambda \cos \chi \\ z=\cos \mu \cos \psi \sin \chi \\ w=\sin \mu \cos \psi \sin \chi \\ v=\cos \nu \sin \psi \sin \chi \\ u=\sin \nu \sin \psi \sin \chi \\ \end{array}$

Conclusion

In conclusion, thinking about how mundane phaenomena would work in higher dimensions is an excellent way to develop higher dimensional intuition, and is also quite fun for those who are mathematically inclined. In fact, even though I've been researching 4D planetary mechanics since I got interested in 4D during July 2024, I only made some specific realizations about planetary mechanics when writing this page in late May 2025.

Anyways, hope you enjoyed reading this and hopefully you learned at least something from this! Might make a video reading this out.