The problem of sphere packing in higher dimensions is well known. In summary, the volume that equal-sized spheres (or hyperspheres) can fill inside a cube (or hypercube) decreases with increasing the number of dimensions. This page shows the range and how higher dimensions affect that.

As can be seen, the packing density at 24 dimensions dropped to around 0.0019, which means that if we tried to fill a hypercube with hyperspheres in the 24th dimension, we would only be able to fill ~0.002 of the volume.

Possible Problem

Although the shapes of subatomic particles are not exactly spherical and electrons follow different distribution around the nucleus, a sphere in general seem to be a unique object for which its center is at the same distance from any points at its surface. It is somehow understandable why many approximate the shape of atoms and subatomic particles as sphere, although they might not be exactly so


If we try to envision matter as we know it in higher dimensions using spheres as approximations for particles, it is obvious that this matter will be of much lower density than what we are used to. In even higher dimensions, it will probably be so low that we might even end up with something closer to vacuum that to ordinary matter.


What are some ideas that can be thought of to justify the existence of normal density matter at say the 100th dimension taking into consideration the packing problem?

  • $\begingroup$ Please read the tag description. Hard science and science-based are mutually exclusive $\endgroup$
    – L.Dutch
    Mar 17, 2023 at 11:32
  • $\begingroup$ By far the biggest problem with more than three spatial dimensions is that calculus doesn't work, or, better said, the way it works is very different from what we are familiar with in three dimensions. Which means that your entire physics must be very different, by necessity. In particular, you cannot have stable orbits with more than 3 spatial dimensions, with rather serious consequences for the existence of everything from atoms to planets. $\endgroup$
    – AlexP
    Mar 17, 2023 at 12:48
  • $\begingroup$ @AlexP a similar problem is that you can't have knots in more than three dimensions. Every crossing of strands can be removed by pushing one strand along the fourth dimension to move it around the other. Effects on protein folding (or an equivalent using atoms that could actually exist in four dimensions) would be similarly catastrophic. $\endgroup$ Mar 17, 2023 at 21:55

2 Answers 2


It Affects Everything Equally

In higher dimensions, your hypercube of flour from the hypermarket weighs less than it would in boring old 3-space. But you also weigh less. You are less strong and the hypercube feels just the right weight to you. Also less heavy is the house you live in and the planet you stand on. The difference in density is uniform and affects everything equally.

You only run into problems when you think about gravity. If everything is less dense then gravity is weaker. So planets and stars and their orbits are bigger.

However this is well above the pay grade of the question. In higher dimensions, gravity and the other forces dwindles much faster with distance. You cannot even have stable orbits. So there are bigger problems to solve.

  • $\begingroup$ ... Ignoring the small problem that gravitation cannot possibly work the way it does with three spatial dimensions; which puts the phrase "you weigh less" in serious need of clarification. $\endgroup$
    – AlexP
    Mar 17, 2023 at 12:50
  • 1
    $\begingroup$ @AlexP Yes, ignoring that. See final paragraph. $\endgroup$
    – Daron
    Mar 17, 2023 at 12:58

Option 1: The closeness of things in 3-d is an optical illusion.

The only way for a lower dimensional thing to exist in a world where more dimensions are real is if the things that exist in the lower dimensions are actually part of the higher dimensional reality. This is to say that a hyper sphere does not actually take up less space than a 3-sphere, but that a 3-sphere APPEARS to take up more space than it really does in N-d reality. If your universe has 100 dimensions, then the reality we 3-d beings experience is dictated by the physics of 100-d space, not the other way around. So the assumption then is that physics in N-d space works because they work at that scale, and the real question becomes why do they also work in 3-d space?

Consider the following orbital diagram. In 3-d space, the orbital particle is sometimes closer and sometimes farther away depending on where in the orbit it is, and if you were to animate this, you would see the blue balls moving faster at some points and slower in others to compensate for the rules of 3-d space. You would see all the same physics that apply to 3-d space "warped" to work in 2-d space, and everything would be consistent between 3-d and 2-d.

enter image description here

What this means for a 3-d being in N-d space is that the he will experience a LOT of apparent randomness at the subatomic scale since there are so many ways that N-d space can twist and turn, but at larger scales, everything will average out into laws that seem consistent. This is why string theorists believe our universe is made of more dimensions than we can perceive because quantum physics appears too random compared to macro-physics to be explained otherwise.

So, there is no such thing as a "3-d being" in N-d reality. Instead what you have is a N-d being who only understands its environment in 3-d because all the physics of the N-d reality average out in a way that they become consistent at 3d.

Option 2: Space is simulated and follows different laws of physics in different numbers of dimensions.

A computer is able to simulate more or less dimensions than we actually live in. The computer itself is a 3-d construct, but can simulate 2-d space, 3-d space, or even hyper dimensional space. It does not care because a program can follow different physics based on how many dimensions you are working with. Because of this, a game like Super Mario 3 can have internally consistent 2-d physics, and Super Mario 64 can have internally consistent physics 3-d physics, and they both work more or less the same despite the extra dimension... but you can not just load a character file from Super Mario 3 into Super Mario 64 because they are not compatible data models. A Super Mario 3 character does not just have a Z-axis of zero, there is literally no Z-axis at all; so, the physics of Super Mario 64 literally do not and can not work with it. This means a 2-d simulated being can not visit 3-d simulated space, and vise versa; so, there is not point in worrying about the irregularities in the physics models. At best you can make a facsimile of your 2-d being, but that facsimile will have "made up" data to fill in the data points that don't exist in the 3-d simulation, and a lot of the data you do have on the 2-d being will need to be replaced with data that is more rational to the new world.

As your 3-d simulation can use very different coefficients, variables, and constants to achieve similar outcomes as a 2-d simulation. Likewise, an N-d simulation can adjust its coefficients, variables, and constants to achieve similar outcomes to 3-d space despite the ways that it is totally different from our 3-d space.


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