# Q: What would life be like in higher dimensions?

The original question was: Assuming we had four (or more) spatial dimensions in which to freely move around, like say a 4+1 dimensional universe, how might one extend our 3+1 dimensional physics to that universe?

Side note: When someone says “3+1 dimensions”, what they mean is “3 regular space dimensions, and one time dimension” which is exactly the situation we live in (apologies to our pan-dimensional readers).

Physicist: Right off the bat, more dimensions means more freedom of movement.  One of the more mundane effects of that is that in 4 dimensional space there’s an extra direction you can move and/or fall over in.  So if you want to build a working bar stool you’d need at least 4 legs instead of just 3.  In fact, in D-dimensional space bar stools need at least D legs, or they’ll fall over.  Just one of the subtle economic effects of higher dimensional living.

You’d also find that in 4 or more dimensions, you’d be able to do a lot of tricks impossible in 3 dimensions, like creating Klein bottles or (equivalently) taping the edges of two Möbius strips together.  Sailing knots could take on stunning complexities.  In fact, they’d need too!  All of the knots that work in 3 dimensions fall apart immediately in 4.

In four dimensions you could make this surface without worrying about it intersecting itself.

Most physical laws are already written in a dimension-free form.  For example, in Newton’s second law, $\vec{F}=M\vec{A}$, $\vec{F}$ and $\vec{A}$ are both vectors, but they can be vectors in any number of dimensions.  So you can use $\vec{F}=M\vec{A}$ for objects on a line (1-D), on a table-top (2-D), in space (3-D), or whatever (whatever-D).

There are some laws that are usually written in a 3-D form, but that’s generally a matter of convenience more than necessity.  For example, we talk about the “angular momentum vector”, which is defined to be perpendicular to the plane of rotation.  It’s convenient because in three dimensions there’s always exactly one perpendicular direction to a plane, whereas in 4 dimensions (for example) there are 2.

In 3-D we can formulate laws about spinning things in terms of the one direction that isn’t spinning (h), the “axis of rotation”. But we can always formulate laws in terms of the two directions that are spinning, regardless of dimension.

This is pretty easy to fix and generalize, it just becomes a little more difficult to work with.  All that said, while our physical laws can be generalized to any number of dimensions, the manifestation of those laws are wildly different.  So, living in higher dimensions would be pretty alien.

Based on our understanding of gravity (gained from studying this podunk universe), gravitational force should drop by $\frac{1}{R^{D-1}}$, where D is the dimension and R is the distance between the objects in question.  It so happens that because of the nature of orbits, a stable orbit can only exist in 2 or 3 dimensions.

The “effective potential” representing the balance between the gravity and centrifugal forces of an orbiting object.  Orbits can be stable in 2 and 3 dimensions. In all other dimensions planets and moons will always either spiral in or fly away.  Shown here is the potential energy from gravity and the centrifugal force combined.  If there’s a “cup” you can form a “bound orbit” in it.

In 4 or more dimensions orbits are always unstable, and in 1 dimension the idea of an orbit doesn’t even make sense.

Most physicists consider light to be native to only 3 dimensions, because light is an EM wave and it’s direction of propagation is perpendicular to both its Electric and Magnetic fields.  (Fun fact: the direction that light points is called the “Poynting vector“, after John Henry Poynting.  Life’s funny.)  In 4 or more dimensions this direction isn’t unique, and in two dimensions there’s no direction at all.  However, you can express EM waves just in terms of “E” in any dimension without problem.

Assuming light can exist in higher dimensions, it would behave very strangely.  Sound waves too.  In odd dimensions other than 1 (3, 5, 7, …) waves behave the way we normally see and hear things: a wave is formed, it moves out, and it keeps going.  However, in even dimensions, and 1 as well, (1, 2, 4, 6, …) waves “double back” on themselves.  You can see this in ripples on the surface of water (2-D waves).  Ripples are more complex than just a ring; the entire circle within the ripples is disturbed.

In even dimensions (like the 2-D surface of water), waves propagate in a more complex way than we’re used to.  Instead of a simple pulse, you get an “area filling” wave.

If you set off a firecracker in 3, 5, 7, etc. dimensions, then you’ll see and hear the explosion for a moment, and that’s it.  If you set of a firecracker in 4, 6, 8, etc. dimensions, then you’ll see and hear the explosion intensely for a moment, but will continue to see and hear it for a while.  For light the effect would be fairly subtle, except for extremely long-distance effects, like somebody reflecting a bright light off of the moon.  You probably wouldn’t notice the effect day-to-day.  However, it would ruin the experience of sound.  In 4 dimensional space the firecracker, even in open air, would sound like thunder; loud at first, and leading into a drawn out boom.  It may not even be possible to understand people when they speak.

All the fundamental particles should still exist, but how they interact would be pretty different.  Which elements are stable, and the nature of chemical bonds between them, would be completely rearranged.  Some things would stay the same, like electrons would still have two spins (up or down).  But atomic orbitals, which are determined by spherical harmonics (which in turn are more complicated in higher dimensions), would generally be able to hold more electrons.  As just one example (for our chemistry-nerd readers), you’ll always have 1 S orbital in every energy level, but in 4 dimensions you’ll have 4 P orbitals in each energy level, instead of the paltry 3 that we’re used to.  This messes up a lot of things.  For example, in 4 dimensions Magnesium would be a noble gas instead of a metal.  Every element after helium would adopt weird new properties, and the periodic table would be longer left-right and shorter up-down.

So, while the laws of physics are actually the same, if you lived on a four-dimensional Earth in a four-dimensional universe you’d find that (among other things): your bar stool may need an extra leg, Earth wouldn’t be able to orbit anything, you’d never be able to hear anything crisply, and the periodic table of the elements would be seriously rearranged.

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### 48 Responses to Q: What would life be like in higher dimensions?

1. Andrew says:

GREG EGAN DIASPORA SPOILER!

Ever read Greg Egan’s Diaspora? I’m wondering how accurate his depiction of a 5 + 1 universe was. He mentioned that there would be no atoms, since no orbits are stable, including those of the electron around the nucleus.

2. steve says:

What about string theory with all of its extra dimensions? Do string theorists have a way to account for all the weird stuff that doesn’t happen but should?

3. The Physicist says:

The extra dimensions of string theory are much smaller than the smallest particles. So, nothing has “room” to move in the other directions. It’s a little like being stuck in a pipe that’s as big as you are: there may be other dimensions, but you can only move back and forth in one direction.

4. Gaurav Bhattacharjee says:

You didn’t mention one thing : the effect of gravity.
The more the dimensions, the more the force of gravity changes with respect to time. This means that if two or more subatomic particles collide, they will form, so called, micro black holes because of the increase in the strength of gravity . So, there will be no electrons, protons, neutrons nor atoms. Only photons and neutrinos will be present.
So, there will be no life in higher dimensional spacetime, as there will be no atoms to clump and form matter.

5. Gaurav Bhattacharjee says:

Even if you are ignoring the increased strength of the gravitational interaction between elementary and subatomic particles, life in higher dimensions would not be possible. Planets won’t exist, because they will be swallowed in by the star around which they are orbiting due to the increased strength of gravity.

6. The Physicist says:

It’s not quite as straight forward as gravity just being stronger. In higher dimensions gravity increases more sharply at small scales (even in 3 dimensions it increases very sharply), but that doesn’t immediately mean that atoms would collapse into black holes. In fact, since the units are different you can’t really draw any conclusions about the gravitational constant in 4-D given the gravitational constant in 3-D. That is, in 3-D, the units of G are m3/kg/s2. In 4-D it’s m4/kg/s2.

7. Gaurav Bhattacharjee says:

Okay. Thanks

8. Jason Goodman says:

I think we need to worry about the existence of atoms and molecules. Electrons are bound to nuclei because the electric force permits stable orbits, just like gravity. (There’s quantum weirdness here, but it doesn’t change the basic picture.) In higher dimensions, you’ve shown that gravitational orbits could not exist: electrical orbits might not either. To work this out properly, you’d have to re-derive quantum electrodynamics in extra dimensions, which I can’t even begin to do, but if you naively assume that gravity and electricity always obey the same inverse-power law, atoms and molecules could not exist above N=3.

So to heck with how many legs a stool would need: you might not be able to hold a stool together in the first place.

Because the strong nuclear force has a very different dependence on distance, I’d guess it doesn’t suffer from this problem. So atomic nuclei may exist, drifting freely in space, surrounded by a sea of unbound electrons. Not a nice universe.

9. Will says:

Be a good place to harvest raw materials though.

10. Locutus says:

I know the term “3+1” is used to mean 3d plus time, but isn’t time a “space” of its own? In other words, what is the difference between 3+1 and 4d, because wouldn’t 4d include time?

Love Long and Prosper

11. theSpleen says:

Could you perhaps explain the thing about even-D (and 1-D) waves being much different from odd-D waves? It seems interesting, but I really don’t understand it…

Also, I would be very interested to know some of the math behind why higher-dimensional orbits can’t exist!

12. The Physicist says:

Time is similar to space, but pretty different. There’s a post here that’s far more specific.

13. Constructing a theory of electromagnetism in four dimensions of space will be difficult. Hell, the only higher dimension where it might be possible, excluding three, is seven. This has to do with the mathematical properties of rotation.

That is, only the two-dimensional, three-dimensional, and seven-dimensional space has a rotation described by a “division algebra” — the complex field, the quaternions, and the octonians, respectively.

I suppose, however, that one way to construct electromagnetism in “N”-dimensions is by a well-known construction of the magnetic field which seems to be dimensionally-invariant (not the Kaluza-Klein construction). That is, to assume the four-dimensional Coulomb’s law and Lorenz invariance. I am not sure where this would leave, but I can guess that the resultant equations would not resemble the canonical Maxwell’s equations in any sense.

14. StringTheoryDropout says:

Octonions are very strange things: quaternion multiplication isn’t commutative, but octonion multiplication isn’t even associative!

It seems to me the better path is the gauge theory approach. That is, in however many dimensions you want, local gauge invariance of charge conservation implies the existence of a magnetic vector potential (conventionally named A). Note that for all they may have told you in E&M that A wasn’t real, it was just a notional thing whose curl was B, that’s actually wrong: the Aharonov-Bohm effect offers experimental proof that what electrons actually feel is A, not B (in accord with their classical Hamiltonian, as used in nonrelativistic quantum mechanics).

Then the field strength tensor (F, which is the proper Lorentz-invariant expression of both E and B together) is dA (the exterior derivative — div, grad and curl all at once), and all of Maxwell’s four equations at once are written d*F = 4pi *J (those stars are Hodge dual operators, not multiplication) for charge density (source term) J. There seems little reason that couldn’t generalize directly to any dimension you like (well, at least 1+1). F wedge F would no longer have enough rank to be an energy density, but supergravity folks often resort to things that look like F wedge F wedge F wedge A, which would be an energy density for 6+1 dimensions…

15. There is a book called Flatland by Edwin A. Abbott that considers how life would be different with more or fewer dimensions. It’s old, but still good.

16. szefunio says:

> electrons would still have two spins (up or down)
That is questionable. In 4D objects will rotate around two axises simultaneously (planets would have 4 geographical poles for example, geographical coordinates would consist of 3 numbers) and electron probably will have 2 spins simultaneously, so 4 different values of spin will be possible (1/2, 1/2), (1/2, -1/2), (-1/2, 1/2), (-1/2, -1/2).

17. Alyster says:

I don’t know if this should be formalized as its own question, but can you please explain how the extra dimensions in string theory are “smaller”? The way I conceptualized extra dimensions is through the linear algebra approach of n-spaces, which from my understanding do not have limits. Does this mean that there are “bounds” to the extra dimensions in string theory?

What if 4+1 (or more) dimensions discredited the egotistical 3+1 dimensional laws of physics? The term three-dimensional is also representative to “Space, Time, and Matter” rather than just relating to spatial form? What if, while in our three-dimensional bodies all we might explain is theories regarding a spatial fourth dimension? What if transcendent dimensions existed within this universe where physical form or linear time is irrelevant? Suppose, maybe we know nothing of how things work in other dimensions; otherwise, wouldn’t we be experiencing these dimensions rather than only theorizing about them? Assume, in another universe, that our laws of physics even apply. Is this possible? Or, does our earthly laws-of-physics embody all authority over all unknown? Assume, a physicist simply said “I don’t know”; then, might have we transcended into another dimension?

19. 4D in 2014 says:
20. DC Ambrose says:

What the philosopher says makes sense from a relativistic point of view, if you think about it. What would the effect of antigravity and antimatter be in relation to higher dimensions of reality?

21. slick rick says:

i do agree with the philosopher, just because we can’t measure another dimension, doesn’t mean it doesn’t exist or the possibility of it existing is zero. Anything is possible there could be an infinite number of dimensions but the question is can matter exist in any higher dimensions is a whole different ball game? what if the black hole exists in some different dimension after all it does leave a messed up patch of geometry in space and talking bout space what are the dimensions of space?

22. Anders says:

Why is it that waves ripple back in on themselves in an even number of dimensions?

23. Heather says:

I am confused. This article describes things as they “would be” in a universe of higher dimensions…… But I thought many physics theories say we do in fact live in a universe of many additional dimensions.

So why aren’t we seeing these or even more strange outcomes?

24. The Physicist says:
25. Heather says:

Do we ‘see’ the universe through ‘glasses’ based on the number of dimensions we exist in. So while all the effects that were described are actually occurring, but our perception of them is limited by the dimensions that we can experience.

I guess then the question is are we in a universe of many more dimensions, but we ‘exist’ in only 4 of them. I am not sure how you can be in a universe of many higher dimensions but then also not in it.

26. Anders says:

One of the things you didn’t mention is that in four spatial dimensions it’s possible for things to have two independent directions of spin. In fact in any odd number of dimensions the number of possible independent ways of spin is (n-1)/2 and the number of independent ways something can spin in an even number of spatial dimensions is n/2.

27. swapan bose says:

i have q. that if a another dimmention is possible is there all the law we know is work there,is gravity will higher or lower.

28. stu says:

What would beings look like in the 4th D.

29. fictioner says:

I’ve been hashing over a scifi/fantasy story setting involving 4D space. I’m struggling with implications of the unraveling knot in 4D. It seems to follow that a 3D body would or could similarly unravel. Can a 3D being survive in 4D space?

30. fictionwriter says:

I’ve been working on a fantasy story with many different worlds/universes existing simultaneously. Now, to my story it is rather essential that in some of said universes there is an additional “dimension”. Since I’m rather poor at maths and physics, the possibilities of higher dimensions don’t explain themselves so easily to me, but from what I understand the dkmensions we live in (3) describe directions, so all higher dimensions would be additional directions? And the time (+1) is similar to direction, but also very different. So, technically, three directions+time, right?
So, while everything (gravity, orbit, etc) would change as described in this article in higher dimensions (4,5,6,…), is it clear how it would change if we added something similar like time (+2,+3,+4…)? Wouldn’t most remain the same but for something abstrat like time added (for the purpose of my book say something as abstract as magic) in a 3+2 dimensional world?

I just wanted to note regarding the sound issue and not hearing things…this would be assuming that 4D beings are only limited to the senses experienced in 3D…this ties in with “telepathy” and “telelkinesis” and other such paranormal things…the ability to touch, move, and experience things from senses in 4D instead of 3D

Its like how a 2D being wouldnt be able to experience colour for example, because it would have had no evolutionary purpose to gain such abilities as in 2D all you see are a bunch of vertical line segments without any real way of ever knowing what is in front or behind you (no beside in 2D)….unless using the same logic in the first paragraph that 2D beings have different senses they have evolved to be able to “know” what is around them…etc etc

I also wanted to share one of my theories….call it crazy if you want, and please leave feed back so i can read your thoughts……….

Here in 3D we really only “see” things in 2D technically, we experience the 2D frames of our reality, but have evolved to learn to see the shadows and lighting of these “frames” to create the illusion of depth, but more so what we are really experiencing is the motion between the movement of 2D frames of 3D objects containing within its geometric perfections some sort of sentient life forces within it….studying the movement of 3D objects within some higher space — or the rate of change of space — will create some 4D space-time manifold containing within it a static “screenshot” of all 3D experiences which has been observed yet, with some dark anti-matter containing some dimension containing all possible values of existence (infinity or negative infinity or zero) (think of quantum physics) which will be brought into existence once it has been “experienced” by us 3D beings….just as our knowledge of the world around us is slowly being pieced together bit by bit like a jigsaw puzzle….

anyways im sure this sounds crazy, or theres some huge flaw im missing using physics or math….but i wanted to share, so leave constructive feedback that i can read and learn your viewpoints too

33. Carrie says:

This is a really interesting article. I got interested after reading Flatland by Edwin Abbott Abbott (really good book, the movie is on youtube)
now although, I am sure there are some things that are correct…I sort of feel like in the comments you guys are overreaching. The reality is- None of us have been in the 4th dimension and since there are such extreme changes…I feel it is one of those things that we simply can not know until we are actually there. I dont know if that is possible or if it ever will be but just like the book, those in the 2D were limited in their mathematics and science because of the reality they are in. I think we may suffer the same bias.
If there can be a 4th dimensions with such extreme changes…I think it may call for new forms of math and new laws of science. Not complete new rules

34. tristan says:

A while ago my friend did this drug DMT and he said he saw a bright light and shot off into it and went through a tunnel and he saw his past and future and he said he woke up on a planet with little alians staring at him with faces that said “what are you and where did you come from” could alians just live in different dimension’s?

35. Jim says:

To the guy who said 2D creatures would not know infront and behind…. I don’t believe that is true. In 2D there is depth, just not height.

You can easily imagine living in a 2D world. It would be like seeing everything through a slit where there is no difference in height anywhere. Lines have immense value in a 3D world. In a 2d world points have immense value. An open door in 2D would be an area (line) in your field of “vision” where there is a depth cue (depth “haze” or similar). You could only see objects from a maximum of 2 sides. (In this case joined horizontal lines, tapering away in depth to varying degrees. A computer screen in a 2D world would just be a smaller horizontal line in your field of vision with horizontal lines and dots showing various degrees of depth haze.

All in all I think it is a little easier to imaging a 4D world for us than it is for a 2D object to imagine 3D. 4D simply has an extra in out dimension. Where 4D gets confusing for me is that the basics of 3D geometry stop being familiar. In 3D a flat wall at the end of a corridor is a simple 2D plane (wall). In 4D a corridor ends in not a plane (wall) but in a cube. It ends in 3D. I have a hard time imagining the geometrical environment of a 4D world, movement within that world is learnable with a little practice.

The brilliant (and free) little 4D maze game helps alot in learning to traverse a 4D environment.

36. Jim says:

Edit: I say “horizontal” but in a 2D world that is obviously arbitrary.

37. Shojin Bateman says:

I would hypothesis that we already live in 4-space dimensions. That our space is really hyperspace. Our observations of electromagnetic fields for instance, in 3-space dimensions the lines appear to be unequal length. However, if we put the north pole in the center of one cube and the south pole in the center of the polar opposite cube in our hypercube, would not the longitubes all be of equal length much like the longitudes between opposite poles are equal length?
Also, I would argue that 4-space dimensions explain why electrons appear as a cloud. I hypothesis that electron(s) do not ‘orbit’ a proton so much as ‘vibrate’ to stay at the polar opposite of the proton(s) hypersphere of influence.
Also, would not a 4-space dimension wave form appear as a sphere when observed from a 3-space dimension perspective? For instance a photon.
Also, does this not explain why the CMB appears as a sphere around us rather than as a point?
Also, our universe existing in hyperspace may help to explain why space is not bound to c for max acceleration.
There are more hypotheses, but this should be enough to contemplate for now.

38. Andy Russell says:

A simple question from a layman – assuming that higher (larger) dimensions exist and that some day we might be able to access them, could they be used to bridge the massive distances (and light speed restrictions) to other stars and galaxies. If Distance is one of the 3 dimensions we perceive, which makes things ‘distant’ then might we be able to access other dimensions where a shortcut is possible?

39. Ok, 3 dimensional objects have weight. Even if they’re tiny. 2 dimensional objects have absolutely no weight. Hear me out, paper IS indeed 3 dimensional, and that does have weight. So would 4 dimensional objects weigh infinitely more than 3d objects?

40. Rajesh Swarnkar says:

” All of the knots that work in 3 dimensions fall apart immediately in 4.”

So basically the Klein bottle is a 3D KNOT NOTATION used by a 4D Creature. Hilarious !!

41. Rajesh Swarnkar says:

“Most physical laws are already written in a dimension-free form. For example, in Newton’s second law, F=M.A; F,A are both vectors, but they can be vectors in any number of dimensions. So you can use F=M.A for objects on a line (1-D), on a table-top (2-D), in space (3-D), or whatever (whatever-D).”

I guess 2D object Cannot have mass. Because for an object to be PERFECTLY 2D will not have 3rd Dimension and will not have mass (I am talking about in common sense, For time being ignore the fact that mass arises from Higgs Interaction and blah blah, which I still find fishy.) even if we align a single layer of atom which is 3D, isn’t it?

42. Stefan says:

I just got struck by this curiosity…

Seeing that, as far as we know, in our “3+1” universe it took almost 14 billions of years for all the right “variables” to fall into place in order for self-aware entities to arise (humans; again, as far as we know), would a 4+1 and above universe take longer than that, since the number of variables is exponentially higher? I would guess the more dimensions there are in a universe, the longer it takes for it to “develop” an intelligent entity of it’s own (that is, an entity that would exist in all those dimensions simultaneously, just like humans do in their 3+1 universe).

43. Anders Gustafson says:

Would the radius of a hydrogen atom be the same in higher dimensions or would that change? Would the temperature required for nuclear fusion be the same in higher dimensions?

44. Eli says:

I was wondering because planes are made of lines and cubes are made of planes, are tesseracts made of cubes? If that rule applies then that means that we are all apart of a 4th dimensional object floating throughout time