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

  1. Anon says:

    Ok, this is a dumb question, but the dimension number does NOT effect the magnitude of a force vector or the total energy right? As in 10 J in 1-D is the same as 10J in 3-D which is the same as 10J in 10-D in terms of magnitude?

  2. The Physicist The Physicist says:

    @Anon
    That’s a great question! Kinetic energy shouldn’t have anything to do with the dimension of the space.

  3. Anon says:

    Thanks a lot
    there are a few people i know that just don’t listen to this distinction
    they think a higher dimension immediately = infinitely more energy

    oh also, mass too is a dimensionless quantity right?

  4. The Physicist The Physicist says:

    @Anon
    Maybe? We’d need to look at some 2 or 4 dimensional matter and see how it behaves. My guess is that it’s a dimensionless quantity.

  5. damian says:

    Well the important difference is dispersion, advection. transport. because in 2d radiation from a point result in 1/dist. using the law of triangles, and particle model, foe example, 10j will do more damage. Or transmit more information, as in music , loud, and, reverb, and echo, since it lingers. if gravity is 1/dist not 1 /dist sq, orbits can be stable, but not in 4 d. So, it is the same but scales differently with other qualities such as distance..unless theres is some string theory to change this, it is a different universe with flat molecules..its first have to have our laws exactly. I only think life is possible in 2d space and 3d. 4d. Seems too far from the sweet spot.. Also doesn’t help me understand 3d of space and time.

  6. Nalyd Srednas says:

    If matter took on strange new properties then who knows what life can be made of. For all we know a 4d universe’s life could have gold or uranium as a base instead of carbon.

  7. Damian says:

    in our world, most of the heavy metals are formed from supernova.. then they end up in the cores of planets when they are cooling down.. ( gold is heavier than iron)… so carbon, is common and has many connectors, so its used by life. After reading the Planiverse and working on the 2d simulator, i’m convinced 2d life can and maybe does exits, although we cannot interact with it. The mass is flat. There are no physical laws that would govern interaction between 3d matter and 2d matter. ( someone its I watched this video, i became more skeptical on 4d space creatures, because, there are more things that have to go right. https://www.youtube.com/watch?v=vDmMnSKEYnIv=vDmMnSKEYnI Also I think 2d creatures can look cool, or weird, but how a 4 creature looks i cant imaging at all.. I guess i go back to 2d mans shoes and try to imaging him thinking about what we look like, he could certainly have a set of set points, or a surface for our face, wih depth info..To visualize it i guess he would slide it along the symmetry where one eye is above the other. That is wrong he would soon turn it and see the L and R symmetry.. We could look a bit like him ..But i havent any proposals on 4d creaturse ( 5d, is needed, if can move or have poses) , expresented as a 3d image or a 4d database. I have invented my plausible 2d one though. Flatland has none, but Planverse has plently. I just think the creature would be aboble to jump , not waddle around. Especially the way things scale with strength. 2 d is stronger bones betcause less ways to break..flux is 1/r. In 4d spacetime, our world, smaller means stronger ( relative to body lenght because of volume scaling with r cubed, and strenght with cross section, thats r squared. so the bigger the slower. Im sure the rules can change but its more enlightenting to me to make analogies, keep the rule similar, and of course consistent, no magic.

  8. Nalyd Srednas says:

    I was wondering. What would happen if a creature of 2-dimensions went into a 3-dimensional universe? Would it die because of a total new set of laws of physics or would it adapt to 3-D?

  9. Damian says:

    Most “alternate universe” thinkers say there could be no physical interaction at all. Just pass right through. They would not even see a cross-section of the 3d world. This is because their atoms and molecular forces are flat and very different. Seems like even if a 2D cross section of a 3d atom, with its electron cloud, passed through their plane, isn’t going to give them something they could see, even if the scales by luck, matched enough. But , it could..just -maybe-When an electron crosses the plane, they could it guess bounce from one of their atoms. But, tunnelling could be the issue. Very likely the matter on a big scale, would enter into the middle of a planet and disrupt and and destroy it. The chance of it being near the edge, and the creatures, without disturbing the inner planet and causing massive volcanic leaks or breaking the planet before it can repel out, is unlikely unless steered by something or someone very nice to the flat peoples.

    When “The Planiverse” came out A. K. Dewdney said that some sort of electronic resonance occurred between the worlds, and brain of one of the creatures, a special one with a desire to know beyond his limits, became linked, so the 3d people , via computer, could see the world and speak to the creature. A lot of people thought it was a true story because it was written in that style, without any silly fiction.

    For a 2d creature, we should be able to see its future and past if time were a 3d space line dimension.. but not if the scale was light years to a creatures year. So that was not in the book. We could see inside the creature though. It is possible we are just data in a simulation. We wouldn’t know unless we we very clever. Some think we found clues as to limits of the resolution of the data, but that proves nothing.

    On the very popular “Flatland” has that the square “creature” he gets to see 3d..feels gravity for the first time .. (in 2d land gravity for disk planets, you walk around is not much different than ours, so they would not be impressed with gravity), Flatland, was written in 1884, it is not educational science fiction, its OK, but so silly that i’m disturbed it always comes up. The square with no feet? the triangle creature ? the Yndrd in Planiverse, is not even that strange in the planiverse. I thin Planiverse is too scientific for most people who don’t see beauty in science and math because the teachers failed to show them. I simulated some fish for my online game project Kontrol and they are similar to 3d fish.. Wiggle side -to- side and move, using the 2d versions of our laws of physics equations.

    As for the Planiverse creatures affecting us, I cant imagine a way unless their time flowed in sync with ours ( things with any sort of interaction do tend to sync, if time were like a flow and such fluids) so that to keep them near some part of us, say, so one creature can see part of my left eye when its near the computer as usual. Then the creature keeps chopping and otherwise messing with the mysterious thing that changes shape, until I go insane or blind.. or… they spoke to someone via brain waves, then we might probably lock him up and medicate him.

  10. jo black says:

    the fact that i cannot picture a 4-d universe only makes me want to try harder. how much torture must the 3-d universe be going through, just to try a imagine if we existed and this torture keeps going on until you get to the nth-d. i believe in a 4-d universe the concept of time becomes a reality i.e in the 3-d universe both time is perceived through movement and expansion of space, thus, spacetime. but in a 4-d universe time exist independent of space and vice versa. i believe if life was to exist in a 4-d universe it would be able to move through space and time separately.

  11. [email protected] says:

    You said more dimensions means more freedom of movement, so will there be any kind of blink or teleportation? Can somehow 3-D creature like human sneak into a higher dimensional space?

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