I was reading Flatland, and the thought occurred to me that interaction between an n-dimensional being and a n+1+-dimensional being may not only be difficult but dangerous, that Flatland contains errors of logic.

Assuming that our n-dimensional universe is a brane wrapped around an immaterial n+1-dimensional sphere (a 2-d universe would be wrapped around a sphere, its matter stuck to the brane like an atom-thick fridge magnet to a steel surface), an n+-dimensional being could move through the n-brane at will and interact with the inhabitants of the n-brane, potentially removing the n-dimensional being from their n-brane to an n+-brane.

So, the question: How could n- and n+- dimensional beings interact, and what would the hazards be for each being, including removal of an n-dimensional being from the n-brane to a higher dimensional brane. What would each experience assuming that the interaction could occur safely?

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    $\begingroup$ I worry this may have no answer. The real answer would be "whatever the laws of physics say should happen." Consider just how many different ways we interact with 2d surfaces, each with its own rules. Sometimes we put ink on the surface, other times we bend the surface, other times we walk through like it doesn't exist. $\endgroup$
    – Cort Ammon
    Feb 23 '15 at 0:52
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    $\begingroup$ @CortAmmon, given your comment, there may be many answers, however we are talking about n-branes, not paper or metal, which is just an analogy to assist thought. $\endgroup$
    – Monty Wild
    Feb 23 '15 at 1:01
  • $\begingroup$ Are you specifically looking at branes from string theory? As in the theory which supports Ekpyrotic scenarios where entire universes are born when branes touch (much less allowing entities within those branes interacted)? $\endgroup$
    – Cort Ammon
    Feb 23 '15 at 4:24
  • $\begingroup$ @CortAmmon, I'm considering branes as a limiting factor in a >n+1 dimensional universe, but I'm after a generic answer for, say 2d <-> 3d & 3d <-> 4d interaction scenarios. I had string theory in mind, but not the new universe part. I have my own ideas of what might happen, but I don't want to contaminate others answers by posting them just yet. $\endgroup$
    – Monty Wild
    Feb 23 '15 at 6:14
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    $\begingroup$ Your "Assuming that..." statement in paragraph 2 is a pretty big assumption! $\endgroup$
    – KSmarts
    Feb 23 '15 at 20:15

The space we live in has to be complete, with the "outside" not simply another spatial direction of the same geometry. You might postulate another orthogonal $w$ axis that works the same as $x, y, z$, and we're not able to move that way because everything is lined up at $ w=0$ now so nothing can get get beside you to push in that direction. There might be some field that attracts everything to that $ w $ position, and it might be very strong.

However, you don't have to stand beside something at $ w=$ 5inches for example to push in the $-w $ direction. The physics is not “closed” over the domain of the dimensions of the current arrangement of particles. Effects can operate at right angles to the participants. That is the general thing you see with cross products. Gyroscopes would produce torque in that direction, electromagnetic effects would have more right angles to reach out to.

We also have the problem of space and time getting mixed up. You can't move only within a line or plane since relativity will change the spacial axes. In fact, a combination of boosts (accelerations, thrusts) cannot be expressed as a single resulting boost vector, as we can with Euclidean geometry and Newton’s laws. You get a boost and a rotation when you try that with special relativity.

So being stuck to a 3D manifold (free in x,y,z but w constant) won’t work since the meaning of $ w=0$ will change depending on your motion, so (1) what am I really stuck to? Or (2) the universe will blink out of view if you move.

If forces and motion in the forbidden direction are corrected via an attractive force, that will cause the force that got turned around (via precession, magnets, whatever) to be apparently lost. Being stuck to a position doesn’t change the fact that the axis exists and that will change things from how we know them.

So, the spacetime we have is complete with the laws of physics only involving that 3+1 dimension spacetime. The “out there” must be something different.

The D-branes in M-theory (which were discovered by working through the math, not put into the theory because it was part of the idea) anchor the ends of a string so it vibrates like a guitar string. Imagine if the sound depended on whether the string was vibrating in the plane parallel to the face vs perpendicular to it, as well as different frequencies. Vibration modes of strings cause them to be whichever particle or force. If you took a guitar string playing its note with its pleasant sounding harmonics and timbre, and said that’s an electron, what happens if you unhook the ends from the instrument?

Well, you could restring it on a different instrument and if everything is the same, you have an electron in a different, disconnected universe.

You need to grab not just the objects from our universe but a piece of the substrate too, to keep them in their context. You can’t pick, up Mr. A. Square from flatland; you have to cut out the little patch of flatland that he's existing in.

So the higher space beings can do stuff to us. What happens analogously, if you stick your hand through Flatland’s plane? If the 4D hand is made up of string that had different D-brane endpoints and presented a 4D space to manifest the particles in, we’re heading to the idea that all metaspace is the same stuff and the universe and its dimensionality and laws are due to the specific strings present. How would electrons interact with the truly alien string it came into close proximity to? Let’s say that the beings engineered a string that manifests like an electron in profile and also interacted meaningfully with their native particles.

I see two approaches now. The classic idea of the higher-d object passing through and being perceived as a slice of the full object could be contrived by design. Instruments would pierce the lower-d universe and provide live interaction and a communication bridge.

The other approach would be to build a little patch of lower-d space filled with stuff, and graft it into the lower universe. That's the inverse of being able to pick up a patch with its contents.

But, how do they figure out what’s in the patch they sampled, if not with the probe method? Maybe the probe works with the sample in a large machine after modifying it: freezing and “staining” it. Now how would that affect contact? You would not meet A. Square in his home, but but destructively analyze him by blasting the sample apart in the LCLS !



The main risk that I see with an n-dimensional creature being freed from their n-dimensional brane to an n+1-dimensional brane is that in n+1 dimensions, structures that would be closed in n-dimensions would no longer be closed.

As an example, imagine a 2d creature with a closed circulatory system, which would in its most simplified form be two concentric circles with circulatory fluid between them. If we describe the 2D space as having dimensional axes X and Y, and the 3D space as adding a Z-axis, then in the Z axis, the creature's internals would be unbounded, resulting in the inside and outside of the circulatory system from separating, along with the circulatory fluid. In other words, the creature would literally fall apart.

Going from 3D to 4D, the risks are similar. 3D creatures' organs are all connected in the macroscopic realm and none are free-floating as may be necessary in a 2D organism, however most organisms have cells with free-floating organelles. Given 3D axes X, Y & Z, when the W-axis is added, neither the cell contents nor the creature's circulatory fluids would be in a bounded volume, and would spill out in a fatal manner.

Structural Stability

Assuming that we applied a minimally substantial membrane to the open sides of the n-dimensional creature that we intend to move to n+1 dimensions so that it would not simply fall apart, if this membrane was flexible, it could result in the creature crumpling up along the added axis, as it would have no structural strength in this direction, having obtained the necessary strength from the n-brane binding energy. This crumping might not have any effect on the creature's biology, but it would reduce the possibility of acting in the additional axis.


Should the structural stability matter be addressed, there is the question of what an n-dimensional being could sense in n+1 dimensions. Most senses should work as expected given their dimensional limitations, and we could expect eyes to work the same too. However, while brane-bound, eyes work because photons travel on the surface of the brane, but freed from the brane, photons may intersect the retina without ever having passed through the lens if the means by which the open sides of the body in the n+1st dimension are closed is transparent. Additionally, we could expect that the optical sensing cells would be sensitive to photons coming from the n-dimensional axes more than photons coming from the additional axis. This would be proportional to the angle of incidence.

So, in effect, an n-dimensional creature would see an n-1-dimensional slice through the n+1 dimensional space, plus an unfocussed light aura (since the photons coming from the additional axis would probably not have passed through any lens) caused by light from the additional axis, becoming brighter as its angle of incidence approached that of the eyes' axis, when the image would snap into focus.

Contrary to the novella "Flatland", an n-dimensional individual transported to a n+1-dimensional location and rotated would not see their whole n-brane as an n-dimensional image, but as an n-1 dimensional cross-section through the n-brane.

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    $\begingroup$ If all all stuff (elementary particles and forces) were promoted to another dimension that gave it another true space dimension while otherwise being the same, it would not just be blood that falls out. Electromagnetism would fall out too, changing the inverse square law to inverse cube. Ah, but the orbitals available to electrons would change too, with for example 4 sets instead of 3 sets of p orbitals in one shell. All the molecules would change with different chemistry. That's not even considering nuclear forces! $\endgroup$
    – JDługosz
    Feb 25 '15 at 4:48

I think Rudy Rucker has a number of flatland themed vignettes in one of his books. One was a retelling of Flatland from the point of view of the human scientist experimenting with the plane. Another has a person fall through the plane and knock off pieces that get stuck to his clothing and other objects.

I like the catastrophic ideas: trying to pass through it like Dr.Sphere would pop it like a soap bubble. More nuanced would be that changing what should be conserved quantities in the plane universe would cause problems that get progressively worse.

BTW, the Planiverse is a better approach than Flatland, if you want to write physics-based hard SF. The fundemental difference is the orientation: they have directions forward-back and up-down. They live on the edge of a disc-shaped planet. If you have gravitating matter in a 2D universe that's what you expect. As hard-SF, he explores the ramifications of physics in 2 dimensions.

  • $\begingroup$ @MontyWild I added to my Answer. $\endgroup$
    – JDługosz
    Feb 24 '15 at 19:18

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