2020-03-21, a Saturday

flatland chemistry

dimensions chemistry physics biology

I was reminded about something I heard a while ago regarding Flatland, a book which I have not yet read, but I have heard of. I don’t remember who said this—probably the author of the book—but they said that in Flatland, the characters couldn’t have a digestive tube from their mouth to their anus because it would split their body into two separate pieces. However, I thought that such a being could still be split into multiple pieces if some attraction held them together like a magnetic clamp. Our cells are semipermeable, according to biology class; perhaps the Flatland creatures could absorb nutrients somehow though selective pores in their membrane. I briefly pondered a different style of cuisine for these nutrient-absorbing unicellular organisms. This wouldn’t be very different from Agar.io.

I wondered if gravity could work in Flatland. I believe in Flatland, the characters move around as if there was no gravity, so one could imagine that what they saw was like a simple raycaster maze with walls of infinite height. However, I think gravity could still work. There are 2D gravity simulations, after all. Planets would just be circles, and the creatures on it would exist as if the world was a side-scrolling game (think Mario, Terraria, Blockheads).

figure comparing horizontal and vertical projection

How creatures in a 2D world would see (left) with no gravity, like in Flatland, or (right) with gravity. (Not exactly to scale.)

I wasn’t sure how they would deal with obstacles though. If there was a tree in the way, the only option would be to climb over it. Evolution of these 2D life forms would probably find a way around over it (perhaps animals have to develop flight in order to navigate a world outside of water); also, tunnels can’t exist very well because there’s nothing on the sides that can support the earth above, so it would collapse. Thinking about this made me wonder about 2D life on an atomic scale.

2D chemistry

A lot of chemistry can be modelled in 2D, but would only two spatial dimensions have any implications on molecular structure? Probably. However, there would still be two electron spin states, and the orbital classes would probably hold the same number of electron pairs (s orbitals hold 1 pair, p holds 3, d holds 5, and so on for each odd number); I don’t think they’re dependent on a third axis. Electrons can still do probability clouds in 2D, though the orbital arrangements will probably look a bit different. Thus, I think in 2D, all the atoms and elements will be the same.

UPDATE 2021-08-30: Though admittedly perhaps there might be some complications in atomic nuclei because the nucleons would be spread further apart because they can’t be clumped together in a ball.

However, I think molecular shapes would be rather different. VSEPR is based on the idea that bonds and lone pairs repel each other, so they distribute themselves as far away from each other as possible. In 2D, bonds and lone pairs can only distribute themselves evenly around a circle—though this might put more stress on an atom with many bonds and lone pairs because they would all be cramped around the molecule, so larger atoms might form less stable molecules than their 3D counterparts, I’m not sure.

figure showing $CH_4$ in 3D and 2D

$\text{CH}_4$ in (left) 3D and (right) 2D. In three dimensions, a tetrahedron spreads four points the furthest apart from each other. When limited to two dimensions, the points will form a square.

One of the larger implications of this is that water might not be polar anymore. In 3D, the bonds and lone pairs would have distributed themselves in a tetrahedral shape because that spreads them as far apart from each other. In 2D, they would arrange themselves around the corners of a square. Water has two bonds and two lone pairs. I think the bonds would be on opposite sides of each other rather than forming a right angle because there’s a similar example in 3D. On an atom with 3 lone pairs and 2 bonds, the bonds stay on opposite sides of each other, so the shape is linear (an example of this is the triiodide ion $\text{I}_3^-$).

3D linear shape (maybe I3-?), 3D H2O, 2D H2O

Left: a 3D linear shaped molecule with 2 bonds and 3 lone pairs. Top right: $\text{H}_2\text{O}$ has a bent shape in 3D because with four orbitals, they follow the tetrahedral shape. Bottom right: $\text{H}_2\text{O}$ in 2D probably has the bonds on either side, making it linear.

Thus, I think water would be linear and thus nonpolar (its polar bonds would cancel each other out), which would have a large impact on its properties. It would probably be a gas at room temperature, like methane ($\text{CH}_4$), and it won’t experience surface tension, which plants in our world rely on for transpiration, which allows water evaporating from its leaves to pull water up from its roots. It also can’t dissolve ionic compounds, though it would be able to dissolve oil. Fortunately, ammonia ($\text{NH}_3$) and hydrofluoric acid ($\text{HF}$) would still be polar, so plants could perhaps use one of those instead and adapt accordingly. The difference in electronegativity between the oxygen and hydrogen atoms in water would still be the same, so the electrons in the oxygen-hydrogen bond will tend towards the more electronegative oxygen, making the $\text{H}^+$ slightly easier to pull apart when dissociating. This means water would still have the same acid dissociation constant of $10^{-7}$ and thus have a pH of $7$. This would also apply for other substances.

However, I don’t think water would be the main solvent anymore since it can’t even dissolve salts. I don’t know much about ammonia, but it might be a viable alternative. It is amphoteric: as an acid, it can donate a proton and become $\text{NH}_2^-$, or more likely, as a base, it can accept a proton and become $\text{NH}_4^+$. This could be like $\text{OH}^-$ and $\text{H}_3\text{O}^+$ for water. If this is the case, then the neutral pH would probably be a bit higher because ammonia is more basic than water, but I’m not entirely sure.

As a final note, I don’t think nitrogen gas can exist in 2D. In 3D, a triple bond can be done by having the first sigma bond be between the atoms, and the other two pi bonds around it, as shown in the figure. There’s no space left for another pi bond, which is why quadruple bonds don’t really exist between nonmetals. In 2D, there wouldn’t be space for a second pi bond, so I don’t think a triple bond would be as common or even possible. Also, this might suggest that in a 4D world, a quadruple bond between two carbon atoms might be possible.

3D triple bond (N2) model, and double bond (O2) in 2D

Top: a triple bond’s orbitals. The pink orbital (the third bond, or second pi bond) is perpendicular to the purple orbital (the second bond, or the first pi bond) in 3D space, so the entire thing cannot be exist in a flat plane. Bottom: a double bond’s orbitals. This arrangement could work in a 2D plane as well.


A lot of things in physics—at least the parts of physics that I have learned—can be modelled in two dimensions, so I don’t think two-dimensional physics would be very different. However, one thing to note is that while things can rotate, their centre axis can’t attach to anything, so there are no wheels in a 2D world.


It’s been a while since I’ve learned biology, but I still remember that the mitochondria is the powerhouse of the cell. They can still exist in 2D; a star at the centre of the solar system could provide light energy to the mitochondria, and through some processes that I have forgotten but remember seeing 2D models of (meaning that they can occur in 2D), the energy is stored in glucose.

I’m pretty sure most of our cells can exist in the same way, though DNA would have to take up more space since you can’t really fold it over itself in 2D when compacting the DNA into chromosomes.

Other things

A writing system in Flatland would probably look like morse code. If there is gravity, then it would be a vertical script, which is pretty cool.

Having an extra eye doesn’t necessarily allow us to see in another dimension. Although having only one eye seems like you’re seeing in 2D and having two eyes allows us to see in 3D, our second eye only gives us the ability to perceive depth. We can’t see through walls with our second eye, and a third eye won’t make you see things four-dimensionally. Thus, a 2D world would still require two eyes to perceive depth, though they would see their world in 1D regardless.

A lot of things I’ve mentioned here about a 2D world might seem like limitations, but this also suggests that compared to a four-dimensional world, our world has many limitations due to a lack of a fourth spatial axis. Perhaps multidirectional ball-shaped wheels are possible in 4D.

See source and revision history on GitHub.