== √Flatland³ === wip
This is my pitch for a new flatland movie [-], based very loosely on the book link(https://en.wikipedia.org/wiki/Sphereland)[Sphereland: A Fantasy About Curved Spaces and an Expanding Universe] [-], called √Flatland³ [-].
[TODO]
Puncto, a hexagon, is a surveyor [-] for flatland. With new advancements in surveying technology (like parabolic mirrors and brighter lights to pierce the world’s fog a greater distance[-]) surveyors beginning to be able to survey greater distances. When doing so, they’re realizing that the interior angles of triangles are not adding up to 180 degrees.
The problem of map distortion has long been an issue. Surveyors would go out to map the lands (which involves building miniature scale models of the land and its important features), but the larger the area of land surveyed the more problems they would have with distortion around the edges. The numbers just wouldn’t add up. And whenever two surveys overlap, the overlapping areas never line up quite right. This was all chalked up to accumulated experimental error, since larger surveys were made by measuring lots of smaller triangles.
The situation was seen as inevitable and an unfortunate reality of life. If the army marched off to war, the quartermaster would take these surveys, called “charts”, with them, with the understanding that they’re only really accurate near their center point. For a large campaign, they’d take a bunch of these charts with them in a collection called an “atlas”.
But now that surveyors are able to measure much greater distances directly, the problem is becoming harder to ignore.
Puncto visits renowned flatland mathematician Alicia Hex [-] with this issue.
They struggle for a while over this problem. They check measurements, equipment, etc. They do many tests in various configurations and learn some interesting facts, like that triangle angles always add to more than 180, never less; and that the amount of distortion has something to do with the triangle’s area, though the relationship seems complicated.
Alicia starts thinking about coordinate systems. For any given chart of flatland, you can endow a coordinate system, $x^1$ and $x^2$. And for any given point $m$ in flatland, there should be able to take $\frac{\partial}{\partial x^1}$ and $\frac{\partial}{\partial x^2}$ and transform them to what that looks like near $m$ using a linear transformation. [TODO: more on this]
They discover two important empirical facts: distances measured along straight lines are consistent with theory, and distortions are rotationally symmetric.
[picture of radial map, with labels showing distortion]
This is a good coordinate system to work in. $\frac{\partial}{\partial L}$ and $\frac{\partial}{\partial \phi}$ are perpendicular, and $\frac{\partial}{\partial L}$ is already correct. This implies that the metric tensor, when written in $(L, \phi)$ coordinates, has the form $\begin{pmatrix}1 & 0 \ 0 & f(L)\end{pmatrix}$. [TODO: more on this].
So Puncto and Alicia begin the work of determining the empirical shape of $f(L)$. They do this by choosing a small angle that would consistently see measurable distortion, and measuring arcs of that angle at various radiuses. Because the distortion is rotationally symmetric, the choice of angle doesn’t matter.
They take a bunch of measurements and get a graph like this:
[TODO: graph]
There is a slight but significant distortion shown here.
They try a couple different curve fitting methods, but the polynomial fit bore the most fruit:
\[f(L) = C_0 + C_1 L + C_2 L^2 - C_3 L^3 + C_4 L^4 + C_5 L^5 + C_6 L^6 + C_7 L^7\]$C_0$ should of course be 0, because there’s no distortion at the center of the circle, that’s the whole point of all this. $C_1$ should be 1 as well, for the same reason. $C_2$ and $C_4$ are pretty small compared to the other coefficients, so they might be 0. $C_6$ and $C_7$ have errors too high to do anything with.
Re-doing the polynomial fit for a 5th-order polynomial, setting $C_0$, $C_2$, and $C_4$ to 0 and $C_1$ to 1 gives:
[TODO: more narrative around this manipulation?]
\[\begin{aligned} f(L) & = L - C_1 L^3 + C_2 L^5 \\ & = L - D_1 \frac{L^3}{3!} + D_2 \frac{L^5}{5!} \\ & = D^0 L - D^2 \frac{L^3}{3!} + D^4 \frac{L^5}{5!} \\ & = \frac{1}{D} \phi - \frac{1}{D} \frac{\phi^3}{3!} + \frac{1}{D} \frac{\phi^5}{5!} \\ & \approx \frac{1}{D} \sin{\phi} \\ & = E \sin{\frac{L}{E}} \end{aligned}\]This form, if true, completes their metric matrix, and thus completely describes how to measure areas, lengths, and angles in this weird geometry.
After some calculations, Alicia shows that this perfectly explains the weird behavior of their surveys. Shown are two paths from point A to B. The first is the “expected” shortest path according to the surveyor, a straight line. The second is the “empirical” shortest path, as measured by the surveyor as someone walks what they perceive to be the shortest path from A to B.
[TODO: picture]
Integrating along this path, $\int_P \sqrt{(dL)^2 + (E \sin{\frac{L}{E}} d \theta)^2}$, for both paths gives:
[TODO]
So the “expected” shortest path is actually measurably longer than the “empirical” shortest path, when measured according to the metric of people walking along the path.
It also explains why light from the survey markers takes the paths that it does. Angles are not preserved in this distortion, so the “expected” straight path looks bent according to those along the path. But along the “empirical” path, at every point the path looks straight! So the light is traveling in a straight line, according to the light ray that’s traveling it.
[TODO: picture of angles getting distorted, picture of these paths and angles]
The mystery seems to be solved. They start sharing their ideas, and while this theory has a hard time gaining traction at first, it’s hard to deny that it predicts experiments, at least the ones that could be done now. It would be a long time before the $L^7$ term would be measured and maybe falsified.
But Alicia is still bothered. A circle in flatland of radius $L$ will empirically measure a circumference of $2 \pi E \sin{\frac{L}{E}}$. Plugging in $\pi E$ for $L$ gives a circumference of 0. This implies that, if a bunch of people were to set off from the same point in many directions, walking straight, then after a distance of $\pi E$ they’d all meet, since the distance between them all is now 0.
Where would they meet? Would they meet back at their original spot? Or is there some other spot that world wraps around to, like a circle in every direction?
And if they kept walking for another $\pi E$ they’d end up at the same spot again. Again, are they now at the starting position, or some other even crazier place?
Well, just extrapolating from the metric that she has, the actual ground area of a circle of radius L is $\int_0^{2 \pi} \int_0^L E \sin{\frac{l}{E}} dl d\theta = 2 \pi E^2 (1 - \cos{\frac{L}{E}})$. Plugging in $\pi E$ for $L$ gives an area of $4 \pi E^2$. Plugging in $2 \pi E$ instead gives an area of 0. This implies that the first meeting point is another place, and the second meeting point is the starting point. So flatland is like a circle with radius $R_F = E = [TODO]$, in all directions.
I’m not sure how to make this next part feel natural, but as the finale I want Alicia to dscover that flatland is realizable in $\mathbb{R}^3$. Maybe she starts with this last point, that it’s a circle in all directions. Like at $\theta = 0$, the natural representation is $\begin{pmatrix}R_F\cos{\phi}\R_F\sin{\phi}\ 0\end{pmatrix}$. At $\theta = \frac{\pi}{2}$, you could try to write it as $\begin{pmatrix}R_F\cos{\phi}\0\R_F\sin{\phi}\end{pmatrix}$. So she could try to combining them as $p = R_F \begin{pmatrix}\cos{\phi}\ \cos{\theta}\sin{\phi}\ \sin{\theta}\sin{\phi}\end{pmatrix}$.
She would then try to re-compute the $\langle\frac{\partial p}{\partial \phi}, \frac{\partial p}{\partial \phi}\rangle$, $\langle\frac{\partial p}{\partial \phi}, \frac{\partial p}{\partial \theta}\rangle$, and $\langle\frac{\partial p}{\partial \theta}, \frac{\partial p}{\partial \theta}\rangle$ calculations from earlier to re-build the metric, this time with explicit forms for $p$. And she sees that it lines up exactly! This, along with the fact that $\langle p,p \rangle = R_F^2$, confirms without any doubt that flatland is a 2-dimensional circle in 3-dimensional space, or a “sphere” as her grandfather mentioned when she was young.
Alicia and Puncto release all of this, and are properly lauded for their impact and brilliance, and their names go down in history [-].
[-]
[-] There is a link(https://en.wikipedia.org/wiki/Flatland_%282007Johnson_and_Travis_film%29#Sequel)[2012 short movie adaptation] of this book already, by the same guys who did the link(https://en.wikipedia.org/wiki/Flatland%282007_Johnson_and_Travis_film%29)[2007 Flatland short movie].
[-] The 2012 adaptation is titled “Flatland²” in the movie’s titlecard, even though people seem to call it “Flatland 2”. I dislike this name. If the implication is that Flatland=2 (because flatland, the world, is 2D), then Flatland²=4, and I don’t know what 4 would be referring to. While 4th dimension does come up, it’s not the main focus of the movie. If Flatland=1 (because Flatland is the first movie in the series), then Flatland²=1 which doesn’t make sense. What, are we suppose to assume that Flatland = $\sqrt{2}$?
I think my title, √Flatland³, more thematically consistent. [TODO]
[-] I keep the book’s profession of Puncto over the movie’s.
And actually, while we’re on this subject, in the movie Puncto is puzzled by the angles he’s measured with radar. But how would he know the angle that the radar beams subtend on the distant planet? He can’t know it. The book even addresses this exact point, so it’s an odd detail to insert into the movie.
[-] IIRC the first book mentions that fog is not uniform everywhere. In this adaptation it might make sense to have a constant fog everywehre. [TODO: check this]
[-] In the narrative of the book, Sphereland is written by “A. Hexagon”. The movies choose to interpret “A. Square”, their grandfather’s name, as “Arthur Square”. I had trouble coming up with a good name for our hexagon that references a famous differential or non-euclidean geometer. Alicia Hex, after Alicia Boole Scott who popularized the word “polytope” and did some early work in high-dimensional geometry, is the best I could come up with.
On A. Hexagon’s gender, I’m choosing to stick to the movies and make her a woman. I have no strong feelings one way or the other, but people seem to be happy about the changes the movies made to gender and pysiology in flatland, so I’m keeping it. A. Hexagon is Alicia Hex, a woman and a hexagon.
[-] In the original book, Square is imprisoned for the rest of his life for his prosthelization of the 3rd dimension. That’s because the book is in large part a social commentary, which means it has to leave you feeling bad about “society”. But the later self-proclaimed sequels, including this one, largely do away with the social commentary.
I think what the first book did with its two subjects was cool and clever. But this pitch isn’t about social commentary. It’s about teaching and celebrating beautiful and clever math concepts. So an ending that reflects this is apropriate I think.
Criticisms of the sphereland movie
Like in the original, the characters have to leave their 2D world and look at it from the “outside”, they don’t show the process of developing a theory of geometry that works within their world. Again, the book goes to great trouble to show the flatlanders developing their ideas in a way that Maybe draw some connections with the development of general relativity?
All the stuff about higher dimensions seemed unnecessary. It’s not important for the plot. Also, they run into the issue that, I guess, they can’t render 4D using their engine, so the whole part where Spherio sees the 4th dimension is all lights and ooOOooOO.
I know what they’re trying to do with the whole plot point of “flipping” linelander/flatlander/spacelanders backwards. But this doesn’t make much sense for flatlanders specifically, because in the movie the flip themselves top-to-bottom all the time! If they’re experiencing the world flipped top-bottom, they can just un-flip themselves. If they’re experiencing the world flipped left-right, then they can again flip themselves top-bottom and experience a rotated, but not flipped world.
[gif of flatlander flipping]
This point is particularly interesting because the Flatland books make a point of specifying the chirality of different beings. [page 87 illustration]
What’s up with the ending? Apparently the “different dimensions” are alternate realities, like in your run-of-the-mill sci-fi flick? And not actual spatial dimensions?
Cool stuff:
A review from the UCI math dept: https://www.math.uci.edu/~asilverb/bibliography/SpherelandSilverberg.pdf https://en.wikipedia.org/wiki/Flatterland https://www.youtube.com/@FlatlandLostMedia