== √Flatland³ === wip
This is my pitch for a new flatland movie [1], based very loosely on the book link(https://en.wikipedia.org/wiki/Sphereland)[Sphereland: A Fantasy About Curved Spaces and an Expanding Universe] [2], called √Flatland³ [3].
Puncto, a hexagon, is a surveyor [4] for flatland. With new advancements in surveying technology (like parabolic dishes and radio waves that can travel through obstructions [5]) surveyors are 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 [5] 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, $(x, y)$ or $(L, \phi)$, etc. If your coordinate systems are smooth, small straight line segments in one coordinate system also look straight in another. So for any area around a given point $m$ in flatland , you should be able to transform them to what that looks like near $m$ in any other coordinate system using a linear transformation.
They discover two important empirical facts: distances measured along straight lines are consistent with theory, and distortion increases with distance from a survey station, and distortions are rotationally symmetric.
img(distortion_2.svg)[Distances measured along straight lines have no distortion, and distortion increases with distance.]
Alicia of course knows about inner products, and that inner products under linear transformations look like $[x, y] = x^T G y$. $(L, \phi)$ is a good coordinate system to work in, because $\frac{\partial}{\partial L}$ and $\frac{\partial}{\partial \phi}$ are perpendicular, and $\frac{\partial}{\partial L}$ is already correct. And because the distortion is rotationally symmetric, $\left[ \frac{\partial}{\partial\phi}, \frac{\partial}{\partial\phi} \right]$ is a function of only $L$. So she knows 3 of the 4 terms of the metric tensor, when written in $(L, \phi)$ coordinates: $G = \begin{pmatrix}1 & 0 \ 0 & f(L)\end{pmatrix}$.
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:
imgcmp(graph.svg)$f(L)$. If there were no distortion, $f(L) = L$ which is what it sort of looks like here.[$f(L) - L$, showing the distortion.]
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 [6]:
\[f(L) = C_0 + C_1 L + C_2 L^2 - C_3 L^3 + C_4 L^4 + C_5 L^5 + C_6 L^6 + C_7 L^7 + C_8 L^8 + C_9 L^9\]$C_0$ 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. By symmetry, she reasons that $f(L)$ must be an odd function, so $C_2$ through $C_8$ should all be zero.
\[f(L) = L - 2.604 \cdot 10^{-15} L^3 + 2.034 \cdot 10^{-30} L^5 - 7.569 \cdot 10^{-46} L^7 + 1.432 \cdot 10^{-61} L^9\]$C_7$ and $C_9$ have errors too high to do anything with [7]. Re-doing the polynomial fit for a 5th-order polynomial gives [8]:
\[\begin{aligned} f(L) & = L - 2.604 \cdot 10^{-15} L^3 + 2.034 \cdot 10^{-30} L^5 \\ & = L - 1.563 \cdot 10^{-14} \frac{L^3}{3!} + 2.441 \cdot 10^{-28} \frac{L^5}{5!} \\ & = (1.25 \cdot 10^{-7})^0 L - (1.25 \cdot 10^{-7})^2 \frac{L^3}{3!} + (1.25 \cdot 10^{-7})^4 \frac{L^5}{5!} \\ & = \left(\frac{1}{8 \cdot 10^{6}}\right)^0 L - \left(\frac{1}{8 \cdot 10^{6}}\right)^2 \frac{L^3}{3!} + \left(\frac{1}{8 \cdot 10^{6}}\right)^4 \frac{L^5}{5!} \\ & = \frac{1}{E^0} L - \frac{1}{E^2} \frac{L^3}{3!} + \frac{1}{E^4} \frac{L^5}{5!} \\ & = E \frac{L}{E} - E \frac{L^3}{E^3} \frac{1}{3!} + E \frac{L^5}{E^5} \frac{1}{5!} \\ & \approx E \sin{\frac{L}{E}} \end{aligned}\]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 $A=(1.1 \cdot 10^7, -0.5)$ to $B=(1.6 \cdot 10^7, -2)$. 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.
img(shortest_path.svg)[Blue is the “expected” shortest path, orange is the “empirical” shortest path.]
Integrating along this path, $\int_P \sqrt{(dL)^2 + (E \sin{\frac{L}{E}} d \phi)^2}$, for both paths gives $1.562 \cdot 10^7$ for the “expected” path and $1.271 \cdot 10^7$ for the “empirical”. 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.
img(walk.svg)[As an observer walks from the survey point to point A, their perspective of lengths and angles changes. When they reach point A, they see that the “empirical” path is actually straight.]
She realizes that if the interior angles of a triangle whose sides are geodesics is always greater than 180 degrees, then the exterior angles must always be less than 360 degrees. This feels very weird to Alicia, but her calculations agree with the empirical measurements. She does some experiments with isosceles triangles, and learns that n-gons of radius L (as measured by a surveyor in the circle’s center) will have a total exterior angle of $2 \pi \cos{\frac{L}{E}}$ as n gets large. So she now knows how vectors get transformed as one walks along a path: the $\phi$ component gets scaled by a factor of $\frac{1}{1 + \cot{\frac{L}{E}} \frac{dL}{E}}$ and both components rotate by an angle of $(1 - \cos{\frac{L}{E}}) d\phi$.
This explains in two different ways why light from the survey markers takes the paths that it does. On the one hand, the path that the light beams take are the shortest paths from the marker to the surveyor. On the other hand, they’re also “straight” paths, from the point of view of the light’s velocity vector getting transformed as it moves in its own direction.
Maybe she could also show the more traditional form of the geodesic equation as well? Maybe by thinking about straight lines in traditional geometry. A straight path $c(t)$ in cartesian coordinates would satisfy $D^2 c = 0$. If that path were transformed into another coordinate system by $c(t) = T(\gamma(t))$, this translates to $0 = D^2 c(t)(u, v) = D^2 \gamma(t)(u, v) + M(\gamma(t))(u, v)$, where $M(\gamma(t))$ is a multilinear map. So she might theorize that a similar property holds in flatland. After establishing the above transformation laws it’s not too huge of a leap to conclude that the derivatives of those transformation rules are the components of $M(\gamma(t))$. Though, even in traditional geometry $M(\gamma(t))$ is complicated, involving factors of $D^2T \circ \gamma$, $DT \circ \gamma$, and $D \gamma$, so I don’t know how obvious this is.
Anyways, 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 $\int_0^{2 \pi} \int_0^L E \sin{\frac{l}{E}} dl d\phi = 2 \pi E^2 (1 - \cos{\frac{L}{E}})$. 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 $R_F = E \approx 8 \cdot 10^6$, in all directions [9].
I’m not sure how to make this next part feel natural, but as the finale I want Alicia to discover 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 [10].
Alicia and Puncto release all of this, and are properly lauded for their impact and brilliance, and their names go down in history [11].
[1] I’m unsure of the value of a movie like this in 2026. When I was a kid, in the era of syndicated television, there wasn’t much in the way of truly educational TV programs for math. But today you can go YouTube and find hundreds of differential geometry explainer videos, taught from hundreds of different perspectives on the subject, many of which are quite good. If a kid today wants to learn about parallel transport using the video medium, is this Flatland framing narrative a valuable addition to that pile? Maybe that narrative was valuable back then, but not so much now?
[2] 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].
I didn’t like it. It’s nothing like the book, and none of the new plot points are good. All the stuff about the Hypersphere doesn’t contribute anything. The plot about flipping flatlanders doesn’t make sense in-universe. And the ending contradicts everything the first movie was about, that higher dimensions are spatial.
[3] 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. The characters start out thinking that flatland is fundamentally 2D. and eventually realize that it’s actually 3D.
[4] 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.
[5] 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 physiology in flatland, so I’m keeping it. A. Hexagon is Alicia Hex, a woman and a hexagon.
[6] When I was a kid, I’d watch Mythbusters or PBS NOVA or whatever, and they’d sometimes flash equations on the screen like “ooo look at how complicated this is”. I’d pause the DVR when they were on screen and stare at them, or jot them down. I wanted to know how to do the calculations that they were able to do. But of course I rarely learned anything from that because those equations weren’t there to be learned from, they were decoration.
I think if you’re going to make an educational show, and you display something on the screen that you think a large part of your audience doesn’t understand, you have to explain it. Otherwise you’re not educating. You have to explain how you arrived at any equations and what you can conclude from them.
[7] I wrote some simulations for this and it’s pretty hard to get the error terms to be just right. In order for the plot to make sense the radius of the universe needs to be big enough that no one has noticed this yet, but also small enough that it can be measured by a couple surveyors traveling reasonable distances, like a couple weeks of travel. Idk, the numbers might just have to be made up.
[8] In the books, Spherio always comes down and gives out answers to the heros’ questions, or at least direct confirmation of their theories. But the point of this pitch is to teach a little differential geometry. But there’s still room for some divine inspiration here I think. Like a voice or a dream saying “that number is the square root of that other number”?
The problem with cutting Spherio is that the main characters no longer have anyone to ask the question of “is your 3D universe a hypersphere?”, and start conversations about, say, what laws of physics still work in a universe with curvature.
[9] The original books, and the movies, are all allergic to numbers. They go out of their way to avoid any calculations or equations. These are stories about professional geometers and astrophysicists or whatever, but the stories are crafted in such a way that the only reasoning that is ever needed is basic geometry knowledge and logic. I understand why they’re written like this, but it also leads to ridiculous in-universe situations. Like in the second book, after they realize that flatland is a sphere, A. Hexagon (the professional mathematician) never once wonders what the radius of that sphere is.
[10] The book’s logic of “if a curved line doesn’t curve left and doesn’t curve right, it must curve in a direction we can’t see” is so delightful, but I’m not sure it works here because we’ve already emphasized that those curved paths are actually straight to the people walking them.
[11] 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” or whatever. 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 appropriate I think.
