2529: Unsolved Math Problems

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Unsolved Math Problems
After decades of studying the curve and the procedure that generates it, the consensus explanation is "it's just like that."
Title text: After decades of studying the curve and the procedure that generates it, the consensus explanation is "it's just like that."


Math has many problems that remain "unsolved." This is not simply a matter of finding the correct numbers on both sides of an equal sign, but usually require proving or finding a counterexample to some conjecture, or explaining some property of some mathematical object. Sometimes this might involve extending an existing proof to a wider range of numbers like reals, complex numbers, or matrices.

A concrete problem is one that is very obviously connected to a real world process, while an abstract problem is one which seems unconnected to actual problems. In modern math, many problems tend to be very abstract, requiring complicated notation to adequately state the problem in the first place, like many of the millennium problems. On the other hand, many unsolved problems are very concrete, such as the Collatz conjecture. Additionally, there are the many problems related to packing objects into spaces which are often very difficult to solve though quite easy to state. Finally, Randall describes a third category of "cursed problems," that have strange, seemingly random behavior, such as the behavior of turbulence or the distribution of prime numbers.

In the first panel, Ponytail describes a weird abstract problem. Her description seems to be a meaningless jumble of terms that are either mathematical or just sound mathematical. And the mathematical terms are from disparate branches of mathematics: group theory, topology, and calculus.

  • Euler field: An Euler vector field represents a space where every point is rotating with its own speed and direction. The name "Euler field", however, is something like "John Smith" - fields are very common algebraic structures, and Leonard Euler was a prolific Swiss mathematician who influenced so many areas of study that some of his discoveries are named after whoever wrote about them next, just to avoid naming everything after him.
  • Manifold: A manifold is a topological space which is locally Euclidean - the shortest distance between two points is a straight line, the ratio between a circle's circumference and diameter is always pi, parallel lines are always the same distance apart, everything generally behaves the way you'd expect. A globe is a two-dimensional manifold, because a small-enough area is indistinguishable from a flat map. Using manifolds as an example of impenetrably occult maths may be a nod to the Tom Lehrer song "Lobachevsky", which makes a similar joke about "the analytical algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds (Bozhe moi!)".
  • Hypergroup: An algebraic structure, like arithmetic, is a set of well-defined operations (addition, subtraction, multiplication, division) mapping inputs to outputs over a domain of elements (the real numbers). A hyperstructure is an algebraic structure including an operation that maps a single input to multiple outputs - the simplest example is the square root, which maps a positive number like 4 to both positive and negative 2. A hypergroup is a hyperstructure with an operation that takes a pair of input elements, and, depending on which pair, can output every element or combination of elements in its domain... but also preserves association (1 + 2 + 3 = 6 whether you start by adding 1 + 2 or 2 + 3) and reproduction (if either input is "the entire domain", then the output will still be the entire domain). It's a decent indication of how abstract a hypergroup is that it takes at least three to five sub-definitions to make it remotely understandable.
  • Isomorphic: Isomorphism describes whether all the attributes of one structure can be mapped to properties of another structure. The structures usually have to be of the same type; it is unclear how a hypergroup would map to a "conjection".
  • Gödel-Klein: Kurt Gödel was a 20th-century mathematician who studied logic and philosophy (he's most well known for Gödel's incompleteness theorems) and Felix Klein was a 19th century mathematician who studied group theory and geometry; the two probably never collaborated.
  • Meta-algebra: Not a real term, though derived from the real term Metamathematics.
  • ϵ<0: Another joke term. In analysis, ϵ is usually defined to be an arbitrarily small positive number.
  • quasimonoid: A malamanteau, combining the prefix "quasi" (meaning "partially" or "seemingly") and "monoid" (an object from group theory) and is probably meant to evoke the character Quasimodo from The Hunchback of Notre-Dame (although quasimonoids are a type of algebraic object, namely a non-associative monoid)
  • Sondheim Calculus: This refers to Stephen Sondheim, one of the most successful composers and lyricists of American musical theatre -- the producer of his musical "Into the Woods" once remarked that "Singing Stephen Sondheim is like calculus for singers and actors."
  • conjection: This may combine conjecture and conjunction, or be a joke on pros and cons plus projection.

Finally she asks whether the problem statement is ill-formed; considering that it's mostly gibberish, this may be true.

Many real unsolved math problems appear similarly abstract. One example is the Hodge conjecture, a Millennium Prize problem. It states "Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X." These words may appear nonsensical to a layperson. And even to an expert, the question is `abstract'. (Given a specific manifold, even an abelian fourfold, how on earth do you determine if a given 2,2 class is a cycle?)

In the second panel, Cueball describes a concrete random walk problem, and then mentions that this somehow has applications in three unrelated fields. This is actually not uncommon. The Wikipedia article says that "random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. Walking randomly on a grid never visiting any square twice is known as a self-avoiding walk." This panel may have been inspired by some of the tricky unsolved problems about self-avoiding walks. Many of these problems have to do with rigorously proving properties of random walks that have been guessed by physics intuition, so these problems are connected to physics. The part about the maximum number of points in a line is reminiscent of problems in combinatorial geometry, which often involve counting points lying on different lines. Python code simulating this situation can be found here. C++ code simulating this situation can be found here.

In the final panel, Megan is looking at a strange curve that seems to have no consistent pattern. At the bottom it's mostly straight, with a few little wobbles. In the middle it looks like a wild, high-frequency wave that suddenly bursts and then dies down. And the top is a spiral that looks like a question mark or a Western-style Crosier. She wonders if this could even be mathematical. On one hand, considering the weird shapes that come from plotting some mathematical processes (e.g. the Mandelbrot set), it could well be. For example the unsolved Riemann hypothesis, another Millennium Prize problem, concerns the properties of a weird and at-first-glance random curve. In number theory, the term "cursed curve" has been used to describe the "split Cartan" modular curve of level 13, which resisted attempts for many years to compute its set of rational points. On the other hand, the question if could even be mathematical suggests that this may indeed not be a mathematical symbol. The curve looks like the unalome symbol, which is a Buddhist symbol which represents the path taken in life, or the journey to enlightenment. It could be argued that this indeed represents an unsolved problem, although not a mathematical one - which might then be part of the humoristic meaning.

The title text states that, despite decades of intensive study on the properties of the cursed curve, the best anyone's been able to come up with to explain its strangeness is "it's just like that." This lack of a satisfying explanation is commonplace with advanced math topics. As one famous example, the monster group (explanation video) is known to be the largest of a category of objects called sporadic groups. Similarly to the cursed curve in the comic, the monster group has a bizarre and complex structure which has, so far, managed to elude any logical explanation aside from "it's just like that."


The Three Types Of Unsolved Math Problem

[First box:] Weirdly Abstract
[Ponytail stands in front of an equation.]
Is the Euler Field Manifold Hypergroup Isomorphic to a Gödel-Klein Meta-Algebreic ε<0 Quasimonoid Conjection under Sondheim Calculus?
Or is the question ill-formed?
⬙ℝ̇ℤ/Eℵ₅ [The Z is raised and underneath it is a double-ended arrow bent at a right angle. One points toward the R the other toward the Z. The ₅ is double-struck (𝟝) like the R and Z.]
Second: Weirdly Concrete
[Cueball stands in front of a grid with 6 columns and 7 rows]
If I walk randomly on a grid, never visiting any square twice, placing a marble every N steps, on average how many marbles will be in the longest line after N*K steps?
Somehow the answer is important in like three unrelated fields.
[The path starts in the 3rd row and 3rd column, a small circle indicates the start. It takes the path: North, East, North, East (a black dot representing the 1st marble is placed here, so N=4), South, East, South, South (2nd marble), West, South, West, North (3rd marble), West, South, South, South (4th Marble), West, North, West, West (this goes offgrid to the West. There is no visible line or marble outside the grid). The 1st, 3rd, and 4th marbles are colinear and there is a dotted line connecting them. The line's slope is 3.]
Third: Cursed
[A Megan with unkempt hair stands next to a curve]
What in God's name is going on with this curve?
Is it even math?
[The curve starts at the bottom of the screen, rises straight upward, begins to wobble left and right a little. It lists to the left and the left-right motion increases, then decreases. It begins a large counter-clockwise arc, spiraling inwards twice, then ends]

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Does anyone have any clue whether the writing on the board in the weirdly abstract panel means anything? Maybe add an explanation about it? 13:23, 17 October 2021 (UTC)

I think it is also "ill-formed" in the sense that it is not very carefully written.

Center panel possibly related to "The drunkards walk" and theories on randomised motion. https://www.quantamagazine.org/random-walk-puzzle-solution-20160907/ More references https://mathworld.wolfram.com/RandomWalk.html

Someone's gotta point out that "walking randomly on a grid, never visiting the same square twice" would rapidly trap you in a corner (even the example has a 50/50 chance of that happening on the next move) 04:29, 16 October 2021 (UTC)

Not if it's an infinite grid.

I think there's two different ways to interpret the question - as a uniform random element of the set of all non-self-intersection NxK length paths, in which case it's fine, or as a path defined by a random walk in which moves onto your own path are not allowed, which doesn't seem well defined, since you might end up in a situation where you are surrounded by your own path and cannot continue for all NxK steps.

An early example of a cursed problem is the Cantor Function. https://en.wikipedia.org/wiki/Cantor_function

I admire whoever wrote the description of the curve in the "cursed" panel. Barmar (talk) 05:36, 16 October 2021 (UTC)

"Algebreic" is a misspelling of "algebraic". Could Randall really have made this mistake, or is it another malamanteau? What does "breic" come from? Barmar (talk) 06:10, 16 October 2021 (UTC)

I wonder if Randall was actually referring to that quote about "Into the Woods", or he just thought "Sondheim calculus" sounded cool and it was a total coincidence. I found it when I googled "sondheim calculus" to make sure it wasn't a real thing. Barmar (talk) 06:29, 16 October 2021 (UTC)

In panel 2, what would 'k' be? 08:00, 16 October 2021 (UTC)

'k' would represent the number of marbles placed on the ground. 08:09, 16 October 2021 (UTC)

Though probably correct, I think the implied state is that an integer multiple (k) of N steps is made (s=N*k), with that number of marbles dropped, not s=(N*k)+c steps (for N>c) which would have the same result (uselessly) for all values of s where c ranges 0..N-1. It just introduces inflections into the graph (with s as an axis) that needn't be there (with just a k-based one). Or, in other words, selectively poll all s-values that are exactly divisible by N, and forget all the rest. (That divisor is k, and hence k is the number of marbles. Or perhaps k+1 if you leave one on the starting spot too.) 21:59, 16 October 2021 (UTC)

To me, the cursed curve looks a bit like a crosier https://commons.wikimedia.org/wiki/File:Crosiere_of_arcbishop_Heinrich_of_Finstingen.jpg

--> I had the same impression and added it. -- 11:40, 16 October 2021 (UTC)

No explanation of the "Euler Field Manifold Hypergroup (Isomorphic to a)..." part?

The cursed curve looks almost like someone took a graph of the Binet formula in the complex plane, stretched it out a bit, and rotated it onto the i axis.

This was my first thought too when I saw it. 17:16, 16 October 2021 (UTC)
It looks like Vulcan script to me. LtPowers (talk) 13:51, 16 October 2021 (UTC)
That's what it looks like to me too; recognized it from that Numberphile video on Fibonacci numbers in the complex plane 07:36, 17 October 2021 (UTC)
It looks to me like someone has raised a dark spirit, which is about to manifest from a column of black smoke. 10:25, 20 October 2021 (UTC)

Could the cursed curve be a reference to the logistic map?

Can someone produce a high resolution image of the Cursed Curve? It needs to be on a T-shirt Avimimus (talk) 21:39, 16 October 2021 (UTC)

Is someone going to mention the title text?

I swear I've seen that third plot, I thought it was in XKCD, but a quick run through tagged entries didn't find anything... unfortunately I consume a lot of math media so I can't place it. It's bugging me so I hope this note will serve as encourgement to someone that DOES remember 21:29, 16 October 2021 (UTC)

I'm sure I've seen components of the cursed-curve, not sure if they fit together like that, easily, though. The differentiation of dy/dt (which is odd in itself) of the first (lower) bit looks discontinuous, followed by a chaotic oscilation (may just be the culmination of the less frenetic chaos that created the first set of x=f(y) - again, an unusual way round) that then settles into a pattern where regardless of the 'prime axis', you have multiple real roots on the other, towards some great-attractor value.
In more standard x/y (or y=f'(x)?) notation, it is clear that there are multiple real roots for various values of x within a range, and possible none at all beyond that (or it's a plotting error insofar as x tends to ±infinity it has a very narrow range of y that is never sampled properly, but should connect to that pulse 'randomness'). If it's a plot of real vs imaginary components of a complex function to a different continuous value, I suspect someone is playing silly-buggers with multiple (perhaps nested?) trigonometric functions, polynomials and variable-shifted powers. But it's nearly thirty years since I did mathematics at the level needed to disentangle this neatly (back when Mandelbrots and Julias were still a staple wall-poster for any student not more into the likes of Iron Maiden skull-motifs or <insert your favourite classic film here>, and even then it might be) so don't ask me where to start. 16:48, 17 October 2021 (UTC)

To me the curve in panel three looks like a cursed (ha) mixture of an oscillatory time responses of dynamic systems with either an Nquist plot or simply trajectories of eigenvalues (of a stable system) at the end. Links: https://en.wikipedia.org/wiki/Nyquist_stability_criterion#Nyquist_plot , https://electronicscoach.com/time-response-of-second-order-system.html https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Domi (talk)Domi

Are there any examples of "cursed" math problems? I've seen "weirdly abstract" and "weirdly concrete" ones, but not "cursed" ones. 01:03, 17 October 2021 (UTC)

Some functions definitely make graphs that look weird to humans for reasons that are not immediately obvious (see sin(cos(tan(xy))) = sin(cos(tan(x))) + sin(cos(tan(y))): https://www.desmos.com/calculator/mt08x3yqxj). I suppose to be cursed in the sense I take here, it would have to be unsolved as to why it looks weird to humans, which is probably not the case in my example, but I imagine there are examples. 19:41, 20 October 2021 (UTC)

The symbol in the third panel looks like an unalome, which is not a mathematical symbol but a Buddhist or mystical one https://en.wikipedia.org/wiki/Yantra_tattooing#Types_and_designs

There is at least one paper on arxiv defining quasimonoid, 1401.7748. It's from 2014 so it existed long before the comic. -- 14:04, 17 October 2021 (UTC)

Links, please! Not all of us are mathematicians. If you mention something that you think the cursed curve might represent, please provide a link to something describing that something so the rest of us can read it and attempt to learn more. Shamino (talk) 17:44, 17 October 2021 (UTC)

That cursed squiggle reminds me of the zeta function: https://www.johndcook.com/blog/2019/11/29/near-zeros-of-zeta/

That cursed squiggle sure looks like the sort of thing that used to flow from Saul Steinberg's pen, as seen in the pages of the New Yorker back in the 60s. The most relevant example I can find right now is from 1965: https://fineartamerica.com/featured/new-yorker-february-20th-1965-saul-steinberg.html 03:38, 18 October 2021 (UTC)

So, is the middle one an actual unsolved problem? -- 16:07, 18 October 2021 (UTC)

In this comic ponytail is obviously not Dr. Adams. (Discussion came up two comics ago.) -- 21:42, 18 October 2021 (UTC)

Is Panel #2 a real unsolved problem? It reads like one.

Adding to the first comment - should we include an explanation of the formula in the first panel as well? The denominator means "is an element of aleph-5, the fifth infinite cardinal number". The numerator is less clear; a dot over a variable usually indicates a derivative, but I haven't seen a dot over a set. Raising R (set of real numbers) to the power of Z (set of integers) refers to the set of all functions from the integers to the reals. I don't recognize the diamond with a line through the bottom or the two arrows.

Shout-out to whoever wrote the incomplete tag. 20:21, 19 October 2021 (UTC)

Is there a reason why the article uses "millennium" (correct) and "millenium" (incorrrect)?

Maybe bad spelling by one or more editors? The spelling has now been fixed where it was incorrect. Ianrbibtitlht (talk) 02:20, 21 October 2021 (UTC)

I swear I recognize the Cursed Curve as the art from the story "The Theory that Jack Built", from "The Space Child's Mother Goose", by Frederick Winsor. Haven't been able to find my copy, so still not sure. Elkern (talk) 21:35, 25 October 2021 (UTC)

In the transcript, the first 3 characters of the equation in panel 1 are showing up as basic squares for me. When I go into editing mode, I see the correct rendering of the characters in the wikitext, just not on the page itself. Don't know if this might create/indicate a problem for screen readers. I'm viewing the page on Chrome 94, in Windows 10 version 1909. Dansiman (talk) 21:12, 26 October 2021 (UTC)

I made an attempt to explain Euler fields, manifolds, and hypergroups; I think I hit "understandable", but definitely not "concise". If they could be as concise as the other entries, though, I guess someone else would have written them before me. I'm not sure "meta-algebra" is actually a thing - metamathematics is the application of mathematical principles to the study of mathematics, and algebra is mathematical notation and symbology, so algebra about algebra would be... semiotics? Noaqiyeum (talk) 09:07, 20 May 2022 (UTC)