Editing 2529: Unsolved Math Problems

{{comic
| number    = 2529
| date      = October 15, 2021
| title     = Unsolved Math Problems
| image     = unsolved_math_problems.png
| titletext = After decades of studying the curve and the procedure that generates it, the consensus explanation is "it's just like that."
}}

==Explanation==
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 {{w|millennium problems}}. On the other hand, many unsolved problems are very concrete, such as the {{w|Collatz conjecture}}. Additionally, there are the many problems related to {{w|Packing problem|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 {{w|Leonard Euler}} was a prolific Swiss mathematician who influenced {{w|List_of_things_named_after_Leonhard_Euler|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.
*'''{{w|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!)".
*'''{{w|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:''' {{w|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:''' {{w|Kurt Gödel}} was a 20th-century mathematician who studied logic and philosophy (he's most well known for {{w|Gödel's incompleteness theorems}}) and {{w|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 {{w|Metamathematics}}.
*'''ϵ<0:''' Another joke term. In analysis, {{w|ϵ}} is usually defined to be an arbitrarily small ''positive'' number.
*'''quasimonoid:''' A [[Malamanteau|malamanteau]], combining the prefix "quasi" (meaning "partially" or "seemingly") and "monoid" (an object from group theory) and is probably meant to evoke the character {{w|Quasimodo}} from ''The Hunchback of Notre-Dame'' (although quasimonoids are a type of algebraic object, namely a non-associative {{w|monoid}})
*'''Sondheim Calculus:''' This refers to {{w|Stephen Sondheim}}, one of the most successful composers and lyricists of American musical theatre -- the producer of his musical "Into the Woods" once [https://www.indiewire.com/2015/01/watch-singing-sondheim-is-like-calculus-in-into-the-woods-behind-the-scenes-video-exclusive-189507/ 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 {{w|Hodge conjecture}}, a {{w|Millennium Prize Problems|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 {{w|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 {{w|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: [https://colab.research.google.com/drive/1nWrByCGBckwVdbAwow7tCYTOvqObYXyR?usp=sharing]. C++ code simulating this situation can be found here: [https://github.com/AMindToThink/WeirdlyConcreteXKCD].

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 {{w|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 {{w|Mandelbrot set}}), it could well be. For example the unsolved {{w|Riemann hypothesis}}, another Millennium Prize problem, concerns the properties of {{w|File:RiemannCriticalLine.svg|a weird and at-first-glance random curve}}.  In number theory, the term "cursed curve" [https://www.quantamagazine.org/mathematicians-crack-the-cursed-curve-20171207/ has been used] to describe the [https://annals.math.princeton.edu/wp-content/uploads/annals-v173-n1-p13-s.pdf "split Cartan" modular curve] of level 13, which resisted attempts for many years to compute its [https://www.jstor.org/stable/10.4007/annals.2019.189.3.6 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 {{w|monster group}} ([https://youtu.be/mH0oCDa74tE explanation video]) is known to be the largest of a category of objects called {{w|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."

==Transcript==
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]

{{comic discussion}}
[[Category: Comics featuring Cueball]]
[[Category: Comics featuring Megan]]
[[Category: Comics featuring Ponytail]]
[[Category: Math]]
[[Category:Comics with cursed items]] <!-- Some presumedly mathematical curve -->