Editing 2529: Unsolved Math Problems
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==Explanation== | ==Explanation== | ||
+ | {{incomplete|Created by COLLATZ'S COLLAPSIBLE CONJECTURE CURVE (NON-INJECTIVE) - Please change this comment when editing this page. Finish explaining the "weirdly abstract" section. Do NOT delete this tag too soon.}} | ||
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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. | 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 | + | 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; for example, there are very many problems related to packing objects into spaces that are very difficult to solve although quite easy to state, such as the {{w|Collatz conjecture}}. 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. | 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:''' | + | *'''Euler field:''' |
− | *'''{{w|Manifold}}:''' | + | *'''{{w|Manifold}}:''' |
− | *'''{{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 | + | *'''{{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 a structure with 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". | *'''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. | *'''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:''' | + | *'''Meta-algebra:''' |
− | *'''ϵ<0:''' | + | *'''ϵ<0:''' a joke about how 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}}) | *'''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." | *'''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." | ||
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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?) | 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 [ | + | 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. | 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. | ||
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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. | 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. | ||
− | + | In the title text, the curve in the final panel is further explained based on the consensus of supposedly a group who has studied it and the procedure that generates it, commenting that "it's just like that" as their conclusion, which is really not an explanation at all. | |
==Transcript== | ==Transcript== | ||
+ | {{incomplete transcript}} | ||
The Three Types Of Unsolved Math Problem | The Three Types Of Unsolved Math Problem | ||
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[[Category: Comics featuring Ponytail]] | [[Category: Comics featuring Ponytail]] | ||
[[Category: Math]] | [[Category: Math]] | ||
− | [[Category:Comics | + | [[Category:Comics featuring cursed items]] <-- Some presumedly mathematical curve --> |