2595: Advanced Techniques

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Advanced Techniques
A blow from Emmy's Cutlass of Variations will transport the dragon to a corresponding symmetric position in the Noetherworld.
Title text: A blow from Emmy's Cutlass of Variations will transport the dragon to a corresponding symmetric position in the Noetherworld.


In typical Miss Lenhart fashion, she is teaching a mathematics class where she outlines a process by which a mathematical result is achieved through steps which sound suspiciously like magical RPG logic. She includes both a dragon and arrows to slay it.

One of her students asks if this is a metaphor for the technique, but her rather tetchy reply "Does this look like English class?!" seems to imply that she literally means that dragons and arrows will be employed in the resolution of the problem. It is also clear from the slide she is pointing at that she has drawn a dragon and a man with a bow that is aiming an arrow at the dragon. Whilst metaphor is an important part of many languages, and so is definitely taught in language classes, it is not usually used in math classes.

The caption beneath the comic states that this approach describes "All advanced math techniques." This could be a reference to the now-common approach in higher mathematics in which a problem is transformed into another domain where it is easier to solve, then transformed back. For instance, in Fourier analysis, commonly used for analyzing the behavior of signals or dynamical systems, a problem can be transformed from the time domain to the frequency domain, solved, and then transformed back again. A (much) more complex example is Andrew Wiles's proof of Fermat's Last Theorem, which uses modularity lifting to transform the problem. Here Miss Lenhart says she will transform a math problem into an actual dragon, slay it, and transform the corpse back into mathematics.

An alternative view is that Randall is referring to Arthur C. Clarke's third law that Any sufficiently advanced technology is indistinguishable from magic, as re-framed for mathematics. What Randall would be implying is that all advanced math techniques look like magic to non-mathematicians. (Another advanced and somewhat magical math technique is deployed by Miss Lenhart in 1724: Proofs.)

Invocations are a common classification for spoken or vocalized types of spell. In the logic Miss Lenhart used, 'invoking' Gauss's operator may refer to casting a magical spell with verbal components (such as True Polymorph). The operator is presumably named after the famous German mathematician Carl Friedrich Gauss. There is nothing on Wikipedia called Gauss's operator, but there is both Gauss's law and the Gauss–Kuzmin–Wirsing operator. As neither can transform an equation into a dragon,[citation needed] it's clear Randall is making a joke.

Slaying the dragon with Hilbert's arrow indicates that the arrow has some magical properties. The arrow is presumably named after David Hilbert, known for many mathematical developments including Hilbert's problems and Hilbert spaces. A Hilbert space converts subsets of an infinite vector space into a complete metric space, allowing the use of linear algebra and calculus methods which might otherwise be applicable only to finite Euclidean spaces. Vectors could be compared with an arrow. Magical arrows are frequently used to slay dragons in myth and role-playing games. Magical items in RPGs such as Dungeons & Dragons are often named after a creator or famous user; hence, a magical "Arrow of Hilbert" might traverse infinite spaces or affect targets for which one or more stats are effectively infinite.

There is in fact a class of Dragon curves, which do have the sort of S-shape shown on the whiteboard, but they have no connection to Gauss's operator, and are not actual dragons that need slaying.

The title text contains two puns and a reference. The phrase "Cutlass of Variations" is a pun on the mathematical technique called "Calculus of variations". The word "Noetherworld" is a pun on "netherworld". The reference is to the mathematician Emmy Noether, a giant in the field of abstract algebra which, through more of Ms. Lenhart’s questionable transformations, may become an actual giant in a field of abstract algae bras. Furthermore, Noether's Theorem is used in the Calculus of Variations. She was previously referenced as one of many important women in science back in 896: Marie Curie.


[Miss Lenhart is using a stick to point at a whiteboard behind her while facing, presumably, a crowd of off-panel students. The white board has a drawing of a snake-shaped dragon with wings, flying with it's body in an S-shape. An archer is pointing an arrow up at the dragon above him. Above the drawings there are three and below two rows of unreadable text and equations.]
Miss Lenhart: To solve this equation, we invoke Gauss's operator to transform it into a dragon.
Miss Lenhart: Then we slay the dragon with Hilbert's Arrow, and transform its corpse back into the solution.
Off-panel voice: Just to be clear, this is a metaphor, right?
Miss Lenhart: Does this look like English class?!
[Caption below the panel:]
All advanced math techniques

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The title text refers to Noether's theorem. Trimeta (talk) 04:24, 19 March 2022 (UTC)

This is my first explanation GcGYSF(asterisk)P(vertical line)e (talk) 05:41, 19 March 2022 (UTC)

This sounds a lot like Laplace or Fourier transforms, converting a function into a different where it is easier to manipulate then reversing the transformation. 06:28, 19 March 2022 (UTC)

When I was learning to use fourier transforms in EE, they were very straightforwardly (and accurately) described as "transferring the function to the Spectral Domain". 22:45, 19 March 2022 (UTC)

I'm not sure that it's proper to refer to someone as a "giant" while explaining a comic that references mythological creatures. Unless it was literal of course, but as far as I'm aware giants never existed. 11:28, 19 March 2022 (UTC)

I think she may actually have been a wizard-giant.Iggynelix (talk) 13:41, 19 March 2022 (UTC)

Not my area, but I am passingly familiar with the Gauss–Kuzmin–Wirsing Operator, Dragon Curves, and Hilbert Spaces (guessing that the "arrow" refers to scalar vector?). Some type of iterative/recursive conversion that yields to analysis of the period? Probably not pertinent to the joke which is more about the fanciful names attached to mathematical concepts, constructs, and processes 11:53, 19 March 2022 (UTC)

I find it interesting that despite now being the day after release (or well into the next day, my time, which is usually sufficient — and I'm not in a DST zone yet) the site explanation hasn't explained (or thought it has explained) every single element of the in-comic 'explanation' — even if not established the (probably) nonsensical whole. As an example, I don't yet see the obvious dragon element that is both alluded to and seemingly illustrated upon the board-notes. Leaving this here to help near-future editors who might have time to bullet-point/tabulate/sub-heading these things and just need that extra bit of info. 15:01, 19 March 2022 (UTC)

There is the misquote of Arthur Clarke "All sufficiently advanced [strike]technologies[/strike] mathematical techniques are indistinguishable from magic." Arachrah (talk)

Fix it!
ProphetZarquon (talk) 23:28, 19 March 2022 (UTC)
I'm not sure what Arachrah means, because technologies is included in the original version: Any sufficiently advanced technology is indistinguishable from magic. I have corrected the explanation to that and also added link to wiki. --Kynde (talk) 08:25, 21 March 2022 (UTC)

The explanation should decide whether the teacher is Miss Lenhart, or Blondie. I think it's Miss Lenhart. Nitpicking (talk) 17:02, 19 March 2022 (UTC)

A teaching Blondie is always Miss Lenhart. It has been corrected before I came here. --Kynde (talk) 08:25, 21 March 2022 (UTC)

That dragon looks suspiciously like Trogdor...

It just looks like a normal wyvern to me, though the perspective doesn't give us much detail to help tell those two cases apart. I think if it were Trogdor though, fewer liberties would have been taken with the shape of the dragon's body. (To be confident we would have to figure out the original problem and use Gauss' operator ourselves to get a more detailed look, which seems difficult.) 16:25, 20 March 2022 (UTC)

"Critical Role: Call of the Netherdeep" released this week, for D&D 5e. ProphetZarquon (talk) 23:28, 19 March 2022 (UTC)

Come to think of it, we do use fantasy-sounding expressions in math: e.g. titanic prime, imaginary part, infinite field, ideals, friendly numbers, brute force attack. I'm pretty sure there are many more fun examples. Yosei (talk) 04:16, 20 March 2022 (UTC)

PS: "Sexy primes" and "latus rectum" are real technical terms. Yosei (talk) 04:16, 20 March 2022 (UTC)

Its weird how this fantastic math have failed to solve the 3n+1 problem. Because I do believe I have solved it within this week. 18:31, 20 March 2022 (UTC)

I'm glad the wiki format saves old versions of explanations, because it would be a shame if that incomplete notice would be gone forever once the explanation is complete enough. Made me chuckle! 08:23, 21 March 2022 (UTC)

I suspect there's also an aspect of how, if you don't know the mathematical concepts involved, some of these solution methods can seem more like the author is just casting spells. The context that most immediately comes to me is solving integrals with weird techniques that involve mapping to other planes and such. I would say that solving integrals was the first place I really saw creativity being heavily focused on in my math curriculum. Trlkly (talk) 08:43, 21 March 2022 (UTC)

I couldn't agree more. (talk) (please sign your comments with ~~~~)
I agree as well. "An alternative view" seems like the wrong way to state this: I believe the entire joke is that Randall is comparing the processes described in the preceding paragraph (transforming a function to another domain & such), to the "sufficiently advanced technology" of Clarke's "third law". It's not an either/or proposition: The references to advanced maths are there, to illustrate how fine the line is between complex operations, & "magic"; & the D&D metaphors are there, to bring the "magic" into a context that sounds more structured & math-like, than some arbitrary 'hocus pocus'.
ProphetZarquon (talk) 22:33, 22 March 2022 (UTC)
Agree as well. Reminds me of the Langlands Programme. Guess Randall has been reading that article, too.-- 16:27, 23 March 2022 (UTC)

Re. '[metaphor] is not usually used in math classes.' - it's used a lot more than you might initially assume - there's at least one example in this explanation, where it talks about transforming between 'domains'. 09:18, 22 March 2022 (UTC)

I had an entirely different take. It is not unknown for mathematicians to use whimsical names. In a new field whimsical names are common, because the usual suspects are taken. It is entirely possible to have well defined mathematical objects called dragons and corpses and an operator called Hilbert's arrow. Ms. Lenhart could be giving a dry description of a mathematical technique using the language common to the field. 22:26, 27 March 2022 (UTC)

I agree with the above comment about whimsical names: see for example the Ham Sandwich Theorem. 20:17, 5 December 2022 (UTC)

Plus one to this. Some more examples: in abstract algebra, one speaks about "annihilators" acting on subspaces or rings. My advanced linear algebra professor would routinely refer to minimal polynomials "killing a matrix/linear operator." So it seems perfectly reasonable to "slay" the "dragon" as in the comic, so long as those terms refer to properly-defined operations and objects! I think such expressive terms can help mathematicians convey a tangible intuition for what is happening; they see the interplay of abstract mathematical objects as a real-life dance or drama. Also- there's a whole subfield of geometric topology called Surgery Theory :) 11:42, 17 January 2023 (UTC)