Advanced Techniques
Advanced Techniques
[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

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
