872: Fairy Tales
Title text: Goldilocks' discovery of Newton's method for approximation required surprisingly few changes.
Eigenvectors are a mathematical concepts that can be applied to a matrix. A matrix is mostly displayed as an rectangular array of elements used to describe the state of objects in physics. In pure mathematics they can be much more complex. The most important issue to the understanding of the comic is that a matrix can be transformed through various processes. These transformations can include rotation, movement and scaling of the object described by the matrix. An eigenvector refers to elements of the vector space of the matrix which remain unchanged (except possibly being scaled to be longer or shorter) after the transformation is applied. The prefix 'eigen-' applied to the term is adopted from the German word eigen for "self-" or "unique to", "peculiar to", or "belonging to." As the eigenvector remains unchanged through the transformation of the matrix it can be used to describe something unique about that matrix.
The story of Cinderella includes Cinderella going to a ball in disguise, dancing with a prince and then leaving early and quickly, so that she accidentally leaves a glass slipper behind. The prince then uses the shoe to find Cinderella. Megan says that the way she learned it, the prince used an eigenvector and corresponding eigenvalue to match the shoe to its owner. This is a somewhat logical mathematical connection to make as eigenvectors, unchanged properties of mathematical matrices that may allow for mathematical identification of the changed matrix, correspond to the unchangeable property of the shoe (size) that allowed the prince to correctly identify the owner of the shoe even after the shoe was misplaced. Eigenvectors are sometimes used in facial-recognition software to match 2 faces.
Megan explains that her mother would talk about her work, math, while she fell asleep in the midst of reading bed time stories. The middle panel refers to the story of The Ant and the Grasshopper with the addition of what is likely a reference to the Poincaré conjecture, a (now-misnamed) theorem in mathematics. Megan also mentions two other story changes. Inductive White and the (n−1) Dwarfs is a combination of Snow White and the Seven Dwarfs with the principle of induction, and The limx→∞(x) Little Pigs combines the Three Little Pigs with mathematical limits. It "got weird toward the end" because the number of pigs tends to infinity as the story progresses. Each of the stories has a varied degree of similarity to the mathematical concepts that were mixed in as though the professor began to talk about a mathematical principle that may have been brought to mind while reading the story or already on her mind.
In the title text, Newton's method for approximation is a method for finding successively better approximations to the zeroes (or roots) of a real-valued function. In Goldilocks, the protagonist finds successively better porridge and comfier chairs in a house where three bears lived. In the same way, in the Mom's version of the fairy tale, she would find successively better approximations to zeroes instead of successively better bowls of porridge.
- [Megan sits in an armchair, reading a book.]
- Megan: Are there eigenvectors in Cinderella?
- Cueball: ...no?
- Megan: The prince didn't use them to match the shoe to its owner?
- Cueball: What are you talking about?
- Megan: Dammit.
- [Megan is in bed, mom is sitting on the edge of the bed reading.]
- My mom is one of those people who falls asleep while reading, but keeps talking. She's a math professor, so she'd start rambling about her work.
- Mom:But while the ant gathered food ...
- Mom:...the grasshopper contracted to a point on a manifold that was not a 3-sphere...
- I'm still not sure which versions are real.
- Cueball: You didn't notice the drastic subject changes?
- Megan: Well, sometimes her versions were better. We loved Inductive White and the (n−1) Dwarfs.
- Megan: I guess The limx→∞(x) Little Pigs did get a bit weird toward the end...
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