63. Chain Complexes of Leftovers
Here's a thought that's been rattling around my head for the last decade or so: leftovers - as in, leftover food - obey chain complex rules. Let me explain.
One thing you can do with leftover food rather than just eat it is to use it as an ingredient - the base for a new meal, remixing it to some extent. Turn a hamburger patty into ground meat for a casserole, or use the carrots and celery from a crudite tray as the start of a mirepoix. There's numerous patterns and a whole art to it - for one, fried rice, French toast, and Spanish omelettes are all exemplars of one excellent pattern: "take slightly stale starch, apply some egg, some seasoning, and maybe some fresh produce". (It's no accident that the French for "French toast" literally translates as "lost bread".) There's only a few techniques, and you can learn most of them and start experimenting if you care about avoiding food waste as much as I do: a few others include "use additional fresh produce, especially alliums", "season heavily", and "feel free to combine multiple leftovers into a single new dish". But one thing to avoid at pretty much all costs is to take the remains of a dish that you've made using leftovers as a base and using it in turn as the base for new leftover-cooking. The result will almost invariably be unpalatable, at least without applying enough new ingredients to make it nowhere near worth using leftovers.
Probably this is mostly because I'm a huge nerd, but this reminds me a lot of chain complexes from abstract algebra. Let \(\{G_i\}_{i \in I}\) be an indexed family of groups, where the indexing set has a natural ordering; one example might be that there's actually only four groups, so that \(I = \{1, 2, 3, 4\}\). Then for each \(i \in I\) that's not the last one, let \(\phi_i : G_i \to G_{i+1}\) be a homomorphism - a structure-preserving map, in the sense that if we take two elements \(g, h \in G_i\) and want to look at what happens when we apply the function to the result of combining them under the binary operation of \(G_i\), it doesn't matter whether we combine them first and then apply \(\phi_i\) or if we instead apply \(\phi_i\) to each element first and then combine the results inside of \(G_{i+1}\) instead: that is, \(\phi_i(g \cdot h) = \phi_i(g) \cdot \phi_i(h)\). (One simple example of a homomorphism is the map from \((\mathbb{Z}, +)\) to \((\mathbb{Z}, +)\) given by multiplication by 2. Convince yourself that it doesn't matter when we multiply by 2 - before else after adding together. Another is the exponential map from \((\mathbb{R}, +)\) to \((\mathbb{R}^+, \times)\).) We call such a series of groups and homomorphisms a chain complex if whenever we pick an element in any of the groups that's not in the last or next-to-last group and apply the appropriate maps twice, we always get the identity element: \(\phi_{i+1} \circ \phi_i(g) \equiv 0\).
I've glossed over some complexity here, because we'd probably do better thinking of choices of meal as monoids and not groups, where the identity element of the monoid is "anything inedible, including the empty dish" and the binary operation is "put the two dishes next to each other and eat what you want to", so the punchline is that we have a (monoid) homomorphism from foods to foods given by "cook the leftovers into a dish". ...OK, sure, that only sort of holds up, given how little control we have over the nature of the homomorphism. Still! Don't cook up leftover-based dishes into new dishes; just let it go.
One thing you can do with leftover food rather than just eat it is to use it as an ingredient - the base for a new meal, remixing it to some extent. Turn a hamburger patty into ground meat for a casserole, or use the carrots and celery from a crudite tray as the start of a mirepoix. There's numerous patterns and a whole art to it - for one, fried rice, French toast, and Spanish omelettes are all exemplars of one excellent pattern: "take slightly stale starch, apply some egg, some seasoning, and maybe some fresh produce". (It's no accident that the French for "French toast" literally translates as "lost bread".) There's only a few techniques, and you can learn most of them and start experimenting if you care about avoiding food waste as much as I do: a few others include "use additional fresh produce, especially alliums", "season heavily", and "feel free to combine multiple leftovers into a single new dish". But one thing to avoid at pretty much all costs is to take the remains of a dish that you've made using leftovers as a base and using it in turn as the base for new leftover-cooking. The result will almost invariably be unpalatable, at least without applying enough new ingredients to make it nowhere near worth using leftovers.
Probably this is mostly because I'm a huge nerd, but this reminds me a lot of chain complexes from abstract algebra. Let \(\{G_i\}_{i \in I}\) be an indexed family of groups, where the indexing set has a natural ordering; one example might be that there's actually only four groups, so that \(I = \{1, 2, 3, 4\}\). Then for each \(i \in I\) that's not the last one, let \(\phi_i : G_i \to G_{i+1}\) be a homomorphism - a structure-preserving map, in the sense that if we take two elements \(g, h \in G_i\) and want to look at what happens when we apply the function to the result of combining them under the binary operation of \(G_i\), it doesn't matter whether we combine them first and then apply \(\phi_i\) or if we instead apply \(\phi_i\) to each element first and then combine the results inside of \(G_{i+1}\) instead: that is, \(\phi_i(g \cdot h) = \phi_i(g) \cdot \phi_i(h)\). (One simple example of a homomorphism is the map from \((\mathbb{Z}, +)\) to \((\mathbb{Z}, +)\) given by multiplication by 2. Convince yourself that it doesn't matter when we multiply by 2 - before else after adding together. Another is the exponential map from \((\mathbb{R}, +)\) to \((\mathbb{R}^+, \times)\).) We call such a series of groups and homomorphisms a chain complex if whenever we pick an element in any of the groups that's not in the last or next-to-last group and apply the appropriate maps twice, we always get the identity element: \(\phi_{i+1} \circ \phi_i(g) \equiv 0\).
I've glossed over some complexity here, because we'd probably do better thinking of choices of meal as monoids and not groups, where the identity element of the monoid is "anything inedible, including the empty dish" and the binary operation is "put the two dishes next to each other and eat what you want to", so the punchline is that we have a (monoid) homomorphism from foods to foods given by "cook the leftovers into a dish". ...OK, sure, that only sort of holds up, given how little control we have over the nature of the homomorphism. Still! Don't cook up leftover-based dishes into new dishes; just let it go.
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