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Under a change of perspective definition uu g f = `gfg X under Y is the answer to the question what's the transformation that, when transformed by the morphism Y, becomes X? examples "setup" moves in cubing is performing an algorithm under those setup moves --- https://youtu.be/rZiDvDGHfe8&t=243 the under modifier in BQN --- https://mlochbaum.github.io/BQN/doc/under.html and https://youtu.be/Hecqsl3GG9s?t=8m57s solving "word problems" by translating them into equations, solving, then translating back using fourier transforms to filter out signal noise --- https://youtu.be/gAPlRlmhXyI?t=5m28s using Laplace transforms to solve differential equations --- https://youtu.be/1u5Z1oap1g8?t=6m1s conjugate elements in group theory --- https://en.wikipedia.org/wiki/Conjugacy_class#Definition similar matrixes in linear algebra --- https://en.wikipedia.org/wiki/Conjugacy_class#Definition a vertical flip under a quarter turn is a horizontal flip --- https://youtu.be/eLhsyAu7G-U&t=1125 the Enigma machine. and `f`, the reflector, is an involution, and thus so is `uu g f` duals. boolean › disjunction under boolean › negation is boolean › conjunction examples interesting thing I noticed: *`()(:-#)()`* is an arithmetic mean #todo id `(-)(:-#)(-)` is a harmonic mean `[.](:-#)/.\` is a geometric mean `\./(:-#)[]2` is a quadratic mean `\/3(:-#)[]3` is a cubic mean #todo math notation for cubic root *`()dd()`* is a derivative (gives the `m` in `x -> mx : b`) #todo id `(-)dd(-)` is a "reciprocal" derivative (gives the `a` in `x -> a -- x : b`) `[.]dd/.\` is an exponential derivative (gives the `b` in `x -> ab[x]`) define uu g f = `gfg. then, *`uu () (:-#)` is an arithmetic mean #todo id `uu (-) (:-#)` is a harmonic mean `uu /.\ (:-#)` is a geometric mean `uu []2 (:-#)` is a quadratic mean `uu []3 (:-#)` is a cubic mean `uu () dd`* is a derivative (gives the `m` in `x -> mx : b`) #todo id `uu (-) dd` is a "reciprocal" derivative (gives the `a` in `x -> a -- x : b`) `uu /.\ dd` is an exponential derivative (gives the `b` in `x -> ab[x]`)
© 2026 Emilien Breton ← 🍁 →

Under

a change of perspective

definition uu g f = `gfg

X under Y is the answer to the question what's the transformation that, when transformed by the morphism Y, becomes X?

examples

"setup" moves in cubing is performing an algorithm under those setup moves --- https://youtu.be/rZiDvDGHfe8&t=243

the under modifier in BQN --- https://mlochbaum.github.io/BQN/doc/under.html and https://youtu.be/Hecqsl3GG9s?t=8m57s

solving "word problems" by translating them into equations, solving, then translating back

using fourier transforms to filter out signal noise --- https://youtu.be/gAPlRlmhXyI?t=5m28s

using Laplace transforms to solve differential equations --- https://youtu.be/1u5Z1oap1g8?t=6m1s

conjugate elements in group theory --- https://en.wikipedia.org/wiki/Conjugacy_class#Definition

similar matrixes in linear algebra --- https://en.wikipedia.org/wiki/Conjugacy_class#Definition

a vertical flip under a quarter turn is a horizontal flip --- https://youtu.be/eLhsyAu7G-U&t=1125

the Enigma machine. and f, the reflector, is an involution, and thus so is uu g f

duals. boolean › disjunction under boolean › negation is boolean › conjunction

examples

interesting thing I noticed:

(*)(:-#)(*) is an arithmetic mean #todo id

(-)(:-#)(-) is a harmonic mean

[.](:-#)/.\ is a geometric mean

\./(:-#)[]2 is a quadratic mean

\/3(:-#)[]3 is a cubic mean #todo math notation for cubic root

(*)dd(*) is a derivative (gives the m in x -> mx : b) #todo id

(-)dd(-) is a "reciprocal" derivative (gives the a in x -> a -- x : b)

[.]dd/.\ is an exponential derivative (gives the b in x -> ab[x])

define uu g f = `gfg. then,

uu (*) (:-#) is an arithmetic mean #todo id

uu (-) (:-#) is a harmonic mean

uu /.\ (:-#) is a geometric mean

uu []2 (:-#) is a quadratic mean

uu []3 (:-#) is a cubic mean

uu (*) dd is a derivative (gives the m in x -> mx : b) #todo id

uu (-) dd is a "reciprocal" derivative (gives the a in x -> a -- x : b)

uu /.\ dd is an exponential derivative (gives the b in x -> ab[x])