Wednesday, October 28, 2026
The Invariant
Identity emerges from invariance across allowable transformations. An interactive explainer: rotate a glyph across the identity boundary (X stays X; b becomes q), drag the complex-plane orbit, turn the dihedral group D₄, climb the Erlangen hierarchy from rigid to similarity to af
Ten things
- Primary visual cortex (V1) simple cells behave like Gabor filters — a grating under a Gaussian window — sharply tuned to orientation but sensitive to phase (where the light/dark bands fall).
- Representation theory says it decomposes uniquely into four irreducible pieces, one per character: under a 90° turn r , each piece is simply multiplied by 1 , i , −1 , or −i — the very numbers this whole site is coloured by.
- Slide the vocal-tract length: in raw Hz the whole vowel cloud slides and spreads; switch to normalized and every speaker collapses onto one template. (The same scaling move as the similarity group in §05.)
- Refine this mesh however you like — subdivide edges, add chords, drop in interior points — and the Euler characteristic χ = V − E + F = 2 never budges.
- The §05 ladder — rigid, similarity, affine, projective — is nothing next to the full freedom of topology , where every continuous stretch and bend is allowed.
- X has a 180° rotational symmetry, so it lands back on itself — invariant. b does not: at 180° it becomes q , a different letter with a different meaning.
- The orbit–stabilizer theorem says their sizes always multiply to |G| = 8 — the symmetry you keep and the symmetry that moves you are exact complements.
- The boards you can clear are exactly the column space of the 25×25 toggle matrix over GF(2) , which has rank 23: only one board in four is solvable.
- Emmy Noether proved that each continuous symmetry of a system's dynamics forces something to stay constant: rotation-invariance ⇒ angular momentum , time-invariance ⇒ energy , shift-invariance ⇒ momentum .
- Press mode plays the game (always stays solvable); switch to Paint to flip a single light and build a board by hand — three times out of four you'll land outside the subspace, and a hidden invariant (a “quiet pattern”) refuses it.
Five questions
Which term completes this sourced statement? “Primary visual cortex (V1) simple cells behave like Gabor filters — a grating under a _____ window — sharply tuned to orientation but sensitive to phase (where the light/dark bands fall).”
Which term completes this sourced statement? “Slide the _____ length: in raw Hz the whole vowel cloud slides and spreads; switch to normalized and every speaker collapses onto one template. (The same scaling move as the similarity group in §05.)”
Which term completes this sourced statement? “The §05 ladder — rigid, _____, affine, projective — is nothing next to the full freedom of topology , where every continuous stretch and bend is allowed.”
Which term completes this sourced statement? “The orbit–stabilizer theorem says their sizes always multiply to |G| = 8 — the symmetry you keep and the symmetry that moves you are exact _____.”
Which term completes this sourced statement? “Emmy _____ proved that each continuous symmetry of a system's dynamics forces something to stay constant: rotation-invariance ⇒ angular momentum , time-invariance ⇒ energy , shift-invariance ⇒ momentum .”