Symmetry as a transformation

A symmetry rearranges an object while preserving its rules.

Start with a concrete example

Rotate a square by 90°: its four corners trade places, but the whole square stays the same. A symmetry preserves the relevant structure while it moves parts.

A small check: try the idea

Think with the concrete example above. You can try again.

After rotating a square by 90°, what is preserved?

Read the answer and reasoning

The whole square, although corners exchange places. A symmetry can move individual parts while preserving the object’s relevant structure. Mark one corner to see it move.

Build the idea in three steps

A symmetry is an allowed transformation that keeps specified rules intact. A square can be rotated without losing its distances.
In arithmetic, the relevant transformations preserve addition and multiplication, rather than just a picture.
A part is stable if every allowed transformation keeps its elements inside that part.

A small example

Rotating a square moves a corner to another corner. Stability asks whether an entire chosen subset is preserved by every allowed move.

Lean definition · original excerpt

FLT · aa2d8b3 · Def_FLTPrelim_GaloisRep.lean

68def IsGaloisStable {n : ℕ}69    (N : Submodule (ZMod n) (Submodule.torsionBy ℤ (W'⁄K).Point n)) : Prop :=70  ∀ (σ : K ≃ₐ[S] K), ∀ x ∈ N, σ • x ∈ N

How to read this Lean

∀ σ means every allowed arithmetic symmetry. x ∈ N says x starts inside N; σ • x ∈ N says the transformed point stays inside.

Original source ↗ · L68–70 Open the complete saved source

Where this idea is used

6. The scissors incident 8. The secret duet 9. Two doors to the concert 10. A three-note audition 11. Borrow a better passport 12. The workshops agree 13. Pip’s impossible quilt 14. The shrinking reservation 15. The concert of absolute silence 16. The room with no room

Symmetry as a transformation

Start with a concrete example

A small check: try the idea

Build the idea in three steps

A small example

Lean definition · original excerpt

How to read this Lean

Where this idea is used

Mika’s help desk

One thing at a time

In simpler words

A small clue