🔑 The Naming Rule
A symmetry is an action — a rotation or a flip — not a picture. But every action produces a unique result when you start from the identity position. So we name each action by the arrangement it creates.
Why does this work? Each of the 6 possible vertex arrangements is produced by exactly one action from the start. The arrangement and the action that creates it are two views of the same thing — so the picture serves as a "name tag" for the action.
Click the play button ▶ on any card below to watch the action transform the identity triangle into its named result.
🧩 How Composition Works
When the Cayley table says r ∘ f₁ = f₃, here's what that means:
1
Start at the identity: vertex 1 at top, 2 at lower-left, 3 at lower-right.
2
Do the first action (f₁): Flip through the top position. Now it's [1, 3, 2].
3
Do the second action (r): Rotate 120° clockwise from where you are. Now it's [2, 1, 3].
4
Name the result: Which single action from the identity produces [2, 1, 3]? That's f₃. So r ∘ f₁ = f₃.
The result is always named by asking: "What single action from the start would have gotten me to this same arrangement?"