📖 The 6 Symmetries of a Triangle
A visual reference for the D₃ symmetry group — with animated demonstrations
How to Use This Reference
Each card below shows one of the 6 symmetries of an equilateral triangle — 3 rotations and 3 flips. Click the play button ▶ on any card to watch the identity triangle transform into that symmetry’s result. The position labels (P1, P2, P3) stay fixed while the colored vertices (1, 2, 3) move to their new positions.
Every symmetry is named by the arrangement it produces from the starting position. This works because each of the 6 possible arrangements is created by exactly one action — so the picture serves as a “name tag” for the action.
🔍 Key Ideas
Things to notice as you study the 6 symmetries.
🔄 The 3 Rotations
The identity (e) does nothing. Rotate 120° clockwise (r) cycles all three vertices one position. Rotate 240° clockwise (r²) cycles them the other way. Notice that r and r² are inverses of each other!
🪞 The 3 Flips
Each flip goes through one fixed position. The vertex at that position stays put while the other two swap. Every flip is its own inverse — do it twice and you’re back where you started!
🏷️ Why Names = Arrangements
A symmetry is an action, not a picture. But since each action produces a unique result from the identity, we can use that result as the action’s name. The arrangement IS the name tag.
🧩 How Composition Works
When we write a ∘ b, we do b first, then a. The result is whichever single action from the start would produce the same final arrangement. See the walkthrough at the bottom of the reference card.
🔗 The Big Picture
These 6 symmetries form a group — a set of elements with an operation (composition) that satisfies closure, associativity, identity, and inverses. This is the same kind of structure you explored with colored circles in the algebra worksheets, but with one crucial difference: this group is not commutative. The order you combine rotations and flips actually changes the result!
Companion Tools
🔺 Triangle Symmetry Explorer
Apply transformations interactively and watch the vertices move in real time.
🔬 D₃ Cayley Table Builder
Map every combination of symmetries and build the complete 6×6 composition table.