🔬 D₃ Cayley Table Builder
Build the complete composition table for triangle symmetries — and discover why order matters!
How It Works
In the Composition Calculator below, you’ll pick two symmetry actions (rotations and flips). The tool shows you the result — which arrangement you’d get if you did both actions in sequence. That result fills in one cell of the Cayley table.
The composition a ∘ b means: “start at the identity, do action b first, then do action a.” The result triangle shows you which single action from the start would produce the same arrangement — and that action’s name is the answer.
🔍 What Did You Discover?
After completing some cells, think about these questions.
🎯 The Identity Row & Column
Fill in the entire first row and first column. What pattern do you see? Why does the identity element (e) leave everything unchanged?
🪞 Symmetry Across the Diagonal?
In your earlier worksheets, the Cayley table was symmetric across the diagonal. Is THIS one? Try r ∘ f₁ and then f₁ ∘ r. What does this tell you about commutativity?
🔄 The Rotation Subgroup
Click “Show Rotation Subgroup” to highlight the 3×3 block of just rotations. Does it remind you of anything from the earlier Cayley table worksheets? It has the same structure as adding mod 3 — that’s isomorphism!
✅ The Latin Square Property
Does every row contain each color exactly once? Does every column? This is called the Latin square property, and it’s guaranteed for any group table. Can you figure out why?
🔗 The Big Connection
The three rotations {e, r, r²} form a group with the exact same structure as the 3-color Cayley table from the isomorphism lesson — and the same structure as adding numbers mod 3. Same structure, different labels. That’s isomorphism!
But the FULL group of 6 symmetries is something genuinely new: it’s not commutative. With rotations and flips mixed together, the order you do things in actually matters. This is your first encounter with a non-abelian group — a concept that shows up throughout advanced mathematics and physics.
Companion Tools
🔺 Triangle Symmetry Explorer
Watch how each rotation and flip moves the vertices. Use this tool to understand the actions before building the table.
📖 The 6 Symmetries Reference
Visual reference showing all 6 triangle symmetries with animated demonstrations. Learn why each name matches each arrangement.