🔺 Triangle Symmetry Explorer
Watch how flips and rotations move the vertices — and discover why this forms a mathematical group
How to Use This Explorer
The triangle has three colored vertices labeled 1, 2, and 3 that move when you apply transformations. The gray position labels (P1, P2, P3) stay fixed — they mark the three corners so you can always see which vertex ended up where.
Flips always go through a fixed position, not a moving vertex. “Flip thru P1” always means flip through the top axis, no matter which vertex is currently sitting there. Click any button to apply a transformation, and use Reset to return to the starting arrangement.
🔍 What to Discover
As you explore, look for answers to these questions.
🎯 Identity
Which transformation leaves every vertex at its starting position? What does it look like when “nothing happens”?
🔄 Inverses
For each transformation, can you find another one that “undoes” it and returns all vertices to their starting positions? What undoes a 120° rotation?
🔒 Closure
When you combine any two transformations, do you always get one of the 6 symmetries? Or can you create an arrangement that isn’t on the list?
🪞 Does Order Matter?
Try Rotate 120° then Flip thru P1. Now try Flip thru P1 then Rotate 120°. Same result? What does this tell you about commutativity?
🔗 The Rotation Subgroup
What happens if you ONLY use rotations (no flips)? Do rotations by themselves form their own closed system? How many rotations are there, and what structure do they have?
💡 Self-Inverses
Try applying the same flip twice in a row. What happens? Do any rotations work this way too? An element that is its own inverse is called a self-inverse or involution.
💡 The Big Hint
The three rotations by themselves have the exact same structure as something you’ve already seen — the 3-color Cayley table from the isomorphism lesson, and adding numbers mod 3. But when you add the flips in, something new and surprising happens: order starts to matter. That’s the key discovery waiting for you in the Cayley Table Builder!
Next Steps
📖 The 6 Symmetries Reference
See all 6 symmetries at a glance with animated demonstrations and learn the naming convention.
🔬 D₃ Cayley Table Builder
Ready to map every combination? Build the complete 6×6 composition table and uncover the full group structure.