Unit 5 · Symmetry Lab
The Mathematics of Flips & Turns
A triangle has exactly 6 symmetries — 3 rotations and 3 flips. Combine any two, and you always get one of the 6 back. That closed little universe is called a group, and you're about to hold one in your hands.
🔺 Tool 1: Triangle Symmetry Explorer
Watch the vertices move as you apply rotations and flips. Discover identity, inverses, closure — and the big shock: order matters! Flip-then-rotate is not rotate-then-flip.
🚀 Open the Explorer (start here)🧮 Tool 2: D₃ Cayley Table Builder
Build the complete 6×6 table of triangle symmetries with the composition calculator — all 36 cells. Somewhere inside it, a smaller group is hiding. Can you find the rotation subgroup?
🧮 Open the Table Builder🧭 The suggested path
- Step 1: In the Explorer, try all 6 transformations. Then test whether order matters: apply two moves, reset, apply them in reverse. Same result?
- Step 2: In the Table Builder, use the composition calculator to fill all 36 cells. Hunt for the rotation subgroup hiding inside.
- Step 3: Compare your finished table to the color tables from Algebra's Hidden Structure. What's the same? What's different? (Hint: the color tables were commutative. Is this one?)
🌉 The bridge
The rotation subgroup you'll find is the same pattern in disguise as a three-color table from the algebra course — Pattern Hunter fans will recognize an isomorphism when they see one. Three units, one deep idea: structure is everywhere.