🔬 Symmetry Lab

Flip, rotate, and transform your way to discovering group theory — the hidden mathematics of symmetry

🧭 What You’ll Discover

A triangle has exactly 6 symmetries — 3 rotations and 3 flips. When you combine any two of them, you always get one of the 6 back. This closed system of transformations is called a group, and it’s the same kind of structure you explored with colored circles in Algebra Wizard. Here, you’ll see it come alive through physical movement!

Interactive Tools

🔺

Triangle Symmetry Explorer

Watch vertices move as you apply rotations and flips. Discover identity, inverses, closure, and — most importantly — that order matters!

Interactive

Start Here

Open Explorer →

🧮

D₃ Cayley Table Builder

Build the complete 6×6 multiplication table for triangle symmetries. Compose transformations and discover subgroups hiding inside!

Interactive

Use After Explorer

Open Builder →

Suggested Path

Step 1: 🔺 Open the Triangle Symmetry Explorer. Try all 6 transformations. Then test whether order matters by applying two moves, resetting, and applying them in reverse.

Step 2: 🧮 Open the D₃ Cayley Table Builder. Use the composition calculator to figure out what each pair of transformations produces, and fill in all 36 cells. Can you spot the rotation subgroup?

Step 3: 🔍 Compare this Cayley table to the ones from Algebra Wizard. What’s the same? What’s different? Hint: the rotation subgroup is isomorphic to something you’ve already seen…

🌉 Connection to Algebra Wizard

In Algebra Wizard, every color operation was commutative — order never mattered. Triangle symmetries are your first example where order does matter. Rotate then flip gives a different result than flip then rotate! But both systems share the same deep properties: identity, inverses, closure, and associativity. That’s the power of structure.