The visual entry point -- discover groups through the symmetries of polygons
Rotate a square a quarter turn or flip it over, and it looks the same. Group theory starts by collecting all such moves and asking how they combine.
Choose a shape, then click the rotation and reflection buttons. Watch where each colored vertex lands, and read the result in cycle notation below.
Current state: e
Group theory begins with a simple question: what symmetries does a shape have? Take a regular polygon -- you can rotate it, reflect it, and the shape looks the same. The collection of all such symmetries, together with the rule for combining them, forms a group.
In this lesson, explore the symmetries of polygons hands-on. You will see that every symmetry is a permutation of the vertices, that combining two symmetries gives another symmetry, and that just two generators (one rotation and one reflection) are enough to produce them all.
Choose a polygon and click symmetry buttons to rotate or reflect it. Watch the color-coded vertices move to their new positions. The current state is displayed in cycle notation below.
Key insight: Each symmetry is fully described by where it sends each vertex. A rotation sends vertex 0 to vertex 1 to vertex 2, etc. A reflection swaps certain pairs. This is exactly a permutation.
Pick a polygon and press Step to discover symmetries one at a time from one rotation and one reflection, or Discover All to finish. You always end with 2n of them.
Start with identity e
Discovered: 1 / 6 elements
A regular n-gon has exactly 2n symmetries: n rotations and n reflections. Remarkably, all of them can be generated from just two operations -- one rotation by 360/n degrees and one reflection. Step through the BFS discovery process.
Key insight: This is the dihedral group D_n. It has order 2n because there are n rotational symmetries and n reflections. Two generators suffice to create the entire group through composition.
Pick a First and a Second symmetry and press Compose. The second is applied first, and the result is always another symmetry of the square.
Step 1: Apply r²
Step 2: Apply r
Pick two symmetries and watch their composition animate step-by-step on the square. First the second symmetry is applied, then the first. The result is always another symmetry in the group -- a preview of the closure property.
Key insight: Composing a rotation with a reflection gives a reflection. Composing two reflections gives a rotation. These patterns are encoded in the group's multiplication table.
Spot a mistake or have an idea?