Formalize the four axioms visually using symmetry shapes
Symmetries obey four simple rules. Anything else that obeys them, from clock arithmetic to shuffling cards, is a group too.
Pick any two symmetries of the triangle from the menus. Whatever you choose, the result is one of the same six symmetries.
Pick any two symmetries of the triangle. Their composition is always another symmetry in the group.
The result e is always in the group. This is closure.
You have seen symmetries of polygons and how to compose them. Now let's formalize exactly what properties these symmetries satisfy. A group is a set with an operation that satisfies four axioms: closure, the existence of an identity, the existence of inverses, and associativity.
These axioms capture the essence of symmetry in the most general way. Any mathematical system satisfying them is a group -- from number systems to shuffling cards to quantum mechanics.
Compose any two symmetries of the triangle. The result is always another symmetry in the group -- never something "outside" the set. This is the closure property.
Key insight: Closure means the group is self-contained. No matter which two elements you combine, you stay within the group.
First click the "do nothing" symmetry. Then click an element and the one that undoes it, until every element has its inverse.
Which symmetry is the identity? Click the "do nothing" element.
Every group has a special "do nothing" element (the identity) and every element has an inverse that undoes it. Can you find them all?
Key insight: The identity element composed with anything gives that thing back. An element composed with its inverse gives the identity. Rotations are undone by rotating the other way; reflections are their own inverses.
Pick three symmetries a, b and c, and compare (a * b) * c with a * (b * c). However you choose, the two sides match.
Pick three symmetries a, b, c. Compare (a * b) * c versus a * (b * c).
r * r² = e
e * s = s
r² * s = sr²
r * sr² = s
Pick three symmetries a, b, c and verify that (a * b) * c = a * (b * c). The grouping does not matter -- only the order does. This lets us write products without parentheses.
Key insight: Associativity always holds for composition of functions (and permutations). This is why symmetry groups are automatically associative -- no need to check every triple.
Spot a mistake or have an idea?