Saved on this device

Follow your curiosity

What discovery has been shared with you?

FACTREE gives you something interesting to discover every day — and something worth talking about together.

Start with one fact. Explore it, go deeper, then follow whichever branch catches your imagination.

Choose subjects for a surprise

Exploring any topic

Begin your discovery

Your next discovery is one click away.

Choose one or more subjects above, or leave Any Topic selected and let curiosity decide.

Mathematics

Group Actions and Orbit-Stabilizer Theorem in Symmetry Analysis

Quick fact

The Orbit-Stabilizer Theorem shows that the size of an orbit (the set of all images of an object) multiplied by the size of the object's stabilizer (the group of symmetries that keep it fixed) always equals the total number of symmetries in the group.

Why this is interesting

You've probably noticed that a snowflake looks the same after a 60° rotation. But how many distinct patterns can you create with a colored cube? The answer comes from a surprising counting rule: the Orbit-Stabilizer Theorem.