In 30 seconds
3Blue1Brown introduces symmetry groups using reflection, rotation and the identity action. A sixfold snowflake has 12 symmetries, while an idealized mirror-symmetric face has two.
- 1Symmetry as actionA reflection across a line can leave an idealized face unchanged.
- 2The snowflakeRotations and reflections preserve a sixfold snowflake.
- 3Include the identityDoing nothing counts, bringing the snowflake’s group to 12 actions.
Takeaways
- A symmetry is an action that leaves an object looking unchanged.
- Rotations and reflections can be combined into a symmetry group.
- The identity action, doing nothing, is also a symmetry.
- The illustrated snowflake has 12 symmetries, conventionally called D6.
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Reflection and symmetry
A symmetric face illustrates an action that preserves an object.
When I say a face is symmetric, I mean that I can reflect it across a line and leave it looking the same. Symmetry is a statement about an action I can perform, not just a visual impression.
The snowflake’s actions
Rotations and reflections form a larger collection.
A snowflake can have more symmetries. Rotating the illustrated sixfold snowflake by 60 or 120 degrees leaves it unchanged, as do reflections across several axes. Taken together, the symmetry actions form a group.
Counting group elements
The identity gives the snowflake 12 symmetries; a face can have two.
I also include the action of doing nothing, called the identity. For this snowflake there are 12 distinct symmetry actions: six rotations, including the identity, and six reflections. This group is commonly called D6.
The idealized face has a simpler group with two actions: doing nothing and reflection. It is called C2. These examples are a first glimpse of the many groups used to describe different kinds of symmetry.
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