Groups that are also smooth manifolds — where algebra meets geometry
A Lie group is a group that is also a smooth, curved space. Rotations are the classic example: you can multiply them, and you can move smoothly between them.
Drag the two points and watch the rose product rotation: its angle is always the sum of the two. Nudge one point a little and the product moves only a little.
A Lie group is a mathematical object that is simultaneously a group (with a multiplication operation) and a smooth manifold (a curved space). The group operations — multiplication and inversion — must vary smoothly as you move through the manifold. This marriage of algebra and geometry is one of the most powerful ideas in modern mathematics.
The simplest example is rotations of a circle: SO(2). Each rotation is specified by an angle θ, the product of two rotations adds their angles, and nearby angles give nearby rotations. The space of all rotations forms a circle itself — a one-dimensional manifold that is also a group.
The group SO(2) consists of all rotations of the plane. Each element is a rotation by some angle θ, and the group operation is angle addition (mod 2π). Drag the two points to see how the product rotation (rose) is the sum of the two individual rotations.
Key insight: The group operation (addition of angles) is smooth — small changes in the input angles produce small changes in the product. This smoothness is what makes SO(2) a Lie group, not just a group.
Hover over the ball to apply each rotation to the frame: the direction from the center is the axis, the distance is the angle. Points on opposite sides of the surface give the same rotation.
Every 3D rotation has an axis (the direction that stays fixed) and an angle (how much to rotate around that axis). This means SO(3) can be visualized as a solid ball of radius π: the direction from the center gives the axis, and the distance gives the angle. Crucially, antipodal points on the boundary represent the same rotation (rotating π around k is the same as rotating π around −k).
Pick a rotation and widen ε to grow a cloud of nearby rotations. Notice that they all move the frame in nearly the same way.
To see that rotations form a smooth manifold, pick any rotation and look at all rotations within ε of it. The resulting “cloud” of nearby rotations all produce nearly identical transformations — the group operation varies continuously.
Click any card to learn about that group. The layout follows which groups sit inside which.
Here are some of the most important Lie groups, arranged by their subgroup relationships. Each has a different dimension, topology, and physical meaning. Click any card to learn more.
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