The Erlangen Program, Concretely

Interactive lessons on groups of geometric transformations.

An accessible, modern introduction to the philosophy behind Klein's Erlangen Program: a geometry is defined by its group of transformations, and different groups — isometries, similarities, affine maps, projective/Möbius maps — give genuinely different geometries on the same underlying space.

Each lesson below is a single, self-contained page: no course, no institution, no login, nothing to install. Open it, drag things, and watch what does and doesn't happen — that's the mathematics.

• Groups of translations

  Two draggable generators a, b tessellate the plane by translation, coloring one fundamental parallelogram and reproducing that coloring exactly on every copy — a picture of the quotient, drawn live as you drag. Includes the group's own "picture of itself": the orbit of a single marked point.

• The Riemann sphere

  The unit sphere sits in the complex plane, meeting it along the equator — which is therefore the plane's own unit circle. Join a draggable point z to one pole by a straight line and it marks a second point on the sphere: stereographic projection. The sphere is translucent, so the image stays visible on the far side; push z away and watch it creep toward the pole that answers to ∞.

• Groups of spiral similarities

  A draggable generator a (a jump z ↦ a·z) and a rotation order n generate a group whose fundamental region tiles the plane by logarithmic spirals instead of straight lines. A one-parameter slider glides continuously through the same points the discrete group only hops between — continuous versus discrete, made visible.