the Erlangen program
/ ER-lang-en /
By the late 1800s mathematicians had a confusing pile of separate geometries: ordinary Euclidean geometry, the new non-Euclidean ones, projective geometry, affine geometry, and more. Each had been built on its own axioms and seemed to be its own island. In 1872, when he took up a professorship at Erlangen, the young Felix Klein offered a single organizing idea that tied them all together. The Erlangen program is that idea: a geometry is not defined by its axioms about points and lines but by a group of transformations, and the 'geometric facts' of that geometry are exactly the properties those transformations leave unchanged.
Spell it out. Fix a space — say the plane — and fix a collection of allowed transformations of that space that forms a group (you can do nothing, you can undo any move, and you can chain two moves into one). Then the corresponding geometry studies only those features of figures that survive every transformation in the group. Euclidean geometry takes the group of rigid motions (translations, rotations, reflections), so its facts are about length, angle, and congruence — exactly the things rigid motion cannot change. Allow the bigger group that also includes uniform scalings, and length is no longer meaningful but shape and angle still are: that is similarity geometry. Allow all affine maps and even shape blurs away, but parallelism and ratios along a line remain: affine geometry. Allow all projective transformations and even parallelism dissolves, leaving only incidence and the cross-ratio: projective geometry. Each smaller group sees more invariants; each larger group sees fewer.
The power of the program is that it turns 'which geometry am I doing?' into the sharp question 'which transformation group am I allowing?', and it makes the relationships between geometries precise: a larger group gives a coarser, more permissive geometry sitting above a smaller, richer one. It does not, however, capture every geometry ever invented — Riemannian and differential geometry, with curvature that varies from point to point, do not fit the single-group mold, and Klein himself knew this. The program organizes the classical 'homogeneous' geometries beautifully; it is a unifying lens, not a theory of everything.
Ask whether two triangles are 'the same'. In Euclidean geometry the answer is yes only if they are congruent (a rigid motion carries one to the other). In similarity geometry any two triangles with equal angles count as the same. In affine geometry any two triangles at all are the same, since an affine map carries any triangle to any other. Each verdict comes straight from the size of the allowed group.
What counts as 'the same figure' is decided entirely by the chosen transformation group.
The Erlangen program does not say geometry is 'just group theory' or that axioms are useless; it says each homogeneous geometry can be encoded by a group, and curved geometries like Riemann's fall outside this single-group picture.