the ladder of invariants

The hierarchy of geometries is a tower of nested transformation groups; the ladder of invariants is the matching staircase of what each rung can still measure. As you climb from the strict Euclidean geometry at the bottom to the permissive projective geometry at the top, the group gets bigger and, rung by rung, an invariant drops away. Reading the two ladders side by side is the cleanest way to see the whole Erlangen picture at a glance.

Walk down it concretely, from the largest group with the fewest invariants to the smallest group with the most. At the top, projective geometry, with the projective group, preserves only incidence (which points lie on which lines), collinearity, and the cross-ratio of four collinear points; it does not even know what 'parallel' means. Step down to affine geometry, where the group is smaller: now parallelism survives, along with the midpoint, ratios of lengths measured along a single line, and ratios of areas — but angle and absolute length are still invisible. Step down again to similarity geometry: now angle and shape come into view, and ratios of any two lengths, but absolute length is still missing. Step down once more to Euclidean geometry, the smallest group of all: here at last absolute length, distance, and area become genuine invariants, joining everything inherited from above. Off to the side runs conformal geometry, whose Mobius group is not on this single chain but which preserves angle and cross-ratio while abandoning straightness. The rule governing the whole ladder is simple: every invariant of a larger group is automatically an invariant of every smaller group below it, so invariants only accumulate as you descend.

The ladder is the practical payoff of the Erlangen program: it tells you, for any geometric quantity, exactly which geometry it belongs to, and therefore which transformations you are allowed to use when reasoning about it. If a theorem mentions only incidence, it is projective and you may use any projective map; if it mentions parallelism but not length, it is affine; if it mentions angle, it is similarity or Euclidean. The common error is to read the ladder upward — to use a Euclidean fact like 'this angle is a right angle' in an affine argument, where angle is not an invariant and the claim is simply meaningless. Match the quantity to its rung before you reason.