From slogan to machinery
In the previous guide you met Klein's big idea: a geometry is not a list of figures but a group of allowed motions, and its theorems are exactly the facts those motions leave unchanged. That is a beautiful slogan, but a slogan is not yet a method. To actually do geometry this way we need to pin down one phrase precisely — what it means for a transformation group to act on a space. Everything in this rung, the whole nested hierarchy of geometries, rests on getting this one definition crisp.
Start with the players. A transformation is just a rule that moves each point of the plane to some point of the plane, reversibly — nothing is lost or doubled up. A translation slides every point three units right; a rotation spins the plane about a fixed centre; a reflection across a line flips it over. Each is an invertible map from the plane to itself, and you can undo it. A whole group of transformations is a collection of such maps that is closed under two habits: do one then another and you stay in the collection (composition), and every member has its undo also in the collection (inverse). The identity 'do nothing' map is always in there too.
The word action ties the group to the space. We say a group G acts on the plane when each element g of G is assigned an actual transformation of the plane, in a way that respects the group's bookkeeping: the identity element does nothing, and composing two elements in the group matches composing their two transformations. In symbols, if we write g . P for 'where g sends the point P', then the identity sends P to P, and (g then h) . P equals h . (g . P). That compatibility is the whole content of a group action — the abstract group and the concrete motions march in lockstep.
Orbits: everything a point can become
Once a group acts, the first question is: starting from a single point P, where can I get to? Apply every transformation in the group to P and collect all the landing spots. That set is the orbit of P. The orbit answers 'what is P allowed to become?' — and it depends entirely on which group is acting. Pick the group of all rigid motions (translations, rotations, reflections — the ones that preserve distance) and the orbit of any point is the entire plane, because you can slide any point onto any other.
Shrink the group and the orbit shrinks with it. Take only rotations about a fixed centre O. Now the orbit of a point P is just the circle through P centred at O, since rotating P sweeps out exactly that circle and nothing else. Take only the single map 'reflect across line L'. The orbit of P off the line is the pair of points {P, P'}, where P' is the mirror image — two points, no more. The richer the group, the bigger the orbits; a tiny group barely moves anything.
Stabilizers: what a point holds still
The orbit asks where a point can travel. Its twin question asks the opposite: which transformations leave the point exactly where it is? Fix a point P and collect every group element g with g . P = P — every motion that pins P down. That collection is itself a group (the identity fixes P, and if two motions both fix P so does their composition and their inverses), called the stabilizer of P. The orbit measures how freely P moves; the stabilizer measures how rigidly it can be held.
A picture makes it vivid. In the rigid-motion group, what fixes a chosen point O? You may rotate the plane about O by any angle, and you may reflect across any line through O — but you may not translate, since any nonzero slide would move O. So the stabilizer of O is the group of all rotations and reflections about O: in two dimensions this is the orthogonal group, the symmetries of a point seen as the centre of a featureless disc. Every point has a stabilizer of the same shape, because all points are interchangeable under the full group.
Orbit and stabilizer are not independent — they trade off against each other, and that trade is one of the most useful facts in the whole subject. Loosely: the bigger the orbit (the more places P can go), the smaller the stabilizer (the fewer motions can hold P still), and vice versa, because the total 'size' of the group is split between moving P and fixing P. When the orbit is everything and the stabilizer is the rotations-and-reflections of a single point, you are looking at the Euclidean plane laid bare. Smaller stabilizers signal a richer space to roam.
Invariants: the geometry itself
Here is where Klein's slogan becomes a working definition. A property of figures is a geometric invariant of the action if it comes out the same no matter which transformation of the group you apply first. Under the rigid-motion group, the distance |AB| between two points is an invariant: slide, spin, or flip the pair however you like, and |AB| never changes. So is the measure m(angle ABC) of an angle, and so is the area of a triangle. These survivors — the quantities the group cannot touch — are precisely the legitimate statements of that geometry. The invariants are the geometry.
This reframes a question you have been answering all the way up the ladder without naming it. 'Are these two triangles congruent?' really means 'is there a rigid motion carrying one exactly onto the other?' — that is, do they lie in the same orbit of the rigid-motion group acting on triangles. The familiar congruence tests (SSS, SAS, ASA) are shortcuts for detecting that exact thing. The Erlangen view does not throw your old geometry away; it explains why distance and angle were the right things to measure all along — they are the invariants of the group you were silently working in.
Bigger group, fewer invariants
Now comes the payoff that gives this rung its shape. The groups themselves nest inside one another, and that nesting runs opposite to the richness of invariants. Allow only rigid motions and you keep the most: distance, angle, area, parallelism, straightness. Allow rigid motions plus uniform scaling — the similarity group, which also includes a dilation that blows figures up or shrinks them — and distance is gone (you can magnify |AB| at will) but ratios of distances and all angles survive. Bigger group, strictly fewer invariants.
Loosen further. The affine group adds shears and independent stretching along different directions; now even angle and ratio-along-different-directions break, but parallelism and straightness still hold — parallel lines stay parallel, lines stay lines. Loosen once more to the projective group from the last rung, and parallelism finally falls too (parallels meet at infinity), leaving only straightness and incidence. Each step up in the size of the group is a step down in what you are allowed to measure. The result is the hierarchy of geometries: projective on top, then affine, then similarity, then Euclidean at the base, the strictest geometry because its group is the smallest.
GROUP (gets bigger going down) INVARIANTS IT PRESERVES Euclidean (rigid motions) distance, angle, parallel, straight Similarity (+ scaling) ratio, angle, parallel, straight Affine (+ shear, stretch) - - parallel, straight Projective (+ projection) - - - straight bigger group ==> fewer invariants ==> looser geometry
One sideways step: conformal geometry
The clean four-storey tower — projective, affine, similarity, Euclidean — is the spine of this rung, but the Erlangen idea reaches sideways too, not only straight down. Consider a group built from a surprising extra move: inversion in a circle, which turns the inside of a circle out and the outside in, fixing the circle itself. Throw inversion in with the ordinary rigid and similarity motions and you generate the group of Mobius transformations, the heart of conformal geometry. Its motions can bend a straight line into a circular arc, so straightness is no longer sacred here.
What is sacred instead is angle. Every Mobius transformation, however violently it warps shapes and trades lines for circles, preserves the angle at which any two curves cross — that is what 'conformal' means, angle-preserving. So this geometry keeps an invariant (angle) that affine and projective geometry had already thrown away, yet drops one (straightness) that they kept. It does not slot tidily above or below them on the single ladder; it sits off to the side, defined the same Erlangen way — name its group, read off its invariants.
Step back and see what you have gained. A geometry is now a thing you can specify: hand over a group acting on a space, and out come the orbits (which figures are 'the same'), the stabilizers (how much symmetry each point carries), and the invariants (the legal theorems). The next guide walks deliberately down the main ladder — projective to Euclidean — examining at each rung exactly what is won and lost, and guide 4 settles into affine geometry to see one storey in full. The ladder of invariants you sketched here is the map for the rest of the rung.