A question hiding behind the whole ladder
By now you have studied several geometries that genuinely felt different. Euclid's plane cared about distance and angle; the chapter on the conics still measured eccentricity and focal lengths; the transformations rung let you slide, turn, and flip a shape while every length survived; and the projective plane you just left threw away distance, angle, and even parallelism, keeping only straightness and incidence. They are obviously not the same subject. But a sharp question lurks underneath all of them, one nobody on the lower rungs had to answer: *what is it that makes a geometry a particular geometry, rather than just 'doing some geometry'?*
In 1872 a twenty-three-year-old Felix Klein gave an answer in his inaugural lecture at Erlangen, and it reorganized the whole subject. The idea, now called the Erlangen program, is this: to specify a geometry, do not list its theorems or draw its figures. Instead, name the collection of motions you are allowed to perform — and then the geometry is exactly the set of statements that stay true no matter which allowed motion you apply. Two figures count as 'the same' in that geometry precisely when some allowed motion carries one onto the other. Change the allowed motions and you change the geometry, automatically and completely.
Two ingredients: a group, and an invariant
The whole machine runs on two ingredients you have already met in pieces. The first is a transformation group: a set of motions of the space that is closed under composition (do one, then another, and the result is again in the set), that contains the do-nothing identity, and in which every motion can be undone by another motion in the set. The rigid motions of the plane from the transformations rung form exactly such a group — compose two slides-turns-flips and you get a third, leave the plane alone and that counts too, and any rigid motion has an inverse that puts everything back. That bundle of closure, identity, and inverses is what 'group' means here.
The second ingredient is a geometric invariant: a property of a figure, or a quantity attached to it, that comes out the same after every motion in the group is applied. For the rigid motions, the distance |AB| is invariant — that is literally what 'isometry' was built to mean. So is the measure m(angle ABC), and so is the area of a triangle. These are not random survivors; they are precisely the things a stiff transparent sheet cannot alter as you pick it up and set it down. Klein's insight is to read the relationship backwards: Euclidean geometry is not 'the geometry where distance happens to matter' — it is the study of the invariants of the rigid-motion group, neither more nor less.
How a group acts, and what it stirs together
To make this precise we need the idea of a group action: the rule that says how each motion in the group moves each point of the space, in a way that respects composition. Doing motion g and then motion h must have the same effect as doing the single combined motion 'h after g' — the action and the group multiplication march in step. This is the formal handshake between the abstract group and the concrete points it pushes around. Once an action is fixed, every figure has a precise fate under every group element, and the question 'are these two figures the same?' acquires a sharp meaning.
Here is the picture that makes Klein's idea click. Fix a single point P and apply every motion in the group to it; the cloud of all the places P can land is its orbit. Under the rigid motions, the orbit of any point is the entire plane — you can slide any point to any other point — so from the group's standpoint all points look alike, which is exactly why Euclidean geometry has no privileged origin. Two figures lie in the same orbit when one motion carries one to the other, and that is precisely Klein's notion of 'congruent', 'equivalent', 'the same'. A geometry, in this view, is a space cut into orbits by its group, and its theorems are statements about which orbit a figure lives in.
Loosen the group, lose some invariants
Now watch what happens as we enlarge the group, allowing more motions. Start with the rigid motions and add uniform scaling — dilations that blow a figure up or shrink it without distorting it. The enlarged group is the group of similarities, and it is bigger, so fewer things can survive all of it. Distance |AB| is gone, because a similarity can double it; but the ratio of two distances survives, and so does every angle. That is the precise reason the similarity world is the natural home of 'similar triangles': in it, two triangles with equal angles really are the same object, since a similarity carries one exactly onto the other.
Push the group looser still. Allow any transformation that sends straight lines to straight lines and keeps parallel lines parallel, even ones that shear and stretch unevenly — these are the affine maps, and the geometry they leave invariant is affine geometry. Now even angle and ratio-of-arbitrary-lengths are gone; a shear can turn a square into a slanted parallelogram and a right angle into an obtuse one. Yet parallelism survives by construction, and so does the ratio of two lengths measured along the same line, and so does the idea of a midpoint. Each time the group grows, a layer of invariants peels away — and the remaining, hardier invariants are exactly the content of the looser geometry.
ENLARGE THE GROUP --> FEWER SURVIVING INVARIANTS group of motions what stays invariant ------------------------ ----------------------------------- rigid motions (isometry) distance, angle, ratio, parallel, line similarities -- angle, ratio, parallel, line affine maps -- -- ratio*, parallel, line projective maps -- -- -- -- line (* ratio only along one line, generalizing to the cross-ratio) bigger group = coarser geometry = fewer theorems, more equal figures
The hierarchy, and a sideways branch
Stack these groups and a clean ladder appears — the hierarchy of geometries. The rigid motions sit inside the similarities, which sit inside the affine maps, which sit inside the projective maps: every Euclidean motion is a similarity, every similarity is affine, every affine map extends to a projective one. Because each group contains the ones below it, every invariant of a bigger group is automatically an invariant of the smaller groups inside it — so a projective theorem is true in all four geometries, while a Euclidean theorem about distance need not survive even one step up. The whole tower reads projective > affine > similarity > Euclidean, loosest at the top, strictest at the bottom, exactly mirroring the table you just saw.
Not every geometry hangs on this one main spine, and it is worth being honest about that. There is a sideways branch of great beauty: conformal, or Mobius, geometry, whose group is built from inversions in circles together with the rigid motions and scalings. A Mobius transformation can bend a straight line into a circle, so straightness — the invariant that defined the whole projective spine — is not preserved here. What it does preserve is the angle at which two curves cross; that is the meaning of 'conformal'. So this geometry sacrifices the very thing the affine and projective levels kept, while rescuing the angle that those levels threw away — a genuinely different bargain, not just another rung on the same ladder.
What you now carry, and where the rung goes
Hold onto the reframing, because it changes how you read everything below. A geometry is no longer a grab-bag of theorems about a particular kind of picture; it is a group of allowed motions paired with its invariants, and the theorems are simply the invariants spelled out. Congruence, similarity, and projective equivalence stop being three unrelated relations and become one relation — same orbit — measured by three different groups. That is the single thought you climbed this rung to gain, and it is worth more than any one theorem it organizes.
Here is how the next four guides cash this out. Guide 2 slows down on the group action itself — orbits, and the stabilizer of a point (the motions that pin it in place) — so the machinery becomes something you can compute with, not just admire. Guide 3 walks the full hierarchy from projective down to Euclidean, watching the invariants accumulate as you descend. Guide 4 settles into affine geometry alone to ask precisely what it can and cannot see. And guide 5 follows the sideways branch into conformal and Mobius geometry, where circles and lines become one family and angle is king.
One honest caveat to pack for the climb. Klein's program is a magnificent organizing lens, but it is not the whole of geometry, and saying so is not a knock against it. The curved spaces opened by Riemann in 1854 — surfaces and manifolds whose curvature varies from point to point — generally have no large transformation group at all, so there are no global motions to be invariant under, and Klein's recipe does not reach them directly. Those geometries need a different engine, the intrinsic and infinitesimal one you will meet in the differential-geometry rungs ahead. The Erlangen idea is a powerful searchlight, not a floodlight that covers every field — and knowing where its beam ends is part of using it well.