The Hierarchy: Projective Down to Euclidean

Smaller Group, Richer Geometry

From the first guide of this rung you carry Klein's one sentence: a geometry is the study of the properties left unchanged by a chosen group of transformations. From the second guide you carry the machinery of a group acting on space — a transformation group that you can compose, undo, and let loose on points. Now we put those two ideas under tension and watch something surprising fall out: the size of the group and the richness of the geometry pull in opposite directions.

Here is the intuition in one breath. A property counts as a geometric invariant only if every transformation in the group leaves it alone. So a bigger group is a harsher judge — more maps must agree to preserve a quantity, and fewer quantities survive the screening. A smaller group is a gentler judge — fewer maps need to approve, so more quantities slip through as genuine, meaningful concepts. Shrink the group, and properties that were invisible before suddenly become real features of the geometry.

Building the Tower, Top to Bottom

Now we stack the geometries. The trick is that each group sits inside the one above it — every Euclidean motion is a similarity, every similarity is an affine map, every affine map is a projective one. So the groups nest like Russian dolls, and the geometries form a clean hierarchy of geometries. At the top is projective geometry, governed by the biggest group, the collineations that preserve only incidence — which points lie on which lines. It is the most permissive judge, so it keeps almost nothing.

Step down one rung and you reach affine geometry. We get there by demanding the group fix the line at infinity instead of being free to move it — concretely, we forbid the perspective-style maps and keep only the affine transformations. The reward for this smaller group is two new invariants that projective geometry could not name: parallelism becomes meaningful, and so does the ratio in which a point divides a segment. Parallel lines stay parallel; a midpoint stays a midpoint. Distances and angles, however, are still scrambled — a square can be sheared into any parallelogram.

Step down again to similarity geometry. Shrink the group once more so it must preserve angles and ratios of lengths, though not lengths themselves. Now shape is real: similar triangles are genuinely similar, a circle stays a circle, and angle measure finally means something. What you still cannot pin down is absolute size — this is the geometry of blueprints and scale models, where a scale factor is free but proportion is sacred. One more step, to the smallest group of all, lands you on the floor.

The Floor: Euclidean Geometry

At the bottom of the tower sits the geometry you grew up with. Its group is the smallest of the four: the rigid motions — translations, rotations, and reflections — the isometries that move figures around without distorting them at all. Because this judge is so gentle, almost everything survives as a genuine invariant: not just incidence, parallelism, and angle, but length itself, area, and congruence. This is why a^2 + b^2 = c^2 is a Euclidean theorem and not a projective one — it speaks of lengths, and only the rigid-motion group is small enough to let lengths mean anything.

So read the tower from the floor looking up. Euclidean geometry sees the most — lengths, angles, areas, parallels, incidence, the lot. Climb to similarity and you forget absolute length but keep shape. Climb to affine and you forget shape but keep parallelism and ratios along a line. Climb to projective and you forget even those, holding on to bare incidence alone. Each step up deliberately blurs more of the picture, trading detail for a wider, more flexible group of allowed motions.

GROUP (size)                 GEOMETRY        new things it can finally see
--------------------------------------------------------------------------
projective (biggest)         projective      incidence, cross-ratio
  > affine                   affine          + parallelism, ratios on a line
      > similarity           similarity      + angles, shape, ratios of lengths
          > rigid motions    Euclidean       + length, area, congruence
          (smallest)

bigger group  ->  fewer invariants  ->  coarser geometry
smaller group ->  more invariants   ->  finer geometry

What Projective Geometry Keeps: The Cross-Ratio

It is fair to ask whether the top of the tower keeps anything at all — if projective maps shred lengths, ratios, and even midpoints, is there any number they respect? There is exactly one famous survivor, and it is the crown jewel of the rung you just climbed: the cross-ratio. Given four collinear points A, B, C, D, the cross-ratio is a single number combining their positions, and every projective transformation leaves it unchanged. It is the one quantitative invariant the biggest group could not kill.

The cross-ratio fits the tower beautifully. As you descend and the group shrinks, the cross-ratio is joined by ever simpler companions. In affine geometry, where parallelism is restored, the cross-ratio specializes: send one of the four points off to infinity and it collapses into the plain ratio in which a point divides a segment — the affine invariant. In Euclidean geometry it sits alongside the most basic invariant of all, the distance |AB|. The single projective number is the ancestor; the homely school-geometry quantities are its descendants, unlocked one rung at a time.

A Sideways Branch: Conformal and Mobius Geometry

The four-storey tower is tidy, but the Erlangen idea is not limited to a single staircase — any group of transformations defines a geometry, so there are useful geometries that branch off sideways rather than sitting cleanly between two floors. The most beautiful of these is conformal geometry, whose defining group preserves angles but is allowed to bend straight lines into curves. Conformal maps keep the angle at which two curves cross, even while they distort lengths, areas, and straightness freely.

The cleanest gateway into conformal geometry in the plane is the family of Mobius transformations. These are the maps built by combining rotations, translations, scalings, and one new ingredient — inversion in a circle, which turns a point inside a circle inside-out across it. The combined group lives most naturally on the inversive plane, the ordinary plane with a single point at infinity bolted on so that even straight lines close up. On this stage a Mobius map has a gorgeous defining property: it always sends circles-and-lines to circles-and-lines, treating a straight line as just a circle of infinite radius.

Notice how faithfully this obeys Klein. Mobius geometry is not 'better' or 'worse' than Euclidean geometry; it is simply the geometry of a different group, asking a different question — what is true about figures when you may bend and invert but must keep every crossing angle? Its invariants are angle and, once again, a cross-ratio (now a cross-ratio of four points using a complex-number measure). The same Erlangen recipe — pick a group, study its invariants — generates a whole landscape of geometries, of which the projective-to-Euclidean tower is just the most famous wing.

Reading the Whole Map Honestly

Step back and the Erlangen program reveals its real power: it turns a pile of separate subjects into one organized ladder of invariants. Projective, affine, similarity, and Euclidean geometry are not four rival theories but one tower, each floor obtained from the one above by shrinking the group and unlocking a new invariant. The same lens places conformal and Mobius geometry as a neighbouring branch, and — looking far ahead — even the curved geometries of later rungs as cousins defined by their own symmetry groups.

Two honest caveats keep this picture from becoming a fairy tale. First, the neat 'bigger group, fewer invariants' rule is exact only when one group genuinely sits inside another; geometries on a sideways branch, like the conformal one, are not simply ranked above or below the Euclidean floor — they answer a different question entirely. Second, the Erlangen viewpoint is gorgeously unifying but it is not the final word: it describes geometries that have a transitive symmetry group, and the deeply curved spaces of differential geometry, where curvature can vary from point to point, ultimately need Riemann's richer language, which a much later rung will give you. Klein organizes a vast and beautiful country; he does not fence in the entire world of geometry.