Space · transformation · invariant
Geometry
Geometry studies space by deciding what transformations are allowed and then asking what survives. Distance, angle, curvature, continuity, coordinates, and scale become different lenses on the same question: what makes a shape or space remain the same kind of thing?
Primary navigation · invariant ladder
Change the rules of space, then track what remains meaningful.
The branches below are not six unrelated shape collections. Each changes the assumptions, representation, or notion of sameness used to study space.
EUC · Flat metric
Euclidean Geometry
Flat-space geometry built from points, lines, angles, congruence, similarity, polygons, circles, solids, constructions, and deductive proof.
The geometric move
Fix a flat space and derive what follows from points, lines, distance, angle, parallelism, congruence, and proof.
Keeps visible
distanceangleparallel structurecongruence
Relaxes or translates
curvature held at zero
01AssumptionsChoose what kind of space and transformations are allowed.
02InvariantsAsk which properties remain unchanged under those transformations.
03RepresentationSwitch among diagrams, coordinates, equations, constructions, and proofs without confusing the representation for the object.