The Geometry of Transformation

This flow traces how figures are repeatedly divided compared extended reflected and recombined to reveal invariant structure.

increasingBearing stren­gth in each eventdecreasing

Division

Division Bisected Forms Angles and line segments are divided to create symmetry and perpendicularity. The act of division exposes a hidden centre or balance. Invariant Result What changes in the drawing reveals what remains equal.

Extension

Extension Produced Sides Sides are prolonged beyond a triangle to create external angles and auxiliary figures. The extension enlarges the diagram without abandoning its original relations. Discovery New equality becomes visible only after the original boundary is crossed.

Recomposition

Recomposition Areas Rearranged Rectangles and squares are partitioned and recombined to establish identities. The parts change position while their total area remains equal. Insight Transformation reveals algebraic truth without numerical calculation.

Circular Transformation

Circular Transformation Position and Relation Chords tangents radii and secants are compared through changing positions around a circle. The circle preserves equality while the lines move. Stability Geometric truth resides in invariant relations rather than a single fixed drawing.

Scaled Transformation

Scaled Transformation Enlarged Figures Similar triangles and polygons reproduce form at different scales. Angles remain while lengths and areas transform according to ratio. Completion The book ends by showing how geometry controls change itself.

during the constructions of Book I

during the isosceles and exterior-angle propositions

during Book II's square and rectangle proofs

during Book III's chord and tangent propositions

during Book VI's similarity propositions