Compass-and-Straightedge Construction Geometry(1): From Symmetry to the Boundary of Measurement

This paper takes compass-and-straightedge construction as its starting point and establishes a geometric chain from symmetry to the boundary of measurement. Symmetry manifests as the straightedge; the straightedge generates line segments, and line segments carry measurement. The compass generates circles; two circles intersect, measurement is bisected, producing the sequence 1/2, 1/4, …, 1/2ⁿ, and measurement tends toward the boundary 0. Both 0 and 1 are boundaries of measurement produced by compass-and-straightedge operations. The Cartesian coordinate system inherits both, but in the coordinate system their measurement identity is converted to a coordinate identity, the measurement boundaries become hidden, and infinity (∞) thereby emerges. This paper aims to establish compass-and-straightedge construction geometry as an independent geometric system and to delineate its connection with the Cartesian coordinate system.