Foundations: The Shape of Two

Foundations: The Shape of Two is the opening document of the Developmental Geometry Series. It introduces the central conceptual objects of the series — the developmental interior, the balance axis, and the reconciliation operator — through a geometric demonstration using the oloid, a solid generated by two circles in orthogonal planes. The document is conceptual rather than technical. It contains one proved result: that the midpoint of the segment connecting the two generating circles of an oloid is the unique point on that segment equidistant from both circles. The remaining objects — the developmental interior, the balance axis, the reconciliation operator — are introduced as motivated structures whose formal definitions appear in subsequent arcs. The intended reader is anyone curious about the foundations of the series, with or without a mathematical background. No prior knowledge is assumed.