Foundations

Primitives, Proportions, Rational Invariants & the Golden Ratio

Before constructing 465 propositions of Euclid or 4,000 triangle centers, GEOMETOR examines the elementary primitives from which every figure arises: an unmarked planar field, two starting points, the line as linear proportion, the circle as perpendicular rotation, and Extreme and Mean Ratio (\(\phi\)) as the self-similar bridge between addition and multiplication.

Foundational Essays

• The Planar Field & Two Starting Points
  • Two Givens Establish Unit Measure
  • Continuous Loci vs. Graphical Segments
  • Cartesian Algebra Without Decimal Drift
• Lines, Proportions & Spreads
  • Exact Line–Line Selection
  • Inclination Without Transcendental Angles: Spread
• Circles, Rotation & Quadrance
  • Center vs. Radius Point: Directional Asymmetry
  • Quadratic Extensions & The Equidistant Poles
• The Golden Ratio (\(\phi\))
  • The Defining Quadratic & Conjugate Roots
  • Fundamental Algebraic Identities
  • Continued Fraction, Fibonacci & Lucas Sequences
  • Where to Explore Further in GEOMETOR
• Squaring the Circle, Kepler’s Triangle & the Pyramids
  • Exact Impossibility vs. Remarkable Harmonic Resonance
  • The Kepler Golden Right Triangle \((1, \sqrt{\phi}, \phi)\)
  • Perimeter and Area Circle–Square Proportions