Elementary Pure Geometry, with Mensuration

The Geometry of Pure Spinors, with Applications

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Safe domain and elementary geometry

A classical problem of mechanics involves a projectile fired from a given point with a given velocity whose direction is varied. This results in a family of trajectories whose envelope defines the border of a “safe” domain. In the simple cases of a constant force, harmonic potential, and Kepler or Coulomb motion, the trajectories are conic curves whose envelope in a plane is another conic secti...

Elementary Surprises in Projective Geometry

The classical theorems in projective geometry involve constructions based on points and straight lines. A general feature of these theorems is that a surprising coincidence awaits the reader who makes the construction. A classical example of this is Pappus’s theorem. One starts with 6 points, 3 of which are contained on one line and 3 of which are contained on another. Drawing the additional li...

Conformal Geometry and Elementary Particles.

The Elementary Geometry of a Triangular World with Hexagonal Circles

We provide a collinearity based elementary axiomatics of optimal quantifier complexity ∀∃∀∃ for the geometry inside a triangle and reprove that collinearity cannot be defined in terms of segment congruence, the metric being Hilbert’s projective metric. MSC 2000: 51F20, 51A30, 51A45, 51G05