Elementary Geometry

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.

Automated theorem proving for elementary geometry

Thesis: Automated theorem proving for elementary geometry. This study analyses automated proofs of theorems from Euclidean Elements , book VI, using the area method. The theorems we will be discussing concern Euclidean field theory about equality of non-congruent figures and similarity of the figures [1]. The proofs are generated by the program WinGCLC. My proposed hypothesizes: 1. The way of m...

Some Elementary Aspects of 4-Dimensional Geometry

We indicate that Heron’s formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in four-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms and other elementary four-dimensional solids.