Geometry of Lines on a Cubic Four-Fold

Universal Affine Triangle Geometry and Four - fold

We develop a generalized triangle geometry, using an arbitrary bilinear form in an affine plane over a general field. By introducing standardized coordinates we find canonical forms for some basic centers and lines. Strong concurrencies formed by quadruples of lines from the Incenter hierarchy are investigated, including joins of corresponding Incenters, Gergonne, Nagel, Spieker points, Mittenp...

A Symmetric Representation of the Twenty- Seven Lines on a Cubic Surface by Lines in a Finite Geometry*

Universal Affine Triangle Geometry and Four-fold Incenter Symmetry

We develop a generalized triangle geometry, using an arbitrary bilinear form in an affine plane over a general field. By introducing standardized coordinates we find canonical forms for some basic centers and lines. Strong concurrencies formed by quadruples of lines from the Incenter hierarchy are investigated, including joins of corresponding Incenters, Gergonne, Nagel, Spieker points, Mittenp...

Basic Algebraic Geometry and the 27 Lines on a Cubic Surface

We present an elementary proof of the fact that any nonsingular cubic surface in P3 over an algebraically closed field not of characteristic 2 will contain exactly 27 lines. During the course of this proof, we present a way to label these lines in terms of a fixed line l and another fixed line m which is disjoint from l in such a way that we can determine which of the 26 other lines a given lin...

Geometry of Cubic Polynomials

Imagine a sphere with its equator inscribed in an equilateral triangle. This Saturn-like figure will help us understand from where Cardano’s formula for finding the roots of a cubic polynomial p(z) comes. It will also help us find a new proof of Marden’s theorem, the surprising result that the roots of the derivative p(z) are the foci of the ellipse inscribed in and tangent to the midpoints of ...