J. Chris Fisher

Properties of Affinely Regular Polygons

Intuitively, one might consider an affinely regular polygon of the Eucidean plane to be the result of applying an affine transformation to a regular polygon. These affinely regular polygons, and their kindred that go by the same name in the Euclidean plane as well as in more general affine planes, have been onjects of investigations at all levels of sophistication and in a remarkable variety of contexts. For example, they arise in linear algebra as a set of vectors that are cyclically permuted by a unimodular matrix. Our purpose is to describe this concept and its attributes in a general setting. The main result is Theorem 1 where we present seven equivalent definitions of affine regularity, one of which appears for the first time. We are careful to distinguish these definitions from the weaker intuitive definition. Our work also features an application of Chebyshev polynomials to describe parameters associated with these polygons.

An observation on certain point—line configurations in classical planes

Abstract We define a cotangency set (in the projective plane over any field) to be a set of points that satisfy two conditions (A) and (B). The main result says that a cotangency set can never contain a quadrangle. A number of profound-sounding consequences involving Hermitian curves are really observations that follow quickly from the theorem by way of elementary arguments.

A 5-Circle Incidence Theorem

Summary We state and prove a surprising incidence theorem that was discovered with the help of a computer graphics program. The theorem involves sixteen points on ten lines and five circles; our proof relies on theorems of Euclid, Menelaus, and Ceva. The result bears a striking resemblance to Miquels 5-circle theorem, but as far as we can determine, the relationship of our result to known incidence theorems is superficial.

Finite Fourier series and ovals in PG(2, 2 h)

We propose the use of finite Fourier series as an alternative means of representing ovals in projective planes of even order. As an example to illustrate the methods potential, we show that the set { w j + w 3j + w −3j : 0 ≤ j ≤ 2 h } ⊂ GF (2 2h ) forms an oval if w is a primitive (2 h + 1) st root of unity in GF(2 2h ) and GF(2 2h ) is viewed as an affine plane over GF(2 h ). For the verification, we only need some elementary ‘trigonometric identities’ and a basic irreducibility lemma that is of independent interest. Finally, we show that our example is the Payne oval when h is odd, and the Adelaide oval when h is even.

SEMILINEAR TRANSFORMATIONS OVER FINITE FIELDS ARE FROBENIUS MAPS

In its original formulation Langs theorem referred to a semilinear map on an n -dimensional vector space over the algebraic closure of GF(p) : it fixes the vectors of a copy of V(n, p^h) . In other words, every semilinear map defined over a finite field is equivalent by change of coordinates to a map induced by a field automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results.

A hyperbolic concurrency theorem

We discuss in the context of Euclidean and projective geometry a recently rediscovered theorem concerning the concurrency of the main diagonals of a hexagon, and provide a detailed proof of its hyperbolic version.

Whiskey, Marbles, and Potholes

The 1990 TV movie Gunsmoke: The Last Apache, which resurrected the Matt Dillon role for James Arness, added an intriguing twist to the traditional duel: Six identical shot glasses of whiskey were s...