Infinite Regular Hexagon Sequences on a Triangle

The well-known dual pair of Napoleon equilateral triangles intrinsic to each triangle is extended to infinite sequences of them, shown to be special cases of infinite regular hexagon sequences on each triangle. A set of hexagon-to-hexagon transformations, the hex operators, is defined for this purpose, a set forming an abelian monoid under function composition. The sequences result from arbitrary strings of hex operators applied to a particular truncation of a given triangle to a hexagon. The deep structure of the sequence constructions reveals surprising infinite sequences of non-concentric, symmetric equilateral triangle pairs parallel to one of the sequences of hexagons and provides the most visually striking contribution. Extensive experimentation with a plane geometry educational program inspired all theorems, proofs of which utilize eigenvector analysis of polygons in the complex plane. INTRODUCTION. This paper is an exercise in the geometry of the complex plane—utilizing the 'eigenpolygon' decomposition of polygons in the complex plane—that extends the well-known pair of Napoleon equilateral triangles intrinsic to each triangle to infinite sequences of them. These sequences, in turn, are special cases of infinite regular hexagon sequences on each triangle. Another theme is the benefit of experimental use of computer graphics in plane geometry. Geometric constructions in this study are tedious—often infeasible—for the unaided person, yet the intuitions gained from dynamic interaction with the complicated constructions are powerful. Each theorem is the direct result of conjecture inspired by experimentation with normally unwieldy geometric constructions. The software used is an educational program [Sketchpad]. Napoleon’s Theorem describes a transformation mapping an arbitrary triangle to an equilateral triangle [Chang-Sederberg 1997; Coxeter-Greitzer 1967; Wetzel 1992]. It is actually a dual pair of transformations leading to the so-called outward and inward Napoleon triangles, called positive and negative here for consistency. Fukuta generalizes the Napoleon transformation to a 2-step transformation that converts an arbitrary triangle to a regular hexagon [Fukuta 1996a; Garfunkel-Stahl 1965; Lossers 1997] and then to a 3-step transformation yielding a different regular hexagon [Chapman 1997; Fukuta 1996b] strongly concentric with the first, meaning they are parallel as well (Figure 1). Each transformation is parameterized by . At 0 , the first Fukuta hexagon is the positive Napoleon triangle plus its Star-of-David complementary equilateral. Similarly, all hexagon sequences in the paper can be interpreted as equilateral triangle sequences. At 0 , most include one or both Napoleon triangles. Iteration of the middle step in the 3-step Fukuta transformation is shown to create an infinite sequence of strongly concentric regular hexagons, each being 2 times (the size of) its predecessor (in edge length). The set of two transformations is enlarged to an infinite set by generalizing to what are called the hexagon construction operators, or hex operators, and applying them iteratively or in any order to generate infinite sequences of concentric regular hexagons. One such sequence has each hexagon 3 times its predecessor and rotated 6 from it (Figure 2). Another has