Some Fractals in Goldpoint Geometry

Problems of interest in goldpoint geometry [1] arise from study of tile-figures that are obtained when goldpoints are marked on sides of triangles, squares, pentagons, etc. and joined by lines in various ways. Many combinatoric problems arise naturally in the course of such studies. Another type of problem is to determine how to combine collections of golden tiles in jig-saw fashion, so that they tile a given geometric figure (or the whole plane) with goldpoint marks on touching sides corresponding everywhere. Examples of these types of problems are the following: (i) Find how many different golden tiles can be formed from regular polygons; that is, find how many inequivalent golden triangles, squares, pentagons, etc. there are. (ii) Given a regular hexagon, find how many different ways it can be tiled by equilateral golden triangles, jig-saw fashion. In this paper I introduce a new type of problem into goldpoint geometry. I study a variety of fractals which are achieved by using as base the segment [0, 1], and a motif which involves the goldpoints of that segment. In Sections 2 and 3, the goldpoint dust set and snowflake are defined, and some of their properties are derived. In the following section, I describe goldpoint fractals which I dedicate to the memory of the inspirational American mathematician Herta T. Freitag, who passed away early in 2000 in her 91st year. In the final section, I present studies of fractals which are based on the regular pentagon. It is well-known (indeed the knowledge goes back to extreme antiquity, since it is mentioned in cabalistic literature) that the golden mean occurs frequently in the geometry of the pentagon [3] and its accompanying pentagram star. It is hoped that the results given below on pentagon fractals will add to existing literature on the pentagram.