Generalized Star Polygons and Star Polygrams 1

GENERALIZED STAR POLYGONS AND STAR POLYGRAMS EIDUR SVEINN GUNNARSSON AND KARL THORLAKSSON Abstract. We extend the definition of regular star polygons and other regular star polygrams to be drawn within some non self-intersecting polygon instead of just regular polygons. Instead of only connecting corners some con- stant apart, we allow any two corners of the polygon to be connected to form a figure. These figures can be thought of as permutations consisting of some cycles where each cycle forms a polygon. We define new permutation patterns called interacting cycle patterns that we use to describe properties of the permutations. We describe some necessary properties for a permutation drawn within a polygon, such that its drawing forms a star. Then we describe sufficient properties of permutations, such that when drawn within any available polygon they form stars. Finally we describe sufficient properties of permutation drawings, that form a star in some polygon. 1. Introduction Thomas Bradwardine, an English 14th-century mathematician, was the first to study star polygons in Geometria Speculativa. Later in 1619 Johannes Kepler studied them further in Harmonices Mundi. [2] They both defined them from regular polygons in similar fashion, Kepler defined star polygons as an augmentation of regular polygons, by extending each side of a regular polygon until it meets a non- neighbouring side forming a vertex. [5] See Figure 1.1 for an example of how a pentagon is augmented to a 5-pointed star. Figure 1.1. Scan from Kepler's Harmonices Mundi [5] Star polygons have been studied and used for tilings, tessellations and forming higher order polytopes. Definition 1.1 (Polygon). A polygon is an ordered set of points O0;O1;:::On−1 , f g where consecutive points in the set along with the points O0 and On−1 are connected by a line segment. The line segments are called sides and the points from the set are called corners. The size of a polygon is its number of corners. Definition 1.2 (Regular polygon). A regular polygon is a polygon where all sides are of equal length and the interior angle between each two consecutive sides are equal. An example of a regular polygon can be seen in Figure 1.2. Date: June 15, 2016. 1 2 EIDUR SVEINN GUNNARSSON AND KARL THORLAKSSON 1 2 0 3 4 Figure 1.2. A regular polygon of size 5 The Schläflisymbol notation is used to describe regular polygons, polyhedra, and their higher-dimensional counterparts. The Schläflisymbol n is used to denote a regular polygon of size n. For example, the regular polygonf g in Figure 1.2 is denoted as 5 . Regular star polygons are denoted by the Schläflisymbol p=q where p=q isf ag fully reduced fraction. The regular star polygon p=q has thef sameg f g vertices O0;:::;Op−1 as the regular polygon p . However, now the line segments f g f g connect each point Ok to Ok+q, where indices are taken modulo p. Figure 1.3 shows an example of the regular star polygon 7=2 . f g 2 1 3 0 4 6 5 Figure 1.3. The regular star 7=2 f g The Schläflisymbol p; q; r; : : : ; s , where p; q; r; : : : ; s is a list of rationals, is used to denote regular polytopesf and tilings.g These are regular polygons and regular star polygons (2-polytopes) amongst regular polyhedra (3-polytopes) and so on, along with tilings of these shapes. For Schläflisymbol for higher dimensional objects and tessellations refer to Coxeter's Regular Polytopes. [3] Definition 1.3 (Boundary polygon). A boundary polygon is a non self-intersecting polygon, that is a polygon where its sides only meet at its corners. Definition 1.4 (Figure). Given a boundary polygon, a figure is a collection of vertices given by its corners. Each vertex of the figure is connected to exactly two other distinct vertices by line segments called edges. The edges must not lie outside of the boundary polygon and each edge connects precisely two vertices. The size of a figure is its number of vertices. 12 3 5 0 4 Figure 1.4. A polygon of size 6 GENERALIZED STARS 3 Given the boundary polygon seen in Figure 1.4, we can form a figure. The edges connecting to corner 0 canP be any two of (0; 1); (0; 2); (0; 5) . However we cannot use the edge (0; 4) or (0; 3), since (0; 4) wouldf lie outside ofg while (0; 3) would connect three vertices by also intersecting corner 5. So either edgeP will break the requirements for this to be a figure. A permutation is a one-to-one correspondence from the set Zn = 0; 1; : : : ; n 1 to itself. These will be written in cycle notation, for example thef permutation− g π = (021)(34)(5) maps 0 2 ! 2 1 ! 1 0 ! 3 4 ! 4 3 ! 5 5: ! The permutation π has three cycles (021); (34) and (5) of lengths 3; 2 and 1 respec- tively. The length of a permutation is the number of elements of the set Zn. So the length of π is 6. Permutations are commonly written in one-line notation, where a permutation of length n, σ = σ0σ1 σn−1 maps i σi. So permutation the permutation in the example above is written··· π = 201435! in one-line notation. We will use permutations to indicate which vertices are connected by an edge in a figure. Definition 1.5 (Permutation drawing). Let = O0;O1;:::;On−1 be a boundary polygon and π be a permutation of lengthPn. Givenf the pair (π;g ), a drawing is formed. P As a basis for a drawing use the corners of . Then draw line segments as indi- cated by π, such that when π maps a b thenP a line segment is drawn between ! vertices Oa and Ob. This drawing is a figure if and only if the conditions of Defini- tion 1.4 are met. Remark. Since the edges of a drawing are not directed like the mappings in permutations, reversing any cycle of a permutation will result in the same drawing. In the following section we will explore figures further and use them to define generalized stars. In Section 3 we look at some necessary properties of permutations, allowing us to exclude permutations that never form stars. Then in Section 4 we look at sufficient properties of permutations that, when drawn in a polygon such that they form a figure, will always be a star. In Section 5 we look at sufficient properties of permutations, forming a star in some boundary polygon, which is formed from the permutation instead of being chosen beforehand. Finally in Section 6 we look at some open problems and ways that stars can be explored further. 2. Generalized stars What is arguably most iconic about the shape of regular star polygons is how, when drawn within polygons, their complement area is a collection of triangles. Therefore we define generalized stars as a subset of figures that have triangles between all of their arms. Definition 2.1 (Boundary triangles). The area bounded by a figure and its boundary polygon is a collection of polygons. Those of which are triangles are called boundary triangles. 4 EIDUR SVEINN GUNNARSSON AND KARL THORLAKSSON In Figure 2.1 the boundary triangles of the figure have been shaded. The polygon formed between vertices 3 and 4 is not a boundary triangle because of the edge connecting vertices 2 and 5. 2 3 1 4 0 5 7 6 Figure 2.1. Shaded regions represent the boundary triangles of this figure Definition 2.2 (Star). A figure is a star (figure) if the size of its boundary polygon is the same as its number of boundary triangles. 2 3 1 4 0 5 7 6 Figure 2.2. A figure that is a star Figures that can be drawn in one continuous trace are called unicursal, like the figure in Figure 2.2. multicursal figures are drawn with more than one trace, like the figure in Figure 2.3 which is drawn with two traces. A trace are continuously drawn line segments between the corners of a boundary polygon, where each corner is visited exactly once except the start and end corner. Unicursal figures are polygons, while multicursal figures are polygon compounds. The number of cycles a permutation has tells us how many traces we need to draw a figure. 2 3 1 4 0 5 Figure 2.3. A multicursal star drawn with two traces GENERALIZED STARS 5 Definition 2.3 (Regular stars). Regular stars are stars which consist of the regular star polygons denoted by the Schläflisymbol p=q and regular polygon compounds. The regular star polygons are unicursal whilef theg regular polygon compounds are multicursal. Regular polygon compounds are denoted by the Schläfli symbol g p=q where p=q is a reduced fraction and g is an integer greater than 1. A regular polygonf g compound consists of the same vertices as gp , where each corner Ok is connected f g to Ok+gq by an edge. Remark. The regular star polygon given by the Schläflisymbol p=q , where p=q is a fully reduced fraction, is the figure = (π; ). Where is af regularg polygon of size p and π = (0; q; 2q; : : : ; (p 1)q),F where allP elements areP modulo p. Similarly a regular polygon compound g p=q− is the figure (π; gp ) where f g f g π = (0; gq; : : : (p 1)gq) ((g 1) + 0; (g 1) + gq; : : : ; (g 1) + (p 1)gq): − ··· − − − − Definition 2.4 (Irregular stars).

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