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Star Polygons Research of Star Polygons IB EXTENDED ESSAY Candidate Name: Šimon Rovder Supervisor: RNDr. František Kosper Candidate No.: 000771-032 Word count: 3931 Šimon Rovder – 000771-032 Abstract This essay presents the steps of research of star polygons with the goal of proving the existence of star polygons of any amount of vertexes. The only information this research began with was the knowledge of two star polygons (pentagram and heptagram) and the knowledge of the fact that they are called star polygons. This research, which started from nothing, discovered basic rules behind the existence of star polygons, presents a clear way of how to find them and successfully proves the existence of an infinite amount of star polygons. This essay presents a proof of the existence of all star polygons with the amount of vertexes greater than 5 and different from 6. Exploration of this behavior of the number 6 is later explained and a valid (yet unusual) way of drawing it was eventually found, demonstrated and explained. In the further sections of this essay, the range of the research was expanded to include research and a proof of the fact that any amount of star polygons of any amounts of vertexes can all be drawn together within one larger star polygon. The appendix of this essay contains other information and formulas (related to star polygons) discovered during this research. Some sections of this essay use that information, yet it is not directly linked with the research question. It is included in this essay mainly in reference to any future research based on this work, which is of course encouraged. 2 Šimon Rovder – 000771-032 Contents Abstract ......................................................................................................................................................... 2 Introduction .................................................................................................................................................. 4 Star Polygons ................................................................................................................................................. 5 Star Polygon Definition ............................................................................................................................. 5 Included Star Polygons (the ones this research started with): ............................................................. 6 Finding the star polygons .............................................................................................................................. 8 Empirical search ........................................................................................................................................ 8 Finding the Vertex Jump Variable k .......................................................................................................... 9 Can any star polygon be drawn? ................................................................................................................. 14 Polygon projections within Star Polygons .................................................................................................. 20 Inward Projections .................................................................................................................................. 20 Outward Projections ............................................................................................................................... 25 Can a combination of many different regular or star polygons be projected into a single star polygon? ................................................................................................................................................................ 30 Conclusion ................................................................................................................................................... 36 Bibliography ................................................................................................................................................ 37 Appendix ..................................................................................................................................................... 38 Angles at the vertexes of any order ........................................................................................................ 38 Length of any line segment within a star polygon of radius 1 ................................................................ 39 Area of any star polygon of radius 1 ....................................................................................................... 41 3 Šimon Rovder – 000771-032 Introduction “Can you draw a five point star with one move?” A popular puzzle among younger children and an interesting problem for a mathematician, if of course, slightly expanded. “Can you draw a seven point star with one move” sounds a bit better, but “Can you draw an n point star with one move” sounds just about right for the initiation of actual research. Seven years ago, I came across a seven vertex star drawn in a very similar fashion as the highly famous pentagram. I found it fascinating, yet I did not continue to pursue this beautiful area of geometry. Now I decided to come back to this problem and dedicate my extended essay to its expansion. I decided to explore this problem by myself and with no research of anything already discovered. Thus what this essay presents, are all my own findings, except for the Prime number Theorem used once on page 11 in a proof related to prime numbers. So, Can we draw an n vertex star polygon? 4 Šimon Rovder – 000771-032 Star Polygons Star Polygon Definition In order to even start working with the star polygons this puzzle was based on, there had to be a definition for them first. The puzzle is: Drawing a star polygon by one move (one continuous line), while only changing the direction of drawing at the vertexes of the star polygon. As the whole idea of studying these star polygons originated from the existence of the pentagram and heptagram, the main definition of what will be considered a “star polygon” was derived from them as well. By this definition, a polygon has to fulfill the following criteria in order to be my star polygon: 1. The polygon is a single self-intersecting polygon. 2. The vertexes of a star polygon are located on the perimeter of one circumference 3. Adjacent points are equally far from one another. 4. All vertexes are connected to two other vertexes, skipping a constant amount of vertexes with this connection. By this definition, the range of this topic is clearly defined to the star polygons we want. The second and third criteria apply to all regular polygons and the first and fourth are 5 Šimon Rovder – 000771-032 the main defining characteristics of star polygons. The absence of the circumference (criterion 2) would eliminate any ability to order the vertexes and check the fourth criterion, while the absence of the third criterion would deform the star polygon and ruin equality in lengths of sides. So we can only check if a polygon is a star polygon if we know the second and third criteria are met. Examples So from the definition of a star polygon, we can take a look at a few valid examples of star polygons as well as at some other shapes excluded by this definition. Included Star Polygons (the ones this research started with): Figure 1 : Pentagram Figure 2 : Heptagram 6 Šimon Rovder – 000771-032 Excluded Polygons: Star of David (in violation of the first Tilted Pentagram (In violation of Criteria 2 criterion – made of more than one and 3, and undeterminable for criterion 4): polygon): Figure 3 : Star of David Figure 4 : Tilted Pentagram Investigation of star polygons of an amount of vertexes of 1 and 2 can be excluded as well, since a polygon cannot be formed out of less than three vertexes. Also since a star polygon has all its vertexes connected to different two vertexes than it would if the polygon was regular, star polygons of less than 5 vertexes can be excluded from this research, as they have no way of fulfilling this criterion (not enough vertexes to change all connections). Thus this research will be focused on star polygons of 5 and more vertexes (the ones that correspond to the puzzle), making the pentagram the simplest star polygon there is. 7 Šimon Rovder – 000771-032 Finding the star polygons Empirical search The initial hypothesis was that for every regular polygon should be at least one corresponding star polygon. This was a valid assumption as after excluding the star polygons of 1, 2, 3 and 4 vertexes, it seems fairly plausible. To make orientation easier I decided to define a number n for each star polygon. This number would be the number of vertexes the star polygon had and should give some structure to any data I was to collect. So from all we know by now: At the very beginning, the search for star polygons was empirical. It was a matter of drawing the vertexes and attempting to connect them by definition. Also the differences between the pentagram and the heptagram suggested the existence of yet another specific number for each star polygon. A number that defines the number of vertexes that are to be skipped when connecting vertexes to form the star polygon. This number will from now on be referred to as k. These two numbers n and k should come in specific combinations for each star polygon and should tell us that: “To draw a star polygon of n vertexes, you have to connect each vertex to the next kth vertex in both directions of the circumference on which
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