Islamic Geometric Ornaments in Istanbul

►SKETCH 2 CONSTRUCTIONS OF REGULAR POLYGONS Regular polygons are the base elements for constructing the majority of Islamic geometric ornaments. Of course, in Islamic art there are geometric ornaments that may have different genesis, but those that can be created from regular polygons are the most frequently seen in Istanbul. We can also notice that many of the Islamic geometric ornaments can be recreated using rectangular grids like the ornament in our first example. Sometimes methods using rectangular grids are much simpler than those based or regular polygons. Therefore, we should not omit these methods. However, because methods for constructing geometric ornaments based on regular polygons are the most popular, we will spend relatively more time explor- ing them. Before, we start developing some concrete constructions it would be worthwhile to look into a few issues of a general nature. As we have no- ticed while developing construction of the ornament from the floor in the Sultan Ahmed Mosque, these constructions are not always simple, and in order to create them we need some knowledge of elementary geometry. On the other hand, computer programs for geometry or for computer graphics can give us a number of simpler ways to develop geometric figures. Some of them may not require any knowledge of geometry. For example, we can create a regular polygon with any number of sides by rotat- ing a point around another point by using rotations 360/n degrees. This is a very simple task if we use a computer program and the only knowledge of geometry we need here is that the full angle is 360 degrees. If we ignore even this fact, then by applying rotations about some specific angle a number of times, we can experimentally determine the size of the angle needed to create a regular polygon. This construction may not be very precise, but in such a case a better precision can be a matter of some practice only. Therefore, we have to ask one important question: should we care about classical methods of constructing geometric figures, if we have easier methods to create these figures? There are a few important points that we have to consider. First of all not everybody uses geometry programs. There are many of us who enjoy drawing geometric ornaments on large paper or Constructions of regular polygons | 1 cardboard and cannot apply tools available in computer programs – their tools are a ruler, a compass, and perhaps a few more things for measuring angles or distances. If you are such a person then you will need classical methods for constructing regular polygons and other shapes. Another rea- son is that even a basic knowledge of elementary geometry can help us in understanding the structure of an ornament and become aware of geometric principles not only in geometric ornaments, but also in architecture as well as in the whole nature around us. Finally, we should show some respect to the original creators of old geometric ornaments. These ornaments were created hundreds years ago; their constructors were artists, craftsmen, or architects. They did not have computers. Sometimes they did not have a compass similar to the one that we use in schools. Quite often they used a fixed compass, so called rusty compass, or a piece of rope with two sticks at its end. And with such primitive tools they created incredibly beautiful things. If we wish to appreciate their skills, and show some respect for their creations we should try to use similar methods to those used by ancient artists. Therefore, throughout this book we will use classical geometric constructions to create our ornaments. However, for the sake of simplicity, sometimes we will use some shortcuts. For example, if we need in a few places of our construction a regular decagon, then we will construct it only once and then we will not repeat it again and again. We simply suppose that the readers can fill this gap on their own. If you are using a computer program, like Sketchpad, then you can save the construction of a decagon, or any other figure, as a tool and then use the tool later whenever you need it. If you are drawing your ornaments by hand I suggest developing a template on a semitransparent paper for any of the complex figures and then using templates. Technical note: Many of our constructions will be created in a few steps. Usually there will be no more than three steps. Throughout this book we will use the following strategy: 1. Objects created in step 1 will be always presented using thin dashed lines. 2. Objects created in step 2 will be presented using thin, solid lines. 3. Objects created in step 3 (usually the final step), will be presented using a solid, medium wide line. 4. The initial segment, if we start from a segment, will usually be a medium wide, green line. All lines created in steps one and two will be dark blue. The final shape will usually be red. 2 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m 5. All points will be the same size – medium size, and labeled according to the order of their creation. 6. Sporadically I will use dotted lines for temporary elements, or another color to distinguish the new elements from the ones created before. The strategy described above will help us to explain and follow up particu- lar steps of a construction. In any case it is important to remember that one of our most frequent goals will be to create a kind of grid. Therefore, in the majority of examples a set of vertices of a regular polygon may be eve- rything that we really need. In some of our constructions we will frequently use the three fundamental school constructions – finding a center of a segment, drawing a line perpendicular to a given line and passing through a given point, and dividing a segment into a number of equal parts. Fig. 18 Construction of a center of a segment C Begin with the segment AB. Then draw two circles with centers in A and B respectively, and radius AB. By connecting their points of intersection, here points C and D, you will get the midpoint C of the segment AB. A B Note, the line CD is perpendicular to AB. E Therefore, in order to produce a line perpendicular to a given line and crossing it at a given point E we need to draw a circle with center in E and any radius. This way you will get points A and B, and the rest of D the construction you know already. Fig. 19 Dividing segment into 3 equal parts Draw a segment AB. Draw a circle with center in one of the ends of the segment and E radius AB. Select on the circle a point C – this can be any point. Draw a ray from the point A through C. Then draw two more circles with radius equal to AB and centers D in C and D respectively. Connect points E and B using a segment or a ray or a line. Construct lines parallel to BE through points C and D. Points of intersec- C tion of these lines with segment AB divide AB into three equal parts. The same method can be used to divide a A B segment into any number of equal parts. x1 x2 Constructions of regular polygons | 3 ON AN EQUILATERAL TRIANGLE, REGULAR HEXAGON AND REGULAR DODECAGON The equilateral triangle is the simplest one of all regular polygons. There- fore, its construction will be quite straightforward. From the construction of an equilateral triangle there is only a small step to the construction of a regular hexagon, and then to a regular dodecagon. Therefore, we will de- scribe constructions of these polygons together. CONSTRUCTION OF AN EQUILATERAL TRIANGLE FROM A SEGMENT Fig. 20 Equilateral triangle from a segment C Start by drawing a segment, here AB. At each end of the segment draw a circle with radius AB and the center in A and B respectively. Point of intersection of these circles, here the point C, is the third vertex of the equilateral triangle. Draw lines connecting points A, B and C. This will complete the whole construction. A B CONSTRUCTION OF A REGULAR HEXAGON FROM A SEGMENT Construction of regular hexagon can be carried out from the point where we finished construction of an equilateral triangle or even a small step earlier. For example, in order to create a regular hexagon from a segment we have to take the construction shown in figure 20, remove the two sides of the triangle and continue the construction further. Fig. 21 Construction of a regular hexa- G F gon from a segment Start by drawing a segment AB, and constructing point C, this is what was done for the equilateral triangle. D C E STEP 1: By constructing a new circle with the center in C and radius CA we get two new points: D and E. A B By drawing two more circles with radius equal to AB and centers in D and E respectively we get two more points: G and F. 4 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m STEP 2: It is easy to notice that points A, G F B, D, E, G and F are vertices of a regular hexagon. Now we have to connect these points to obtain a shape of a regular hexagon.

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