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 exploring 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 noticed 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 rotating 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 meth-

ods 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

C o n s t r u c t i o n s o f r e g u l a r p o l y g o n s | 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 reason 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 | A u t h o r : M i r e k M a j e w s k i , s o u r c e h t t p : / / s y m m e t r i c a . w o r d p r e s 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 particular 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 everything 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

E

Note, the line CD is perpendicular to AB. 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 the construction you know already.

D

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 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 in C and D respectively.

E
D

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 intersection of these lines with segment AB divide AB into three equal parts.

C

The same method can be used to divide a segment into any number of equal parts.

B
A

x₂

x₁

ON AN EQUILATERAL TRIANGLE, REGULAR HEXAGON AND REGULAR DODECAGON

The equilateral triangle is the simplest one of all regular polygons. Therefore, 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 describe 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 hexagon from a segment

G

F

Start by drawing a segment AB, and constructing point C, this is what was done for the equilateral triangle.

C

D

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.

STEP 2: It is easy to notice that points A, 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.

G

F

C

D

E

A

B

CONSTRUCTION OF AN EQUILATERAL TRIANGLE INSCRIBED IN A CIR- CLE

We can imagine another situation, when a circle, or sometimes another shape, is given and we have to fit an equilateral triangle inside it. In such a situation we do not have the side of the triangle, but we may have a center of the circle and a radius of it, or a point on its perimeter.

In the next construction we will show how an equilateral triangle inscribed in a circle can be constructed.

Fig. 22 Construction of an equilateral triangle inside of a circle Start by drawing a circle with center A and one point on its edge, here this is the point B. Then draw from B a circle with radius AB, and then we draw two more circles with centers in C and D, both with radii equal to AB.

B A
C
D

After connecting points B, G and F we obtain an equilateral triangle with its center in A and three vertices on the initial circle.

G

F

CONSTRUCTION OF A REGULAR HEXAGON INSCRIBED IN A CIRCLE

Construction of a regular hexagon inscribed in a circle can be carried out from the point where we finished constructing an equilateral triangle inscribed in a circle.

Fig. 23 Construction of a regular hexagon inscribed in a circle Start this construction from the point where we stopped while constructing an equilateral triangle inscribed in a circle (see fig. 22). Remove the two sides GB and FB.

B
D
C

After drawing circles with centers G and F, and radii equal to the radius of the original circle we obtain the last missing point, the point H.

A

This completes the construction of a regular hexagon inscribed in a circle. Now we can draw its sides and create tools to replicate a regular hexagon whenever we need it.

G

F

H

In Islamic art the regular hexagon can be a base for many interesting designs, not necessary geometric ornaments.

Fig. 24 Istanbul, Bayezid II

Mosque, or Beyazıt Camii in

Turkish – a geometric ornament based on multiple regular hexagons

CONSTRUCTION OF A REGULAR DODECAGON INSCRIBED IN A CIRCLE

From a regular hexagon there is one small step to a regular dodecagon, i.e. the twelve-sided regular hexagon. Simply, we have to continue the construction shown in the figure 23 for the regular hexagon inscribed in a circle. In figure 25 we show one of the possible methods how this can be done. In fact, the method presented here can be used to double the number of sides for any regular polygon. We will use it later in a few other constructions.

Fig. 25 Construction of a regular dodecagon

P
Q

Start by removing the shape of the regular hexagon in the construction of a regular hexagon inscribed in a circle (see fig. 23).

B

Now we have to find centers of arcs between consecutive vertices of the hexagon. This way we will obtain six new points on the edge of the initial circle. These points are not labeled in our figure.

D
C

A
R

G
F

By connecting all consecutive points on the edge of the initial circle we get a shape of a regular dodecagon.

H

Rectangular tiles with a regular dodecagon shape in the middle.

P
Q

B

D

C

A
R

G

F

H

The method for constructing a regular dodecagon inscribed in a circle can be used to create any regular polygon with 32ⁿsides, for any n>1.

Regular hexagons and dodecagons, although a bit complex to construct, are often seen in unexpected places. The photograph above shows a geometric ornament created by using triangles and pentagons. As a result of connecting multiple triangles a dodecagon shape was created on the junction of four tiles. This photograph was taken in Istanbul in the Secreci area. From this photograph we can draw one important conclusion – many ornaments can be constructed by connecting together a variety of rectangular tiles. The next figure shows another application of the dodecagon.

Fig. 26 The marble muqarnas in the iwan of the Sultan Ahmed Mosque in Istanbul. We can easily notice that the shape of the muqarnas is based on half of a regular dodecagon. Its rough floor plan (below) reveals clearly a dodecagon frame and all elements of the muqarnas are organized inside of this figure.

ON A SQUARE, AN OCTAGON AND HEXADECAGON

These three relatively easy constructions can be a part of a high school mathematics class curriculum. If we know how to draw a square then we can easily extend its construction to get a regular octagon, and then by dividing each side of the octagon create a regular hexadecagon.

CONSTRUCTION OF A SQUARE WITH A GIVEN SIDE

Fig. 27 Construction of a square with a given side

C

D

Draw a segment AB, construct two lines perpendicular to AB passing through points A and B. Finally draw two circles with centers in A and B respectively, and radii equal to AB. This way we will obtain the two remaining vertices of the square, points C and D. By connecting them we get a square with the given side.

A

B

CONSTRUCTION OF A REGULAR OCTAGON WITH A GIVEN SIDE

Construction of a regular octagon with one side given is a little more complicated task. We have to begin by creating two 45° angles. So, let us start.

Fig. 28 Construction of an octagon with a given side STEP 1: Draw a segment AB, and two lines perpendicular to AB in points A and B.

D
C

Draw two circles with radius AB and centers in A and B respectively. Points of intersection of the perpendicular lines with circles label as X and Y.

B
A

Draw two slanted lines passing through points A, Y and B, X respectively. Obtained this way points C and D are the two next vertices of our octagon.

X
Y

Draw another pair of lines perpendicular to the line AB and passing through points C and D.

STEP 2: Draw two circles with radius AB and centers C and D respectively. Mark their points of intersection with the new perpendicular lines as E and F. This way we have two more vertices of the octagon.

H
G
F
E

Finally draw two more circles with still the same radius AB and centers E and F respectively. Points of intersection of these circles with the first pair of perpendicular lines mark as G and H. These are the two remaining vertices of the octagon.

D

C

Finally connect points A, B, D, F, H, G, E, C, and A. The octagon with a given side is ready.

B

A

X
Y

Construction of a hexadecagon with a given side is similar to the above construction of a regular octagon. This time we have to exercise a bit with creating angles of 22.5°, i.e. 45°/2. I will leave this construction as an exercise to the readers of this book.

CONSTRUCTION OF A SQUARE INSCRIBED IN A CIRCLE

Now, let us have a look again at constructions of a square and an octagon provided that they must be inscribed in a circle. The first step is very easy.

Fig. 29 Construction of a square inscribed in a circle

B

Draw a circle with the center in point A and a point B on its edge. Draw a line passing through points A and B, and then draw another line perpendicular to AB and passing through A. The three points C, D, and E of intersection of the circle with the two lines are the remaining vertices of the square. Connect them and the square inscribed in a circle is ready.

C

A

D
E

Note, orientation of the square depends only on the location of the point B in respect to A.

CONSTRUCTION OF A HEXADECAGON INSCRIBED IN A CIRCLE

Constructions of a regular octagon, and regular hexadecagon, inscribed in a circle are straightforward continuations of construction of the square. We have to divide arcs BE, ED, DC and CB in two equal parts. This way we will get an octagon inscribed in a circle. By dividing arcs between two consecutive vertices of an octagon in two equal parts we will produce a regular hexadecagon. This way we can create a sequence of regular polygons inscribed in a circle with 2ⁿsides, where n=2, 3, 4, … .

Now let us look at the construction of the octagon inscribed in a circle.

Fig. 30 Construction of a regular octagon inscribed in a circle.

B

STEP 1: Start with the square BCDE inscribed in a circle. Remove sides of the square. Draw two circles with radius AB, and centers in points E and D respectively. The point of intersection of these two circles label as J.

F

G

STEP 2: Draw line JA and perpendicular to it line AG. Points of intersection of these two new lines with the circle are the missing vertices of the regular octagon.

C

A

D
E
H

Now, connect points on the circle in the following order B, G, E, H, D, I, C, F and B. The regular octagon inscribed in a circle is ready.