sketch 11 - wordpress.com€¦ · · 2014-12-27in the next figure i show what it looks like after...

G e o m e t r i c o r n a m e n t s f r o m t h e S u l t a n A h m e d M o s q u e | 1

►SKETCH 11

GEOMETRIC ORNAMENTS FROM THE SULTAN AH-

MED MOSQUE1 The Sultan Ahmed Mosque is one of the treasures of Istanbul. It is proba-

bly the most frequently visited mosque in Turkey. It is also my favorite

place. I can spend many hours in this Mosque and every time I am there I

discover something new.

Fig. 47 The Sultan Ahmed mosque is one of the masterpieces of Islamic architecture. The neighborhood in Istanbul around the mosque is today called Sultanahmet.

Sultan Ahmed I is buried in a mausoleum right outside the walls of the famous mosque.

1 This text contains fragment of the first edition of my book “Islamic Geometric Patterns in Istanbul”. The second, updated edition will be available in 2015.

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

The Sultan Ahmed Mosque was built between 1609 and 1616 by the Ot-

toman architect Sedefkar Mehmet Agha, during the rule of Sultan Ahmed I

(1590–1617). Its architecture is exceptional in many ways. This is the on-

ly mosque in the world with six minarets.

It is called the Blue Mosque because of the color of the Iznik tiles used to

decorate its interior. Its design is based on the standing opposite to it the

building of Hagia Sophia (The Church of Holy Wisdom). The Blue Mosque

is the last great mosque of the classical period of Ottoman architecture.

For a mathematician this mosque is particularly very interesting. Its shape

and the shapes of the domes and minarets can be modeled using many

mathematical objects: spheres, cylinders, cones, cubes, arcs, etc. We could

also discuss proportions of the building or its parts, their location, and

many other things. A complete mystery for a westerner are the multiple

muqarnas carved in white marble. However, all these 3D objects are be-

yond the scope of this book. We will concentrate only on geometric orna-

ments. A few of them can be found in this mosque. We have already ex-

plored one of these ornaments in the first example in this book. Another,

rather simple geometric ornament a visitor can see is on a small hexago-

nal fountain in the middle of courtyard. Construction of this ornament is

so simple, that we can skip it or treat it as an evening exercise after a long

busy day.

Fig. 48 The small hexagonal fountain in the middle of courtyard of the Sultan Ahmed Mosque

Here we see a very simple geometric ornament. In the next figure I show what it looks like after creating it on paper or a computer screen.

http://symmetrica.wordpress.com/


Fig. 49 The ornament from the fountain in the Sultan Ahmed Mosque

It is very easy to create this ornament and there are many ways to do it. For example you can create an equilateral triangle, find the center of it and then connect the center with centers of each side. Then using reflections about sides of the trian-gle create the whole ornament. While creating this ornament, I used a different method – I created a rectangular repeat unit. Can you find – how I did it?

Now, let us enter the Blue Mosque. The first thing we see inside is an in-

credible space with huge columns, walls covered with blue tiles and so-

phisticated arabesques. Just opposite the main entrance we can see the

minbar. In most of the mosques in Istanbul a minbar is the place where

we can find some interesting geometric ornaments. This one is not an ex-

ception. On both sides of the minbar there is a long strip of a geometric

ornament along the balustrade. Let us have a closer look at it, and try to

construct it?


Fig. 50 The minbar in the Sultan Ahmed Mosque contains a lot of arabesque decorations and quite an intriguing geometric orna-ment along its balustrade.

Fig. 51 A detail of the geometric ornament on the minbar. The two rounded squares show two different square tiles that can be used as a repeat unit to create the whole ornament.

As we can see, the repeat unit of the ornament can be created in a few

ways. One of them could be to create the repeat unit as a square with the

circled star ornament in the middle (rectangle 1). Another way could be to

create a square determined by the vertical bars going through the whole

ornament (rectangle 2). This way on the sides of the square we will have

centers of four circles and inside the square there will be four halves of a

circle with half of a star in each of them. Without any special reason we

will choose the second method. I suggest my readers to construct the re-

peat unit using the first method as an interesting exercise.

The following images show steps of our construction. We start with a

preparation step, and before going to the actual construction we will hide

all unnecessary elements. This way we will have much clearer under-

standing of what we are trying to do.



Fig. 52 Construction of the ornament on the minbar in the Sultan Ahmed Mosque

STEP 1: Construction of the first subgrid

Create a square, mark the center of each side. Draw diagonals of the square as well as lines connecting centers of opposite sides. This will be our first sub-grid.

Draw a circle with center in G and radius equal to GE (length of the side of the square), find intersection point of it with the right side of the square (point L). Draw line GL and mark point K. Note: LGB=60..

STEP 2: Construction of the second subgrid (circular)

Before starting this step hide all unnecessary ele-ments leaving only first subgrid, all points including and point K.

Draw the first circle with center in G and passing through K. Draw another circle with center in F pass-ing through K also. The two remaining circles with centers E and H we draw using the points of intersec-tion of the diagonal lines with the two existing cir-cles.

STEP 3: Use the existing points and the points of in-tersection of the four circles with diagonals of the square to draw a part of the ornament.

Some of the existing intersection points can be hid-den at this stage, but I will leave them in order not to confuse my readers.

K

L

F

G

H

EC D

A B

K

F

G

H

EC D

A B


STEP 4: Construction of the third subgrid (small cir-cles)

In order to create the remaining elements of the re-peat unit we need another subgrid of circles. Create 12 small circles – 4 in the middle and 8 close to the edge of the square.

Circles of the new subgrid are marked here using a thin solid line.

Mark all points of intersection of the new subgrid of circles with the existing subgrids and part of the ornament created in the previous step.

Now we are ready to finish the repeat unit.

STEP 4: Draw the missing lines of the ornament, i.e. the lines shown inside of small circles. Hide all points and the three subgrids: diagonals of the square, large circles and small circles. The repeat unit is ready (below).

Fig. 53 A geometric ornament created using the repeat unit constructed above. This ornament is very specific to Istanbul. We find it in many old-er mosques here, as well as in the Topkapi pal-ace.

The same ornament can be found also in many other mosques in Turkey.



The ornament from the minbar in the Sultan Ahmed mosque is rather un-

usual. In order to create it we needed two circular subgrids. Later we will

have another example where circular subgrids will be used.

In the Sultan Ahmed mosque there are many other geometric ornaments.

Let us have a walk around the interior of the mosque. There are many

window shutters and doors with interesting, sometimes very complex,

geometric ornaments carved in the wood. Constructing some of them can

be a real challenge. Let us try to see how we can approach this task. We

will start with a carving that contains a relatively simple ornament. This is

a very interesting ornament as it can be modified and expanded into many

more complex designs.

Fig. 54 A simple geometric ornament on one of the doors in the Sultan Ahmed Mosque

It is easy to notice that here we are dealing with only a small part of a larger ornament. If we draw a line crossing this ornament vertically exactly in the middle, and another one crossing this ornament horizontally, again exactly in the middle, then we will get lines of a rectangular grid. Each quarter of the picture will be a quarter of a rectangular repeat unit with ten-fold star in the middle.

We could construct exactly what we see it on the enclosed photograph.

However, the central part of the photograph is the least important. The

key to this and many other similar ornaments is the repeat unit with a

ten-fold star. The center of the pattern shown on the photograph is, in

fact, located on the periphery of a ten-fold star ornament. Therefore, we

will construct a repeat unit with the ten-fold star in the middle and then

we will show how to produce the ornament shown in the photograph.

We will start by creating a regular decagon, the 10 sided regular polygon,

inscribed in a circle. Actually we do not need the circle and we also do not

need edges of decagon, but if we leave them on the screen or on a paper,

then we will have a clear view of the location of the future star ornament.


Note also that the top and bottom vertex of the decagon are the points

where the horizontal lines of the main grid go. The location of vertical

lines of the grid we will determine later.

Let us start our construction.

Fig. 55 Construction of the repeat unit of the ornament on the door in Sultan Ahmed Mosque

STEP 1: Creation of the first subgrid

Start by constructing a decagon inscribed in a circle. Draw a subgrid of lines connecting every other vertex of the polygon, e.g. AC, BD, etc. Mark their intersec-tions outside of the polygon area.

Through the top and bottom vertices of the decagon draw horizontal lines. These are horizontal lines of the main grid.

On the right side of the decagon find two intersection points that can form a vertical line. Draw a line through these points. Do the same for the intersection points to the left of decagon. These are the two vertical lines of the main grid.

STEP 2: Creation of the second subgrid

From each vertex of the decagon draw a segment to the vertex that is fourth to the left and fourth to the right from the starting vertex, e.g. AG and AE. This way an internal subgrid will be created. We will need this subgrid for a short while and then we will hide it. Therefore I marked it using a dotted line.

D

B

C

A

EG

A



Now, we are ready to start drawing those fragments of the repeat unit that are based on the existing subgrids.

Using medium line connect each vertex of the decagon with an intersection of the first subgrid with the sup-plementary grid created just a while ago and extend your new lines to the lines of the main grid.

Now we can clearly see fragments of a ten-fold star, and fragments of the ornament outside of the deca-gon.

Before proceeding to the next step hide the first and second subgrid. We

do not need them anymore.

STEP 2 Construction of the third subgrid and internal part of the ten-fold star ornament

Draw the third subgrid (thin solid lines) connecting opposite intersections of the first subgrid with the sup-plementary subgrid. You will obtain a clear outline of the ten-fold star.

Mark internal intersection points of the new subgrid. You will get a ring of 10 points close to the center of the circle.

Now, we are ready to develop the internal part of the ten-fold star.

Using the third subgrid draw internal edges of the ten-fold star ornament.

Your construction is almost ready. You will have to re-move all subgrids, remove the circle and the edges of the decagon.

Finally, you will have to add some missing elements in the corners of the repeat unit.


STEP 3: Construction of the peripheral elements

Draw two horizontal lines passing through the points P, Q, R, S, and through the same points draw slant lines, parallel to the longest segments of the ornament. Use these lines to add short segments connecting points P, Q, R, and S with the boundaries of the repeat unit, e.g. PU and PT.

The same additions should go to the remaining corners of the ornament

Now, you have to do a big cleaning and use your repeat unit to create a larger ornament.

The next figure shows the repeat unit and an ornament created using this

repeat unit. The shaded area marks the part of the ornament that is visible

on the photograph from the Sultan Ahmed Mosque.

Fig. 56 The repeat unit (below left) and the final ornament (below right)

The repeat unit can be extended to the right and left, as well as down or up by adding some more space for the periph-eral part. This way we can create hun-dreds of similar ornaments, all based on the ten-fold star ornament. Some of them can be found in Istanbul mosques.

I suggest my readers experiment a bit with this ornament. Coloring it can

be quite an interesting task. Changing or expanding the peripheral part of

the ornament can be another fascinating activity. Let us see another ex-

ample based on the same ten-fold star ornament with a larger peripheral

part.

U

T

S

P Q

R



Fig. 57 Another geometric orna-ment based on the ten-fold star ornament

This ornament is part of the old door inside the Sultan Ahmed Mosque (right side from the en-trance).

This time the ten-fold star is the center of the ornament and peripheral

parts are where they should be – on the sides of the star. If we compare

this ornament with the one that we created a while ago, we can see that

here the star was rotated 90 degrees. Therefore, if we wish to reuse our

former construction we should rotate the photograph 90 degrees. We can

easily notice that there are another four identical ten-fold stars in the cor-

ners of the ornament. Here we see only a quarter of each of them. This

suggests that the repeat unit of the ornament can have a form of a flat-

tened hexagon shape (see the picture). One can also create this ornament

using a rectangular repeat unit. This method is more convenient if we

wish to produce an ornament covering larger area of the plane.


Fig. 58 The ornament rotated 90 degrees

Two versions of the repeat unit can be used to create this ornament.

The black dashed line shows places where the main grid can be. Lines of the grid can be used as mirrors for the ornament. This is also the place where pieces of wood were joined together.

The solid blue line shows another way of creating the repeat unit. This is the probably how the original tem-plate for the ornament looked. It reminds us of the tem-plates we have seen in the Mirza Akbar scroll. We will use this approach. We also notice that it is sufficient to create only the top-left part of the repeat unit and then rotate it about the point dividing the dashed line into two equal parts. On the picture this is the large yellow point.

Let us start by creating the three subgrids for the ten-fold star ornament.

We will copy some introductory steps from the previous construction.

Fig. 59 All three subgrids necessary to create the ten-fold star ornament inscribed in a circle – the circle and the hexagon are not shown.

We also added two rays from the center of the picture – a horizontal one and a vertical one. These lines are the top and left edges of the repeat unit. The two remaining edges of the repeat unit will be added later.

The subgrids on the picture are exactly the same that we developed for the previous construction. Many of these lines and points are not necessary for our con-struction. But we will keep them for a while in order not to confuse the reader.

Now, we can start creating the ten-fold star ornament. However, we will

only create the part that is inside of the repeat unit.



Fig. 60 Construction of the repeat unit

STEP 1: Construction of the ten-fold star ornament

Follow the subgrids for the ten-fold star ornament to draw the thick lines shown on the picture. This part is easy; you have to repeat almost exactly what you did in the previous construction.

We do not need all these subgrids that we created for the ten-fold star. You can remove them (hide). You should get a con-struction that looks like the one to the right.

STEP 2: Creation of the right boundary of the repeat unit.

First draw the thin solid line that separates the top-left and bottom-right part of the repeat unit. This is easy, you have to draw a line passing through the points labeled on the picture as A and B. Draw another line passing through the center of the star (point D) and the point labeled as C. The intersection of this line with the line AB is the rotation point. Having this point that is half way between the left and right lines of the main grid, makes it easy to determine the location of the right boundary of the repeat unit. Just draw a circle with the center in E and ra-dius EF. Then through the intersection of the circle and line AB draw a vertical line perpendicular to the horizontal edge of the grid.

B

A

B

A

F

E

B

A

C

D


STEP 3: Construction of the peripheral part.

Draw the subgrid (dashed lines) for the peripheral part of the ornament and add the remaining segments of the peripheral ornament.

Note, some of the segments of the peripheral part of the orna-ment do not follow any particular subgrid lines, they connect existing points and they are usually parallel to some of the exist-ing parts of the ornament.

STEP 4: The rotation thing

Remove or hide all subgrids for the peripheral part.

If you are doing this construction in a geometry program then your task at this moment is quite simple – select the whole or-nament, and use the rotation tool. The rotation point is the large blue point. The rotation angle is 180 degrees.

If you are developing this construction by hand, then probably the easiest way is to copy it on a transparent paper, rotate the paper 180° and position it on the original picture in such a way, that the thin solid lines will overlap. You should get a picture similar to the one to the right.

Now, we need a big cleaning. Hide all unnecessary points, lines, etc. The

repeat unit for this ornament, as well as one of the final versions of the

ornament is shown in the next figure. My ornament was created in

Sketchpad.

B

A

C

D



Fig. 61The repeat unit of the ornament from the photograph (fig. 57 rotated 90 degrees), and the whole ornament combined by using four copies of the repeat unit. In this case the repeat unit has a rectangular shape and it is similar to the templates from the Mirza Akbar scroll.

Doors and windows in the Sultan Ahmed Mosque are a real gold mine of

interesting geometric ornaments. Exploring them can be a task for hours

or even days. Constructing them can take even a few weeks. Some of these

ornaments are inaccessible to a tourist, as they are in the sacred space

where tourists have no right to enter. In such case a good camera may be

a very helpful tool.

Analysis of geometric ornaments on the doors and windows of the Sultan

Ahmed Mosque can only be a very bulky topic for a good master thesis of

a student of mathematics or arts. Both of them will benefit from this task

and both of them will have a lot of joy. Let us look briefly at some of the

treasures to explore in this mosque.

The next collection of photographs shows some of these ornaments. The

pulpit shown in the next figure is far away from where a tourist can go.

With a good camera and on a sunny day there is a possibility to take a

good photograph of it.


Fig. 62 Pulpit from the Sultan Ahmed Mosque

The ornament on top and bottom of the pulpit is slightly similar to the one that we have seen on the minbar. Later we will have an opportunity to see and construct it while visiting a few other places.

The ornament in the middle is similar to the two last ornaments that we created a while ago. This ornament contains a very large and complex peripheral part. How-ever, we can easily see the logic of this part.

.



Fig. 63 Two doors in the Sultan Ahmed Mosque

The ornament on the left door is rea-sonably simple and can be a nice tar-get for homework.

The right door probably contains the most complex ornament that we have seen until now. Once more there is a ten-fold star in the middle and a whole cosmos of shapes around it. However, there is again a very clear logic behind this cosmos.

Fig. 64 A row of doors on the right side of the Sultan Ahmed Mosque. Each door contains an ornament simi-lar to the examples discussed in this chapter


CREDITS This text contains fragment of the first edition of my book “Islamic Geo-

metric Patterns in Istanbul”. The second, updated edition will be available

in 2015.

All sketches were created using Geometer’s Sketchpad®, a computer pro-

gram by KCP Technologies, now part of the McGraw-Hill Education. More

about Geometer’s Sketchpad can be found at Geometer’s Sketchpad Re-

source Center at http://www.dynamicgeometry.com/.

All rights reserved. No part of this document can be copied or reproduced

without permission of the author and appropriate credits note.

MIROSLAW MAJEWSKI,

NEW YORK INSTITUTE OF TECHNOLOGY,

COLLEGE OF ARTS & SCIENCES,

ABU DHABI CAMPUS,

UNITED ARAB EMIRATES


sketch 11 - wordpress.com€¦ · · 2014-12-27in the next figure i show what it looks like after...

Documents