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

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.

Constructions of regular polygons | 5

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 B circle (see fig. 22). Remove the two sides GB and FB. After drawing circles with centers G D C and F, and radii equal to the radius of the original circle we obtain the last A missing point, the point H. This completes the construction of a regular hexagon inscribed in a circle. G F Now we can draw its sides and create tools to replicate a regular hexagon whenever we need it. 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. 6 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m

Fig. 25 Construction of a regular dodecagon P Start by removing the shape of the Q regular hexagon in the construction of a regular hexagon inscribed in a B circle (see fig. 23). Now we have to find centers of arcs between consecutive vertices of the D C hexagon. This way we will obtain six new points on the edge of the initial A R circle. These points are not labeled in our figure. By connecting all consecutive points G F 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

D C

A R

G F

The method for constructing a regular dodecagon inscribed in a circle can be used to create any regular polygon with 32n sides, 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 junc- tion 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.

Constructions of regular polygons | 7

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 C D side 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.

8 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m

Fig. 28 Construction of an octagon with a given side STEP 1: Draw a segment AB, and two lines per- C D pendicular to AB in points A and B. 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. Draw two slanted lines passing through points A B 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 G H 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 E F of the octagon. 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 C D first pair of perpendicular lines mark as G and H. These are the two remaining vertices of the octagon. Finally connect points A, B, D, F, H, G, E, C, and A B A. The octagon with a given side is ready.

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.

Constructions of regular polygons | 9

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 C A E square inscribed in a circle is ready. 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 2n sides, 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 F G circles with radius AB, and centers in points E and D respectively. The point of intersection of these two circles label as J. STEP 2: Draw line JA and perpendicular to it C A E line AG. Points of intersection of these two new lines with the circle are the missing vertices of the regular octagon. Now, connect points on the circle in the follow- H I ing order B, G, E, H, D, I, C, F and B. The regular octagon inscribed in a circle is ready. D J

There are many Islamic designs where squares and octagons are used as an invisible grid for the whole design. In the next figure we can see such example.

10 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m

Fig. 31 Istanbul, the Sultan Ahmed Mosque, or the Blue Mosque, the rosette ornament on the wall. The octagonal symmetry is the base for the ornament. The floral ornament was created using circles tangent to the main circle and rays from the center to the midpoints of the sides of an octagon.

ON A REGULAR PENTAGON AND A REGULAR DECAGON The regular pentagon is in particular a very interesting polygon, and there are many ancient, as well as, completely modern ways of constructing it. Here we will demonstrate only one of them where we develop a pentagon inscribed in a circle. It is an ancient method for constructing pentagons. The method has its origins in the Pythagorean School and it was described in details by Euclid. Readers interested in other constructions of a regular pentagon can find them in a small book by Allen John (see [1‎ ]). A good resource of methods for drawing regular polygons is a book by late Issam El- Said (see [2]).

The regular pentagon itself is a very mysterious figure and our ancient ancestors spent a lot of time investigating pentagons, their properties and relations with the world around them. The regular pentagon was studied by ancient Greeks, Christian mystics, and Islamic mathematicians as well. This is probably because its shape is strongly rooted in another famous mathematical phenomenon, so called golden ratio or golden section. We can prove that the side of the regular pentagon and its diagonal can be represented by the formulae:

푅 푅 푆푖푑푒 = √10 − 2√5 and 퐷푖푎푔표푛푎푙 = √10 + 2√5 2 2 Here R is the radius of the circumscribed circle. Moreover Side/Diagonal=Φ, where Φ is the Golden Number (Φ=1.61803…).

Constructions of regular polygon s | 11

Pentagon and pentagram – a simple star with 5 vertices, are frequently used in Islamic art and architecture.

CONSTRUCTION OF A REGULAR PENTAGON INSCRIBED IN A CIRCLE Now let us see how to construct a regular pentagon using the Pythagorean method.

Fig. 32 Construction of a regular pentagon in- B scribed in a circle STEP 1: Start by drawing a circle with center in A and radius AB. Then draw a line passing through points A and B, as well as a perpendicu- F K lar to it line AC. D E A C Construct the midpoint of AC, the point D, and draw a new circle with center in D and radius DB. This way you will obtain point E. Finally draw another circle with its center in B and radius BE. Point F formed by intersecting the last created circle and the initial circle is the second vertex of the pentagon. STEP 2: The three remaining vertices of the pen- B tagon you can obtain by drawing a sequence of arcs with centers in F (the first one), in G and in H, each arc having radius equal to BF. Finally by connecting points B, K, H, G, F and B F K D you will develop a regular hexagon inscribed in a E A C circle, furthermore by connecting every other point we obtain a regular pentagram. Below: (b) final construction of a regular pentagon G H (c) construction of a regular pentagram

B B

F K F K D D E A C E A C

G H G H (b) (c)

CONSTRUCTION OF A REGULAR DECAGON INSCRIBED IN A CIRCLE From a regular pentagon inscribed in a circle there is one small step to the construction of a regular decagon inscribed in a circle.

12 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m

Fig. 33 Construction of a regular decagon B Start from the last step of the regular pentagon N P inscribed in a circle. Remove sides of the pentagon. From each vertex of the pentagon draw a ray passing through the center of the circle. Intersections of each ray with the opposite side K F of the circle create the remaining 5 points of the D decagon. E A C Now draw segments connecting all consecutive points obtained on the edge of the circle to get M Q the decagon shape.

G H L

Although the construction of a regular decagon is quite complicated it is amazing in how many places we can find designs based on a regular decagon. Some of them are incredibly complex.

Fig. 34 Istanbul, the Yeni Camii, or the New Mosque or Mosque of the Valide Sultan – a very beautiful and complex ornament using decag- onal grid

Constructions of regular polygons can be a topic for a book larger than this one. In this book we will concentrate on those regular polygons only that we need in creating geometric ornaments from Istanbul mosques, streets and other places. Therefore, I will demonstrate three more constructions, not very often seen in literature. This will be the construction of a regular heptagon, i.e. a seven-sided regular polygon, a regular enneagon, i.e. a nine-sided regular polygon, and a regular hendecagon, an eleven-sided regular polygon.

Constructions of regular polygons | 13

ON SOME ODD SIDED REGULAR POLYGONS Regular polygons with odd number of sides usually are more difficult to construct. Those where the number of sides is a prime number are espe- cially hard to construct and the results are often not accurate.

CONSTRUCTION OF A REGULAR HEPTAGON Unfortunately there is no accurate construction of a regular heptagon. All known constructions have some minimal error. The construction shown in this book was invented by the author when he was a high school student. It has a very minimal error.

Fig. 35 Construction of a regular heptagon in- N scribed in a circle. B STEP 1: Draw a circle with the center in A and a point B on its edge. Draw a line passing through points B and A; and another line perpendicular to it, here the line AD. O Draw another circle with radius AD and center M in D. Connect points of intersection of both cir- A D cles by the line NM. Segment NM can be used as a measure for the side of the heptagon. Draw a line parallel to BN and passing through the point M and mark point O. Now the length of the segment BO can be used as a measure for the side of the heptagon. F

STEP 2: Now we are ready to construct vertices of the regular heptagon. Point B will be one of B N them. P Draw a series of circles, each one with radius U equal to BO, and centers B, P, Q, R, and S respectively. O STEP 3: Finally by connecting points B, P, Q, R, S, M T, U and B you will construct an almost regular A D heptagon. Side UT may not have exactly the same length as the other sides –it was created by Q T connecting the outlining points of the sequence of circles. In any geometry program we can measure the F length of each side of the heptagon and calculate R S the so called mean absolute error1. In our construction it is 0.016653=1.6653%.

1 There are many ways to calculate an error in a geometric construction of a regular polygon. One of the natural methods is so called mean absolute error. We calculate the mean length, say M, of all edges and then for any edge Ek we calculate its absolute error | M- |Ek| |, where |Ek| is the length of the edge Ek. By taking mean value of all absolute errors we get the mean absolute error. One can also use standard deviation to calculate these errors. 14 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m

CONSTRUCTION OF A REGULAR ENNEAGON The regular enneagon, a nine-sided regular polygon, is another regular polygon that cannot be constructed with a compass and straightedge. Con- struction shown in this book is relatively universal but not very accurate2.

Fig. 36 Construction of a regular enneagon using the perfectagon method B P STEP 1: Draw a circle with center in A and point B on its edge. Draw line AB and an- x4 other line perpendicular to it through the point A. C D A Use points D and E to draw two large circles with centers in P and Q respectively and radii equal to 2AB. Mark the point of intersection of these two circles as F. Divide the distance between BC into 9 equal segments. Finally draw a ray start- E ing from F and through the fourth point, counting from B, in the sequence of points between B and C. F A distance between the point B and newly created point P can be used as a measure for the side of the enneagon. STEP 2: Now, starting from the point B draw down-left a series of consecutive circles with radii equal to BP and centers in points B, Q, R, S, T, U, and V. For the last Q B created point Z we do not need to draw a P circle. Finally, we can draw segments connecting x4 consecutive points of the sequence B, Q, R, Z S, T, U, V, Z, P and B. This way we have con- R C structed a regular polygon with nine sides. D A The mean absolute error of this construc- V tion is 0.014815=1.4815%. S

T U E

Although geometric ornaments basedF on the nine-sided regular polygon are not as frequently seen as those based on decagons, or hexagons, the method presented here has a very significant value. It can be used to con-

2The method of construction described here was described by David Biagini on his web pages at http://www.perfectagon.com/tridecagon.html . This method, so called perfectagon method, can be used to create many other regular polygons. In general it is not a 100% accurate method. Constructions of regular polygons | 15 struct a number of other regular polygons. Every time we divide segment CD into an appropriate number of equal segments, and then we use the fourth from the top point, excluding B, to draw the line connecting points F and x4. A point obtained on the edge of the initial circle will be the point setting up the length of the side of the polygon. This way we can construct some unusual regular polygons, like the one with 11 sides or 13 sides.

CONSTRUCTION OF A REGULAR HENDECAGON INSCRIBED IN A CIRCLE The regular hendecagon, an 11 sided regular polygon, is surprisingly often seen in Islamic designs. I guess the major reasons are the 11 sides that make designs fairly complicated, and the odd number of sides that intro- duces a very specific kind of symmetry in the design.

We could use the perfectagon method to create this polygon. However, there is a very classical construction by Durer that is worth remembering.

Fig. 37 Durer’s construction of a regular hen- B decagon inscribed in a circle STEP 1: Start by drawing a circle with center in A and radius AB. Draw a segment connecting A C1 and B. Divide AB into two equal parts, to get C4 C3 point C1, then divide AC1 again into two equal C2 parts to get point C2, then divide again C1C2 into two equal parts to get point C3. Finally divide C1C3 into two equal parts to get point C4. A

STEP 2: Now, draw a sequence of circles, each with radius equal to BC4, and centers in E, D, E, F, G, F, H, I, J and K. Two more points on the cir- B cle will end this sequence, points L and M. D E Finally connect points B, E, G, I, K, M, L, J, H, F, D and B. The regular hendecagon is ready. Calcu- C1 C4 lations show that the side LM is slightly longer F C3 G C than the ten remaining sides. However, the 2 mean absolute error is only 0.008264=0.8264%. A H I

J K

L M

16 | Author: Mirek Majewski, source http://symmetrica.wordpres s . c o m

The fairy-tale of constructions of regular polygons could go and go forever. We just touched the tip of an iceberg. Our most complex regular polygon was the hendecagon, the one with 11 sides. But what about those regular polygons with 13, 17, 19, 23, and more sides? This is a story that is still waiting to be written.

CREDITS This document was originally published as an appendix in my book “Islam- ic Geometric Patterns in Istanbul”, ISBN: 978-83-231-2696-6, 2011. Nich- olas Copernicus University Press, Torun, Poland.

A new edition of this book is in preparation and it will go to print around July 2015.

All illustrations and text were done by the author. All sketches were com- pleted using Geometer’s Sketchpad®, a computer program by KCP Tech- nologies, now part of the McGraw-Hill Education. More about Geometer’s Sketchpad can be found at Geometer’s Sketchpad Resource Center at http://www.dynamicgeometry.com/.

BIBLIOGRAPHY 1. Allen John, Drawing Geometry – a primer of basic forms for artists, de- signers and architects, Floris Books, 2010. 2. Issam El-Said, Islamic Art and Architecture, The system of Geometric De- sign, Garnet Publishing, Reading, 1993. 3. Majewski Miroslaw, Islamic Geometric Patterns in Istanbul, Nicholas Copernicus University Press, Torun, Poland, 2011.

MIROSLAW MAJEWSKI, PROF. DR. PROFESSOR EMERITUS AT NEW YORK INSTITUTE OF TECHNOLOGY ABU DHABI CAMPUS, UNITED ARAB EMIRATES

Constructions of regular polygons | 17