# Nonagons in the Hagia Sophia and the Selimiye Mosque

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## Abstract

One of the classical topics in geometry is the ruler and compass construction of good approximations of the regular nonagon. We propose a method to choose a desired error of approximation, based on new linear third-order recurrence relations. It is related to patterns shown on the balustrade of minbar of Selimiye Mosque in Edirne (Turkey, 1569–1575), where apparently regular nonagons are placed over a net of regular hexagons. This kind of ornament also occurs in Hagia Sophia in the minbar, in freezes stucco carvings and in window grilles. In Medieval Islamic art and architecture the use of the nonagon is not frequent, but it is remarkable that this particular grid of interlocked nonagons and hexagons appears in the decorations of the works of Ottoman architect Mimar Sinan. The pattern is present in his masterpiece, the Selimiye Mosque, in the Hagia Sophia and in other works in the Istanbul city. Looking at practical ways for the construction of the pattern, we provide simple procedures to obtain angles close to 40° that could have been useful for a craftsman to realize the nonagonal geometric designs. In particular, almost regular nonagons are constructed using some elementary shapes that are related with semi-regular tessellations. We compare the patterns obtained through theoretical considerations to those displayed in the examples given above. Several hypotheses are proposed for the practical construction of the interlocked hexagonal patterns for nonagons.

## Keywords

Ruler and compass construction Nonagon Selimiye Mosque Hagia Sophia Minbar Mimar Sinan## Introduction

*mahfil*of Sultan Murad III (1574–1595; Fig. 3), in the grill on the upper level of the Library of Sultan Mahmud I (1739; Fig. 4), and in the minbar in the rosette decorating the end of the staircase (Fig. 5).

This paper studies the geometry of this nonagonal pattern. It presents original results for the regular nonagon and looks at practical ways to approximate its construction. It gives simple procedures for obtaining angles close to 40° that could have been useful for craftsmen to realize the nonagonal geometric patterns. Observation of the patterns provides mathematical arguments to analyse the hypothetical regularity of the nonagons of the pattern. Finally, the pattern will be constructed in different ways.

## Mathematical Prerequisites

*ABCDEFGHI*, the numbers

*α*,

*β*and

*γ*(

*α*<

*β*<

*γ*) usually denote the ratios between the three diagonals of the polygon and its side (see Fig. 8a). Without loss of generality, we will always consider regular nonagons with side

*AB*= 1, so that

*AC*=

*α*= 1.87938…,

*AD*=

*β*= 2.53208… and

*AE*=

*γ*= 2.87938….

The six diagonals with common vertex *A* dissect the nonagon in seven triangles *ABC*, *ACD*,…, *AHI*, where \( \angle BAC = \angle CAD = \cdots = \angle HAI = 20^{ \circ } \), so that \( \angle BAE = 60^{ \circ } \) and \( \angle ABG = 60^{ \circ } \). Thus, the triangles *ABJ*, *KDE* and *HLG* shown in Fig. 8b are equilateral with side 1 and the quadrilateral *ABDK* is a parallelogram. Hence, Fig. 8b is an illustration of the relation *γ* = *α* + 1.

*ABDEGH*, with angles 120° and with sides of alternating lengths 1 and

*α*. Three congruent isosceles triangles,

*CDB*,

*EFG*and

*HIA*, with sides 1, 1 and

*α*are added to it. Note that for any two lines segments of lengths

*a*and

*b*, with

*b*/2 <

*a*<

*b*, we can construct a general equiangular hexagon with shorter side

*a*and longer side

*b*, deriving a nine-sided equilateral polygon whose

*regularity*depends on the value of the ratio

*R*=

*b*/

*a*where 1 <

*R*< 2 (Fig. 9). Thus, each value of

*R*generates an equilateral nonagon that will be regular only if

*R*=

*α.*The case

*a*=

*b*(

*R*= 1) is not considered because that equality implies the hexagon is regular, and then the derived nonagon is an equilateral triangle. The case

*b*/2 =

*a*(

*R*= 2) is not considered because the nonagon derived in a similar way is the initial hexagon.

For each of these equilateral nonagons an equilateral triangle *PQR* is obtained by extending their sides *AB*, *DE* and *HG* (Fig. 10).

*α*,

*β*and

*γ*satisfy a large number of relations between them (Steinbach 1997:26) thus generalizing the golden section. Sixteen different identities hold involving two or three of these ratios.

*α*= 1.87938524…, while the smallest solution of the second equation is −

*α*. So, both equations of Al-Biruni imply the identity \( \alpha^{3} - 3\alpha - 1 = 0 \):

Also, because of the relation \( \alpha \, = \,\gamma - 1 \), the substitution for *α* in the equality \( \alpha^{3} - 3\alpha - 1 = 0 \), leads to the equality \( \gamma^{3} - 3\gamma^{2} + 1 = 0 \).

*ABC*in Fig. 8 it follows that \( \alpha = 2\cos 20^{ \circ } \). Considering the triangles

*ACD*and

*ADE*, this property, combined with the sine and cosine law, implies that \( \beta^{2} = \alpha^{2} + 1 + \alpha \) and \( \sin \left( {20^{ \circ } } \right) = {{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } {\left( {2\beta } \right)}}} \right. \kern-0pt} {\left( {2\beta } \right)}} \), respectively. Next, as \( \alpha \, = \,\gamma - 1 \), the angle of 20° can be evaluated through the following expressions:

*α*,

*β*and \( \gamma \), are determined as limits of the ratios of certain pairs of consecutive terms that are obtained by

*jumping*over generalized Fibonacci sequences. However, because we are more interested in geometric aspects, we focus on a different sequence

*c*

_{ n }:

*c*

_{ n }: 1, 1, 1, 3, 8, 23, 66, 190, 547, 1575, 4535, 13058, 37599, 108262,…

It is defined by a recurrence relation, such that the ratios of pairs of consecutive terms \( {{c_{n + 1} } \mathord{\left/ {\vphantom {{c_{n + 1} } {c_{n} }}} \right. \kern-0pt} {c_{n} }}\, \) converge to \( \gamma \), similar to the Fibonacci sequence, where consecutive terms converge to the golden section.

### A Trisection of the 60° Angle

*ADE*with angles 20°, 60° and 100°, and with sides of lengths proportional to 1,

*β*and

*γ*. We will use consecutive terms of the sequence \( c_{n} \) to construct approximations of that triangle.

*d*,

*d*and \( c_{n} - 2c_{n - 1} \), and two triangles with sides \( c_{n} \), \( c_{n - 1} \) and

*d*, and angles

*θ*, 60° and 120° −

*θ*. Thus, \( d^{2} = c_{n - 1}^{2} + c_{n}^{2} - 2c_{n - 1} c_{n} \cos 60^{ \circ } \) and \( {{c_{n - 1} } \mathord{\left/ {\vphantom {{c_{n - 1} } {\sin \theta }}} \right. \kern-0pt} {\sin \theta }} = {d \mathord{\left/ {\vphantom {d {\sin 60^{ \circ } }}} \right. \kern-0pt} {\sin 60^{ \circ } }} \). Therefore

*θ*, obtained using the constant

*γ*and one of the ratios \( {{c_{n} } \mathord{\left/ {\vphantom {{c_{n} } {c_{n - 1} }}} \right. \kern-0pt} {c_{n - 1} }} \). The terms \( c_{n} \) can be expressed in the closed form \( c_{n} = Ax_{1}^{n} + Bx_{2}^{n} + Cx_{3}^{n} \), where

*A*,

*B*and

*C*are arbitrary constants, and \( x_{1} \), \( x_{2} \) and \( x_{3} \) are the solutions of the characteristic equation of the recurrence relation \( x^{3} - 3x^{2} + 1 = 0 \). Thus, the error of the approximation \( \gamma \approx {{c_{n} } \mathord{\left/ {\vphantom {{c_{n} } {c_{n - 1} }}} \right. \kern-0pt} {c_{n - 1} }} \) can be evaluated by

Successive approximations of \( \gamma \) using \( {{c_{n} } \mathord{\left/ {\vphantom {{c_{n} } {c_{n - 1} }}} \right. \kern-0pt} {c_{n - 1} }} \) and the corresponding approximations of the angle 40°

\( n \) | \( {{c_{n} } \mathord{\left/ {\vphantom {{c_{n} } {c_{n - 1} }}} \right. \kern-0pt} {c_{n - 1} }} \) | \( 2\theta \) |
---|---|---|

5 | \( {8 \mathord{\left/ {\vphantom {8 3}} \right. \kern-0pt} 3} \) | 43.5735785965236° |

6 | \( {{23} \mathord{\left/ {\vphantom {{23} 8}} \right. \kern-0pt} 8} \) | 40.0679870494440° |

7 | \( {{66} \mathord{\left/ {\vphantom {{66} {23}}} \right. \kern-0pt} {23}} \) | 40.1525538863933° |

8 | \( {{190} \mathord{\left/ {\vphantom {{190} {66}}} \right. \kern-0pt} {66}} \) | 40.0092482554315° |

*θ*= 20.0339…°. This implies a division of the side of the triangle into 23 parts and results in a dissection into the parts 8/23, 7/23 and 8/23. The equiangular hexagon shown in Fig. 8 right is determined by its

*triangular skeleton*and the equiangular hexagon with sides 8 and diagonals 23 will result in a derived equilateral nonagon with sides 8 and third diagonal 23. The three central angles will be 2

*θ*= 40.0679…°, alternating with two angles of \( 60^{ \circ } - \theta \). The dissection 8/23–7/23–8/23 produces nonagons that are perceived as regular by the naked eye (Fig. 16).

From the eleventh century, several methods for approximating a construction of the regular nonagon are known in Islamic architecture. From the elements of the patterns with hexagons and nonagons, one can guess that the craftsmen applied an approximated solution of the trisection of the angle of 60°, consciously or not. However, we have to keep in mind that skilful craftsmen did not make the patterns by means of quotients of consecutive terms of recurrence sequences, nor by exact mathematical constructions, but using practical methods. The lack of accuracy in the geometric construction is obvious in the stucco marble frieze of the Mahfil in Hagia Sophia (Fig. 3) and in the panel of its minbar (Fig. 5). In contrast, the woodwork decoration of the minbar of Selimiye Mosque looks almost perfect (Fig. 2). We will see that the concept of the pattern seems to be the same and so perhaps the difference in the result can be explained by the materials used to execute them (wood, stucco, stone, etc.).

The mutual influence of the methods of mathematicians and craftsmen is well-known. There are several Persian medieval treatises about the ‘conversazioni’ between both groups (see, for example, Hogendijk 1979; Özdural 1996, 2002). Yet, the texts provide little information about what exactly the craftsmen’s method was. For instance, the construction of the nonagon is explained in these treatises without saying how it was actually done, probably because the better construction methods required solving a cubic equation or studying conics. Perhaps these topics were too advanced for craftsmen, who preferred methods like *trial*–*error* or *dissection and composition*. Thus, in view of the practical application, we searched for angles of about 20° in the shapes composed by simple polygons as the equilateral triangle, the square, the regular hexagon, the octagon and the dodecagon. Simple procedures for obtaining angles close to 40° could have been useful for craftsmen to realize the nonagonal geometric patterns.

### Approximations of the 20° Angle

#### The 20.1° Angle, Starting from a Square and an Equilateral Triangle or a Regular Hexagon

*POQ*imply the angle 20.1° can be found in some semi-regular tessellations composed of hexagons, dodecagons, squares and triangles. Figure 17 shows the triangle

*POQ*overlapping two semi-regular tessellations and Fig. 18 the nonagons determined in the corresponding tessellations.

### The 20.1° Angle, Starting from a Regular Octagon and a Dodecagon

- 1.
Draw a regular dodecagon and a chord of 90°,

*AB*. Call its centre*O*. - 2.
Draw a regular octagon such that

*AB*is a chord of 90°. Call its centre*P*. - 3.
Rotate the octagon over 60° around the point

*O*so that*P*is mapped onto*P’*. - 4.
Call

*Q*the vertex in the dodecagon such that the chord*AO*is 120°. - 5.
Drawing the segments

*PO*,*PQ*and*P*′; then \( \angle OPQ \approx 39.9^{ \circ } \) and \( \angle QPP{\prime } \approx 20.1^{ \circ } \).

Using dynamic geometry software (such as Geogebra or Cabri) we can check in Islamic style, that is, by moving geometric shapes, that both preceding approximations of 20.1° are identical (Fig. 19b).

### The 19.1° Angle

*POQ*which is located in the centre of one of the equilateral triangles of the net.

*r*between the side of the hexagon and the nonagon lead to different variations of the pattern that we call

*pattern p*(

*r*). They have different visual aspects, but are geometrically

*equivalent*. Figure 22 shows some patterns belonging on the family of patterns \( \left\{ {p\left( r \right):0 < r < + \infty } \right\} \), ordered by decreasing ratios.

Note that the overlapping of the nonagons doesn’t happen when \( \gamma \ge \sqrt 3 \alpha \beta /3 \approx 2.75 \), and the central star of the rosette becomes like a six-petal shape if \( \gamma < \sqrt 3 /(6\sin 10^{ \circ } ) \approx 1.66 \). Thus, we will only consider patterns *p*(*r*) such that \( \sqrt 3 /(6\sin 10^{ \circ } ) \le \gamma < \sqrt 3 \alpha \beta /3 \). Assuming that the nonagons are regular, we will analyse the theoretical pattern *p*(*r*) from different points of view. To make it easier, we again suppose that the length of the side of the nonagon is 1 and the length of side of the hexagon is *r*.

### Geometric Analysis of the General Family of Patterns *p*(*r*)

*edge to edge*union of three different components or

*elementary tiles*: I, an equilateral six-pointed star; II, an irregular convex pentagon we will call

*house*; III, an irregular concave octagon we will call

*bow*(see Fig. 23).

*FGO*in Fig. 23:

- 1.
The side of the nonagon has length 1. The side of the hexagon has length

*r*, and is divided in three line segments*FA*,*AB*and*BG*; \( BC = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} \). - 2.
The sides of the hexagon and the nonagons are perpendicular in their intersections

*A*and*B*. The points*E*and*C*are vertices of the regular nonagons, and thus

- 3.
*FB*is the apothem of the nonagon, so

Note that the length of the perimeter of the tiles I, II and III, depends on the value of the parameter *r* for given a pattern *p*(*r*), though the angles of the tiles are constant. The angles in the vertices of the star tile I are 80° and 220°; the angles of the house tile II are 80°, 140° and 90°;and the bow tile III has angles of 120°, 90°, 220°, 100° and 140°.

_{1}, L

_{2}and L

_{3}(Fig. 24). Layer L

_{1}is a hexagonal net of side

*r*while layers L

_{2}and L

_{3}are formed by regular nonagons with centres on the vertices of an imaginary equilateral triangular grid with side \( \sqrt 3 r \). Both layers of nonagons L

_{2}and L

_{3}coincide under translation by vectors v

_{1}, v

_{2}and v

_{3}(Fig. 24).

Intersection, union and/or subtraction of the layers L_{1}, L_{2} and L_{3} determines different layers L_{4}, L_{5} and L_{6} (Fig. 25a), generating the same pattern. Each pattern *p*(*r*) can also be generated by edge to edge addition of a basic polygon (a shape that covers the tiling using only isometries). Figure 23 showed one of them, but the smallest polygon with this property is the right triangle V with angles 30°, 60° and 90° and determined by the radius of the hexagon, the apothem of the nonagon and the half-side of the hexagon between them (Fig. 25b). This small shape covers the tilling by reflections only. We can also find a rectangular basic polygon, such as VI. We call these polygons V and VI *practical tiles*, because their triangular or rectangular shape makes it possible to use them as a template for a practical construction, especially for friezes.

### Geometrical Analysis of the Practical Patterns

In the practical realizations of the pattern in the grilles, panels and friezes given in the examples, the nonagons are perceived as regular. However, one can wonder if the nonagons were regular in the mind of the designer: what could their mathematical foundations have been? We will not need measurements, because careful observation of the patterns gives theoretical arguments suggesting the designed nonagons are necessarily non-regular.

*a*and

*a*/2.

This implies that the angles of the nonagons of Selimiye Mosque and Hagia Sophia are \( 2\theta \approx 3 8. 2 2^{ \circ } \) and \( 60^{ \circ } - \theta \approx 4 0. 8 9^{ \circ } \). The same angle was obtained in Fig. 20 by a semi-regular tessellation.

*OABC*and the midpoint

*M*where the diagonals

*OB*and

*AC*intersect

*FD*. It follows that

Note that, in spite of the different locations of the nonagon and the hexagon in Figs. 27 and 28, the calculations are indeed the same, and they twice yield an angle of 19.1° angle. Both graphical properties occur in all of the analyzed practical patterns. There is thus a very simple way to construct the angles of the nonagons of all of the examples. We apply the two preceding observations to the theoretical pattern *p(r)*, where the nonagons are perfectly regular.

*B*is not the midpoint of the side

*AC*of the hexagon. Indeed, the three rectangles

*OAB*,

*DFG*and

*GFH*have angles 20°, 120° and 40°, and thus they are similar; therefore

*AC/AB*is constant in all

*p*(

*r*) patterns and equals \( \alpha \), not 2.

For a regular nonagon, \( \angle POE = 20^{ \circ } \) and this is greater than \( \theta \). Consequently, the radius *OP* of the nonagon does not lie on the diagonal *OB*, and thus the points *R* and *B* are distinct points. Therefore, the construction of the pattern by using exact regular nonagons is not possible.

*a*–

*b*–

*a*the lengths of the three parts determined in the side of the hexagon, we observe a ratio \( {b \mathord{\left/ {\vphantom {b a}} \right. \kern-0pt} a} \) around 1.4 \( \approx \sqrt 2 \), in the Hagia Sophia and the Selimiye Mosque. However, in the grill of the window of the Mausoleum of Shehzade Mehmet this ratio is approximately 1, while it is close to \( {2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3} \) in some similar modern screens.

Finally, according the two preceding observations from Figs. 26 and 27, we construct the pattern of the frieze of the mahfil in the Hagia Sophia, and we provide a second method to construct the pattern.

### Two Constructions for a Pattern in Hagia Sophia

#### Construction 1

*P*, is the intersection of the line segment

*AB*and the line passing by

*O*and

*C*(Fig. 32). We locate the points

*A*,

*B*,

*C*in a Cartesian coordinate system with origin in the center of the nonagon. The

*x*-axis is the line containing

*OB*, and the side of the hexagon is taken as unit length.

*AB*and

*OC*concur in the point

*P*, it follows that

*P*leads to a very easy value of the radius, 3/4, which is three-quarters of the side of the hexagon. Figure 33 visualizes the construction of the nonagon.

*v*= (3, 0) completes the frieze.

The nonagons of this construction have three sides with length *l* _{ 1 } = 3\( \sqrt {21} \) /28. This value of the sides of the nonagon produces dissections *a*-*b*-*a* of the side of hexagon with ratio \( b/ a= \,{(30\sqrt 7 - 56)}/ {(56 - 15\sqrt 7 )} = 1.432 \ldots \). This approximation corresponds to the graphical estimations of Fig. 31.

#### Construction 2

*FGO*defined in Fig. 23. However, it can also be derived based only on the house tile II, over a hexagonal net. Figure 35 shows the sides

*EB*and

*EF*produce the two sides of the irregular nonagon. The angles \( \theta \) and \( \angle EFG = 360^{ \circ } - 2\theta \) determine six angles \( \theta \) and three angles \( \alpha \) in the nonagon, in the order \( \theta - \theta - \alpha \).

The analysis of Fig. 31 suggest a hypothetic ideal ratio \( {b \mathord{\left/ {\vphantom {b a}} \right. \kern-0pt} a} \approx \sqrt 2 \), and thus we assume that the nonagons dissect the side of the hexagon in three segments *AB*, *BC* and *CD* such that *AB* = *CD* and with ratio \( {{BC} \mathord{\left/ {\vphantom {{BC} {AB = \sqrt 2 }}} \right. \kern-0pt} {AB = \sqrt 2 }} \). This choice allows us a simple way to construct the house tile and complete pattern by intersection of straight lines in a hexagonal net.

Of course, we don’t know the exact procedure of the craftsmen who constructed the nonagonal patterns on Hagia Sophia and the Selimiye Mosque, but the hypothesis of a construction in a way similar to our Construction 1 seems quite realistic.

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