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Vol.:(0123456789)
Nexus Network Journal
https://doi.org/10.1007/s00004-022-00621-z
RESEARCH
The Geometric Patterns ofİzzettin Keykavus Tomb
MakbuleÖzdemir1 · SemraArslanSelçuk1 · HasanFevziÇügen2
Accepted: 7 June 2022
© Kim Williams Books, Turin 2022
Abstract
The Islamic architectural heritage permits us to understand the formation and
development of geometric patterns. Many techniques are used to express the
patterns related to the forms, and the geometric principles underlying the designs
are revealed mainly via compass and ruler. Generally, it is not sufficient to express
complex patterns with a single technique, and different methods are used together.
In this context, in the analysis of geometric patterns of İzzettin Keykavus tomb, a
combined technique using simple drawing tools such as point-joining with the
composition technique according to the symmetrical background of the seventeen
wallpaper patterns is used. When the created patterns are compared, it is seen
that the patterns with similar basic units can differ with the change of symmetry
elements.
Keywords Islamic geometric patterns· Design analyses· Symmetry groups· Design
computations· Geometric analysis· İzzettin Keykavus Tomb· Seljuk art
Introduction
As a common feature of Islamic art and architecture, ornaments are known to have
a wide variety; they are applied in different shapes and types as floral motifs and
geometric patterns (Kaplan and Salesin 2004). Geometric patterns, which have
become a remarkable element among these ornaments, constitute the largest class
of Islamic patterns. In Islamic geometric patterns, where mostly complex polygons
are used, regions bounded by circular arcs are used less frequently (Bier 2015).
* Makbule Özdemir
makbuleozdemir@gazi.edu.tr
Semra Arslan Selçuk
semraselcuk@gazi.edu.tr
Hasan Fevzi Çügen
hasanfevzi@gmail.com
1 Department ofArchitecture, Faculty ofArchitecture, Gazi University, Ankara, Turkey
2 Asır Proje, Ankara, Turkey
M.Özdemir et al.
There are different approaches and alternatives for forming patterns (Grünbaum
and Shephard 1987; Abas and Salman 1994; Cromwell 2012; Castera 2016). It is
impossible to know precisely which methodology was used to produce these pattern
groups in the past (Bonner 2017). Traditionally, rulers and compasses are used as
tools for creating geometric patterns. This method, which Bonner (2017) calls the
“point-joining” or “graph paper” technique, begins by dividing the circle using a
compass and ruler. The intersection points selected within these sections and the
lines connecting these points provide a reference for pattern formation and produce
the repetition unit of the design. In this way, it is possible to form regular repeating
units such as squares, pentagons, and hexagons. There are different studies on this
method (Critchlow 1976; Wade 1976; El-Said et al. 1993; Demiriz 2004; Broug
2008; Sönmez 2020).
In the systems that are based on the division of the surface by mathematical
methods, the patterns usually consist of a “repeated unit” and “repetitive structure”
composed of several polygons or other regular shapes (El-Said et al. 1993). The
repeating unit is the smallest region that makes up the geometry. Repetitive units
systematically multiply to form the repetitive structure, while repetitive structures
reveal periodic patterns (Alani 2018) (Fig.1).
As one of the basic elements of the design, repetitive structures provide a grid
system.In grids that offer a design methodology for the completion and production
of pattern composition, the basic units are built on combinations of overlapping and
intertwined repeated circles (Critchlow 1976; Wade 1976; Gonzalez 2001; Broug
2008)and consist mostly of polygonal structures.As an integral part of geometric
pattern designs, the rules form the basis of compositions(Broug 2008).
One of the other approaches seen in the design process of patterns is the focus on
mathematics and symmetry.The classification of a pattern design in symmetrical
strategies is provided by groups of symmetry, which include a combination of
isometric transformations in two dimensions(Knight 1998).In a simple descriptive
way, symmetry is a type of transformation that matches the object’s shape over
itself and leaves it unchanged. Displacement is provided by mirror and glide
Fig. 1 Creation of an Islamic geometric pattern from a repeating volume using a hexagonal structure
The Geometric Patterns ofİzzettin Keykavus Tomb
symmetry as the repetition of a unit in geometric patterns. The variety of styles
in decorations largely depends on these rules. There are multiple researchers
conducting symmetry groups and analysis studies(Müller 1944; Grünbaum et al.
1986; Necipoğlu 1996).Studies focus on the innovative mathematical knowledge
of the time to understand and produce a cultural heritage that spread throughout
the Islamic world over the centuries. In a two-dimensional plane, patterns are
expressed as belonging to one of the seven frieze groups or to one of the seventeen
wallpaper groups. The most comprehensive of these is considered classification
by Abas and Salman (1994) (see also Ostromoukhov 1998). These symmetries
are divided into seventeen symmetrical groups, allowing the same shape to
be expanded with multiple copies. Seven groups of friezes define all patterns
created by offset along a dimension (Shubnikov and Koptsik 1974), while the
seventeen wallpaper groups consist of patterns created by two linearly independent
displacements (Schattschneider 1978). All periodic patterns fall into seventeen
wallpaper groups and can be said to consist of primitive units that we call “cells”
with rectangular, rhombus, square, or parallelogram(Schattschneider 1978).
Within these features, the cooperation between mathematics and design led to
the creation of complex geometric designs in the early and medieval Islamic world
(Özdural 2000). With the increase in the frequency of using geometric patterns in
the Seljuk Period, these patterns became the compositions we frequently encounter
in religious buildings.
It is possible to state that Islamic geometric star patterns have evolved from
simplicity to complexity throughout history. In the ninth and tenth centuries,
this new form of ornamentation was characterized by a general geometric matrix
containing primary stars or regular polygons located on the vertices of a repeating
grid. Geometric star patterns from this early period have either three-fold or four-
fold symmetry, with the former being formed through hexagons or six-pointed stars
at the vertices of a triangular or hexagonal repeat unit and the latter being expressed
by eight-pointed stars, octagons, or squares placed at the vertices of a square repeat
unit.
The tomb architecture of the Anatolian Seljuk Period suggests that the Great
Seljuk tradition was sustained. In the Anatolian Seljuk tombs, which were built more
modestly than the tomb architecture in Central Asia due to geographical conditions,
the stone and brick materials were collectively used as in the Great Seljuks.
However, these materials were later replaced with cut stone (Diez and Aslanapa
1955). Even if the first Muslim Turkish states contradicted the understanding of
Islam, they partially continued their traditions, which created different types of
tombs depending on the conditions of new beliefs, customs, artistic ideologies, and
regions (Diez and Aslanapa 1955; Sözen and Akşit 1983; Aslanapa 1993).
İzzettin Keykavus Tomb is one of the important architectural structures that
has survived to the present day, and there is no detailed study on the ornamental
infrastructures of the building. In this context, the motivation of this study is to
seek answers to the question, “How should the constructs regarding the geometric
patterns of the building be expressed understandably?” In the light of the hypothesis
“Using different methods together plays an important role in better understanding
geometric pattern compositions,” a field study was conducted on the İzzettin
M.Özdemir et al.
Keykavus Tomband its ornamental infrastructures. The last reparation was carried
out in 2013 in the tomb, which has undergone more than one reparation from the
past to the present. To guarantee the accuracy of the restoration, precise images
taken prior to the repair were consulted. The ornaments in the tomb were examined,
and their geometrical infrastructures were deduced. There are studies examining the
role of basic geometry in pattern infrastructures (Kaplan 2000; Kaplan and Salesin
2004; Broug 2008; Bonner 2017; Sönmez 2020). However, in this present study,
each structure is reviewed specifically; symmetry classes and the formation steps of
the patterns are also examined.
Method
In the analysis section of the article, a mathematical approach is applied to
the decorations on the exterior and interior of the tomb. Repeated units and
substructures were revealed, and the design phases were analyzed. During the
analysis, the focal point is the use of mathematics and symmetry knowledge and
the practical geometry knowledge of the Islamic geometric patterns found in the
building. Initially, the formation stages of the patterns are shown using the point-
joining method, using the traditional tools of ruler and compass to create geometric
patterns. Then, the repeating units of the completed design are revealed. Repetitive
units are handled in two ways. The first one is repetitive units consisting of
polygons as design elements produced by the point-joining technique and forming
the basis of geometric compositions. The second is to determine the unit lattices
consisting of parallelograms, square rectangles, and rhombuses belonging to the
seventeen wallpaper groups in pattern designs in order to understand the symmetry
applications comprehensively. These structures are indicated by marking them in the
completed figure of each pattern. In addition, there are several patterns in the tomb
in which no-repeat units are formed. There are aperiodic patterns where repetition is
not possible. Different alternative approaches to these patterns have been proposed.
Next, these basic units were evaluated according to seventeen wallpaper groups.
Finally, patterns, geometric drawing methods, repeating units, and geometric
features of their substructures are compared with each other. The sequencing and
analysis of the figures, starting at the entrance door, continues to the cone section
and the interior. All geometric structures and drawings were created by the authors.
Field Study: İzzettin Keykavus Tomb
The tomb is located in the southern iwan of the İzzettin Keykavus Hospital
(Darüşşifa), which is located across the Double Minaret Madrasa in Sivas city
center and today known as Şifaiye Madrasa (Fig.2). The building located opposite
the Double Minaret Madrasa was built as a hospital by Seljuk Sultan İzzettin
Keykavus I in 1217 (Cantay 2002; Uzunçarşili and Edgüer 2014). After the death
of İzzettin Keykavus I, the southern iwan of the building was turned into a tomb,
The Geometric Patterns ofİzzettin Keykavus Tomb
and Keykavus’s body was buried there in 1220, three years after the construction
(Cantay 2002: 354).
The courtyard of the building, which has a single story, open courtyard, and three
iwans, is surrounded by porticoes on three sides (Fig. 3). The main construction
material of the building constructed with the masonry technique was stone and
brick. The main walls of the building, the main iwan and adjoining rooms, the
entrance iwan and the volumes opening to it, rectangular porticos, and the interior
walls facing the courtyard were entirely made of cut stone blocks. While brick was
used in the partition walls of the building, the eastern and western partition walls of
the tomb in the southern iwan were built with stone. Brick was used in the vaults
covering the cells and porticoes, the partition walls of the northern iwan and other
cells, the entrance facade of the tomb, and the decagonal drum. Along with the
Fig. 2 Top view of Şifaiye Madrasa (Archive of Asır Proje)
Fig. 3 Plan of the A. Şifaiye Madrasa B. View of the tomb from the 1963 restoration project (Archive of
Asır Proje)
M.Özdemir et al.
brick, tile is used on the arch surfaces, the niches in the northern iwan, the tomb
façade, and the sarcophagi. There was an iwan on the north and south axis of the
courtyard, and the arch openings between the stone-coated massive legs were kept
wide to emphasize the iwans. The dividing wall in the south of these iwans, which
were reached through a door and two windows, was turned into a tomb section and
covered with a decagonal rimmed cone.
The tomb was formed by closing the iwan arch opening with a brickwork wall.
This section behind the barrel-vaulted portico draws attention to the intricate
decoration of the facade with turquoise, cobalt blue, and white tiles (Fig.4).
The transverse rectangular tomb was covered with a dome with Turkish triangles
on the inside and a pyramidal cone on the outside. Entrance to the tomb was ensured
through a longitudinal rectangular door opening to the north facade center. It is
believed that the sarcophagi inside the tomb were covered with turquoise-colored
flat hexagonal plate tiles. Relevant sources indicate that the sarcophagus in front of
the mihrab belongs to İzzettin Keykavus (Yetkin 1972; Öney 1976; Önkal 1996). In
addition, it is thought that the tomb’s floor was paved with hexagonal brick material,
as seen in the other iwans and cell rooms. The stone altar in the middle of the south
wall, on the other hand, was decorated with writing, geometric and floral motifs and
was unpainted and plaster-free (Fig.5).
It is safe to state that the entrance (north) facade of the tomb facing the courtyard
was decorated with brick and tile mosaic as it is today. This part was divided into
three rectangular sections, starting from the level of the arch springing, at the
bottom, broad in the middle, and narrower on the sides. There was an entrance
opening in the middle of these niche-shaped partitions, which was not very deep,
and a window opening on each side. The upper parts of these openings, framed
by geometric patterned border arches, were animated with blind pointed arches.
These areas were decorated with inscriptions made in the tile mosaic technique and
geometric compositions made of plain bricks (Fig.6, left).
From both the material used in the body and from the comparative period examples,
it is understood that the cone of the tomb in this period was made of brick. The decagon-
shaped body surface of the tomb was animated with shallow blind niches. According to
Fig. 4 Theview from the tomb in the southern iwan of the hospital (Archive of Asır Proje)
The Geometric Patterns ofİzzettin Keykavus Tomb
the existing traces and documents, the surface of the niches here was filled with hedesi
compositions in which plain and turquoise glazed bricks and turquoise-colored tiles
were used together. These compositions were handled differently in each niche (Fig.6,
right).
Fig. 5 Ground and upper cover plan of the tomb, general view of the tomb from the north and south
walls, and details from the stone altar(drawings and photograph archive of Asır Proje)
Fig. 6 Views of the tomb from the north (left) and south (right) facades (Archive of Asır Proje)
M.Özdemir et al.
Analysis ofPatterns
In the analysis of the patterns, the tomb entrance facade and the cone surfaces are
discussed respectively. The entrance facade of the tomb is divided into three bays
featuring point arches. The upper part of the arch surface is divided into three
longitudinal rectangular panels (Fig. 7). All the building photographs used to
analyze the geometric patterns belong to Asır Projearchive.
The pointed arched surface on the left side of the facade consists of two
interlocking decagons, one large and the other small (Fig.8). Turquoise five-pointed
stars are seen in the center of the composition and the lower corners. The pattern has
a background based on multiples of five. The pattern extends from the central axis
of a ten-sided polygon with a radial grid system. The formation stages of the pattern
are expressed in six steps. A decagon is reproduced by taking mirror images at an
angle of 36°, following the central axis. In the second step, lines are drawn at the
same angle from each junction point. Thus, a ten-pointed star is achieved. A circle
of radius C is drawn from the center point to the corner point of the first star. Two
circles of diameter A are then drawn to center the distance between the two circles.
A decagon is formed through the intersection of these two circles. In the next step,
a circle with radius B is drawn so that the second star formed in the center is at the
corner point of the pattern. The same circle is copied perpendicular to the line drawn
at an angle of 18°. The copied circle is divided into ten equal parts. Another decagon
is formed, passing through two straight lines coming at an angle of 108°. Then, the
star pattern is combined regarding this polygon, followed by breaking on a second
decagon line drawn from the middle point with the extension of the star pattern
lines. This geometric pattern belongs to the class of aperiodic patterns. Although
it has fivefold rotational symmetry, it does not have translational symmetry. For
this reason, an expansion can be achieved with a finite translation. The pattern
multiplied radially, but it is possible to derive the pattern in such a way as to achieve
translational symmetry. The same ‘repeating unit’ is reproduced using translational
Fig. 7 Detailed view of the architectural decorations at the entrances (Archive of Asır Proje)
The Geometric Patterns ofİzzettin Keykavus Tomb
symmetry in the lower right corner. In pattern analysis, the shaded regions represent
the repeating unit. It can be included in the CMM and P2 symmetry groups with
180° rotation.
The substructure of the pattern on the surface above the door is based on a
multiplication of hexagonal units (Fig.9). The entire pattern is created by taking
translational symmetry on each side of the hexagon. The method of making interior
patterns can be described in six steps. Initially, lines are drawn from the center of the
hexagon to each vertex. The resulting triangles are bisected twice by their bisectors,
and each side is divided into four equal parts. In the first step of the pattern, the first
bisector points from each corner point of the hexagon are connected by a line (points
A and B). Then, in step 2, points A and B are joined by another line at an angle of
120°, as shown in Fig.9. In the next step, a circle is drawn from the center, and lines
are created parallel to the corner lines of the hexagon and, secondly, passing through
the intersection of the circle and the median side. In the last step, the excess lines
are deleted, and the repeating part of the pattern is created. This pattern is created
through adaptation from the study. The six-fold pattern is completed using rotational
symmetry. The symmetry type is P6M.
Fig. 8 Analysis of the ornamentation detail on the right pointed arch surface of the windows on the
entrance facade (Pattern-1)
M.Özdemir et al.
The surface of the windows on the right is arranged in the form of a composition
of strips that cut the decagon formed by the turquoise strip (Fig.10). The turquoise
pentagonal star in the center of the composition and turquoise Y motifs are added
to the other spaces. The pattern is begun by drawing a decagon. Two pentagons
are placed inside, passing through each corner point. Lines are passed through the
intersection points of these pentagons and are divided into ten equal parts. Two
lines are drawn at 36° angles from each intersection point, and a ten-pointed star
is formed. Then, a line is drawn from the center indicated by A to the vertex of the
inner star, and a circle is formed with radius A. In the next step, another ten-pointed
star is created, passing through the intersection points of this circle. The pattern is
created by extended these last star lines. This is a variant of an aperiodic pattern
with fivefold rotational symmetry. Finite reproduction can be achieved in a periodic
manner. The symmetry type, using the translational symmetry of the repetitive unit,
is CMM.
There are two cuboctahedrons on the capitals of the columns next to the
entrance door (Fig. 11 above). These represent polyhedra formed by the
combination of triangles and squares, decorated with a tile mosaic composition
in which turquoise-colored stripes form quadruple knot motifs on rectangular
surfaces and triple knot motifs on triangular surfaces.In the cuboctahedron, the
tile mosaic design on each face consists of a repeating motif.These polyhedral
surfaces are connected to each other, creating the appearance of a seamless
pattern.The unfolded form of the cuboctahedra shown in Fig.11. indicates the
relationships of the square and equilateral triangle faces and the patterns inside.It
is formed by repeating a square and a triangle motif.The geometric structure
of this unit is illustrated in four steps.A square is divided into three from two
Fig. 9 Analysis of the ornamentation detail on the surface of the pointed arch above the door on the
entrance facade (Pattern-2)
The Geometric Patterns ofİzzettin Keykavus Tomb
sides.The new square formed in the middle is rotated at an angle of 45°, and the
line extensions are combined.Drawing the design in a holistic way creates the
pattern seen on the right in Fig.11.The symmetry type is P6M.
There is a braided Kufic script on the panel above the pointed arch of the
window (Fig. 12). The composition of the panel is formed by the turquoise-
colored strips cutting each other to create circular and angular knots. The
geometric structure of this unit is formed by repeating a rectangular unit pattern.
The geometry of this unit begins with a square grid and then is completed with a
rectangular scheme. The creation of the unit cell consists of 6 steps. The first step
started by dividing a square into twelve parts at an angle of 30° from its center
point. Then, a circle is obtained from the intersection of the line at a distance
of 1/8 from the center of the square, each side of which is divided into eight
equal parts, and from the line that came at an angle of 30°. A circle is then drawn
with its radius at the second 1/8 line. A six-pointed star is created in the center.
In the third step, a part of the motif is formed by taking the extensions of the
lines. Then, the 30° lines are continued to create the continuation of the pattern,
and the infrastructure is completed. The repetitive pattern consists of four axes of
reflection. The symmetry type is PMM.
Fig. 10 Analysis of the ornamentation detail on the right pointed arch surface of the windows on the
entrance facade (Pattern-3)
M.Özdemir et al.
The pattern in Fig.13 shows a pattern beside a window. To draw the substructure,
a circle is first drawn inside a square, and an octagon is then formed inside this
circle. Furthermore, squares are formed between interlocking octagons. Then the
square segments are drawn to center the octagons, as shown in the figure. After
that, the excess lines are erased, forming interlocking shapes. This pattern is created
through adaptation from the study. The symmetry type is PMM.
The body of the columns near the doors, on the other hand, consists of a geometric
composition of intersecting hexagons and six-pointed stars at the intersections. It is
multiplied with a repeating unit based on a hexagon. Infrastructure stages are shown
in Fig.14. First, a hexagon and a second hexagon passing through the medians are
drawn. Next, two equilateral triangles that met at the vertices of the first hexagon are
placed. Next, the second hexagon is divided into six 60° angles from its center point.
In the next step, these lines are shown in blue. Reflections of the lines are created
so that a six-pointed star is formed in the middle. In the last step, the reproduction
pattern of the repeating unit is shown. The symmetry type is P6M.
Another pattern near the entrance door is the composition of six-knots and six-
pointed stars in the center, with the intersection of the red brick strips. The stages
Fig. 11 Analysis of the cuboctahedron in the column header (Pattern-4)
The Geometric Patterns ofİzzettin Keykavus Tomb
of the pattern consist of five steps. In the first step, two interlocking hexagons are
drawn, similar to the designs shown in the past figures, and the inner hexagon
is divided into six triangles. The formation of the basic structure with sixfold
symmetry continues by dividing each side into equal parts. Then, extensions
Fig. 12 Braided Kufic on the panel above the pointed arch of the window (Pattern-5)
Fig. 13 Analysis of window side patterns (Pattern-6)
M.Özdemir et al.
of six-pointed stars and stripes are added to the center to complete the design.
Finally, a dodecagon that intersects the medians of the second hexagon is created.
All tessellations can be done by offsetting the decorated hexagon. The symmetry
type is P6M (Fig.15).
The brick-knitted decoration is also seen on the decagonal surfaces of the
pyramidal cone.Each face of the decagonal surface is animated with superficial
niches opened in rectangular form. The interiors of these recesses are decorated
Fig. 14 Analysisofdoor side columnpatterns (Pattern-7)
Fig. 15 Analysis ofdoor sidepatterns (Pattern-8)
The Geometric Patterns ofİzzettin Keykavus Tomb
with a geometric composition in which different patterns are created with
brickwork (Fig. 16). Among these is a composition consisting of swastikas
coming out of eight-pointed stars in the arch facing the southwest corner of the
cone. On the surface immediately to the left (west facade) of this, there is a large
octagonal composition consisting of eight swastikas formed by eight-pointed
stars and double swastikas enclosing them.The other arch surface on the left side
of this arch is in the form of a swastika composition created in the hexagonal
version of the previous one.
An intricate composition on the first arch surfaces consists of the rays
emerging from the six-pointed stars spreading towards infinity to form hexagons
and diamonds around them. The pattern is drawn based on a hexagon. First, a
hexagon is divided into six triangles. The central motif, which has sixfold
symmetry, begins in a way that is similar to the previous pattern. The repeating
unit can be rotated around the center of the hexagon to cover the entire shape.
The symmetry type is P6M (Fig.17).
Similar to the designs shown in the previous figures, in Fig. 18 a second
hexagon is created within a hexagon that intersected at the median. The hexagon
is divided into six equal parts at 60° angles. A six-pointed star is formed to merge
at the intersection of the blue and red lines in the second step. Two circles with
radius A are drawn between the hexagon and the center, and another six-pointed
star is drawn on the inner circle. Moreover, the repeating unit of the pattern is
formed. All tessellations can be done with a decorated hexagon offset. As seen
in the figure, the designer placed a turquoise tile mosaic and star patterns in the
center of the hexagons. The symmetry type is P6M.
Another pattern formation on the cone surface begins with a dodecagon. It is
divided into twelve equal parts with lines extending at a 30° angle from the central
axis. Next, a hexagon is obtained with a line passing through the intersection of both
lines. In the next step, a six-pointed star that ended in the median of the hexagon is
created. A second twelve-pointed star is then obtained, with each arm having a 30°
angle. The basic units shown in shaded form are obtained in the fourth step. Then,
the excess lines are deleted, and the repeating unit is obtained. In the last step, the
combination of the three basic structures occurred. The motif has a center of rotation
Fig. 16 View from thecone surfaces (Archive of Asır Proje)
The Geometric Patterns ofİzzettin Keykavus Tomb
Pattern 4 is formed by multiplying a hexagon as a repeating unit. To draw the
substructure, it a circle is divided into twelve segments of 30°. Two concentric
hexagons are then formed to pass a line through each point at 60° angles. The
repeating unit A of the first hexagon is denoted by unit B of the second hexagon.
In the next step, straight lines are drawn from the intersection point of the hexagons
perpendicular to the parallel line. A six-pointed star pattern is formed in the center.
In the next step, six equilateral hexagonal patterns are created at the intersection
points of the lines with the hexagon. In the last step, a mirror image is taken to make
the symmetry of the region A. In this way, six rotational symmetries are created,
and a star pattern is obtained in the center. It can be composed of a multiplying
unit circle grid; a hexagonal textured honeycomb grid can also be preferred. The
symmetry type is P6M (Fig.20).
Pattern 5 consists of hexagon-based star patterns.It is an aperiodic pattern. Its
eight-pointed star is combined with five-pointed and six-pointed stars.There is no
regular flow in the pattern.The creation of the repeating unit, on the other hand, is
handled with a different approach.First, a hexagon is created.In the next step, each
edge is divided into three, and these points are connected with the help of lines.The
intersecting points are joined at a 60° angle.Thus, six hexagons represented by the
shaded part are obtained.Moreover, a six-pointed star is formed at the midpoint.The
symmetry type is PMM (Fig.21).
On the arch surface of the cone on the east side, there is a geometric composition
formed by the intricate interconnection of twelve-pointed stars. The pattern is
formed by repeating a square-shaped motif.It has two centers with 90° rotation and
Fig. 19 Pattern analysis of cone surfaces (Pattern-11)
M.Özdemir et al.
a center with 180° rotation.There are no glide reflections.The motif is one-quarter
the size of the lattice unit.Considering these features, the symmetry type is P4M
(Fig.22).
Fig. 20 Pattern analysis of cone surfaces (Pattern-12)
Fig. 21 Examination of the cone surfaces (pattern 13)
The Geometric Patterns ofİzzettin Keykavus Tomb
In the arch at the north-eastern corner of the cone, there is a large composition
of octagons and smaller pentacles, merging the ends of the eight-pointed stars.
The turquoise tiles are placed inside the eight-pointed star in the middle, and
the six-pointed stars are positioned around it. When the symmetry centers of
the pattern are evaluated, it is found that the pattern has similar features to the
previous one. The symmetry type is therefore P4 (Fig.23).
On the arch surface on the northwest side of the cone is found a composition
of larger interlocking hexagons formed by the intersecting and merging of rays
created by six-pointed stars in the middle of the hexagons. The pattern has a
60° center of rotation.It also has two centers of rotation that differed by a 180°
rotation.The motif is one-sixth the size of the lattice unit.The symmetry type is
P6 (Fig.24).
The arch surface in the center of the south facade of the cone is completed with
intricately braided ornaments and eight-pointed stars formed between them.The
symmetry type is PG (Fig.25).
Fig. 22 Pattern analysis of cone surfaces (pattern 14)
Fig. 23 Pattern analysis of cone surfaces (pattern 15)
M.Özdemir et al.
Evaluation
In the evaluation section, the ‘repetitive unit’ of each pattern with analyzed
geometric substructures and the substructure of each unit are presented in Table1.
As mentioned in the method section, these basic units are evaluated according to
seventeen wallpaper classes from symmetry groups. The sequencing and analysis
of the figures begins at the entrance door and continues with the cone section.
The substructure of each unit is divided to express the reflection properties of the
symmetry group. The dotted line surrounding the unit shows the boundary of the
repeating unit. Aperiodic patterns are also specified, and the repeated pattern of
the pattern with translational symmetry is discussed. Each basic unit shows the
rotation, mirror, and glide reflection symmetries of the symmetry group it belongs
to. The analysis of the ornaments in the table is reviewed, and the drawing
methods of the geometric ornaments on İzzettin Keykavus Tomb are categorized
according to the similarities in the geometrical features of the repeating units and
substructures.
Fig. 24 Pattern analysis of cone surfaces (pattern 16)
Fig. 25 Examination of the cone surfaces (pattern 17)
The Geometric Patterns ofİzzettin Keykavus Tomb
Table 1 Analysis of the patterns
Geometric pattern Symmetrical substructure Repeating unit Symmetry
features
1 Aperiodic.
The repeating
units belong to
two alternating
cells.
Above: CMM
Below: P2
2 Six-fold rotation.
Reflection
Symmetries.
P6M.
3 Aperiodic.
The repeating unit
belongs to the
alternative
pattern.Reflections
in two
perpendicular
directions.
CMM.
4Six -fold rotaon
with reflecons.
P6M.
M.Özdemir et al.
Table 1 (continued)
5 Two axes of
mirror reflection.
Two-fold rotation.
PMM.
6 Two axes of
mirror reflection.
Two-fold rotation.
PMM.
7 Six-fold rotation.
Reflections. P6M.
8 Six-fold rotation.
Reflections. P6M.
The Geometric Patterns ofİzzettin Keykavus Tomb
Table 1 (continued)
9 Six-fold rotation.
Reflections. P6M.
10 Six-fold rotation.
Reflections. P6M.
11 Six-fold rotation.
Reflections. P6M.
12 Six-fold rotation.
Reflections. P6M.
M.Özdemir et al.
Conclusions
Seventeen patterns, including the cone section and the entrance section of the
tomb, have been examined. Patterns mainly consist of repetitive structures. Three
of them belonged to the aperiodic pattern class. These non-periodic systems do not
provide a systematic reproduction. Patterns included in this tiling class are evaluated
Table 1 (continued)
13 Aperiodic.
The repeating unit
belongs to the
alternative
pattern. Four-way
reflection.
PMM.
14 Four-fold
rotation. Reflectio
nP4M.
15 Four-fold
rotation.P4.
16 Six-fold
rotation. No
reflection or glide
symmetry. P6.
17 Glide symmetry.
PG.
The Geometric Patterns ofİzzettin Keykavus Tomb
in the basic unit category, and alternative growths are provided using translational
symmetry. Patterns are classified according to seventeen symmetry groups. Eight
of the analyzed patterns belonged to the P6M group. Accordingly, it is safe to
state that the patterns mostly have rotational symmetry and reflection symmetry
in six directions. Three patterns are included in the PMM class. These patterns
are similar thanks to their two-way reflection properties. Two patterns have CMM
symmetry. Other patterns are included in the classes P6, P2, PG, P4M, and P4. The
study allows us to understand that different patterns consist of similar repeat units.
Therefore, it is possible to create the same pattern with other repeating units. This
is because the repeating units of parallelogram and hexagon form the infrastructure
of the same pattern, as seen in patterns 1, 2, 4, 7, 8, 9, 10, 11 in Table1. In addition,
as seen in patterns 3 and 16, it has been observed that a pattern can consist of both
rhombic repeating units. Patterns mainly consist of a repeating infrastructure of
squares and parallelograms. The symmetrical infrastructure indicates that this unit
constitutes the basic building block in the completion of the pattern. It provides
the repeating unit with mirror symmetry. It can be said that different patterns
with similar simplicity and symmetrical infrastructure are formed, especially in
geometric patterns based on hexagons. As can be seen, the use of methods together
provides a better understanding of the patterns. Although İzzettin Keykavus Tomb
is a small-scale structure, it has rich, diverse ornamental patterns. The present study
has indicated that similar patterns can be diversified with different reproduction
methods. Accordingly, patterns with similar basic units can differ with the change of
symmetry elements. This feature can be a useful guide for designers.
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and institutional affiliations.
Makbule Özdemir is researcher and teaching assistant in the Department of Architecture at Gazi
University. She completed her B.Arch in the Department of Architecture at Selcuk University. She
received her M.Sc in Architectural Program in Gazi University and presently, she is a Ph.D. candidate
in the Architectural Program at the same university. Her research interests include computational design
thinking, machine learning, computer-aided design and digital reconstruction of geometry in the history
of architecture.
Semra Arslan Selçuk graduated from the Department of Architecture, SU, Konya. She received her Ph.D.
from the Department of Architecture, METU, Ankara. Her research interests involve computational
design and manufacturing technologies in architecture and architectural education, biomimesis in
architecture, form-finding processes in architecture, and sustainable design approaches by using
biomimetic approaches. She has been on duty in the Department of Architecture, Gazi University as
an Assoc. Prof. since 2014. Her recent publications are on biomimesis in architecture, performative
architecture, digital design technologies and CAD/CAM processes, collaborative design knowledge,
modeling, information and communication technologies in design education, cognitive processes in
design education and practice.
Hasan Fevzi Çügen completed his bachelor degree in architecture at Anadolu University, Eskisehir
in 2003and master degree at Gazi University in 2022. In 2007, he founded ASIR PROJE architectural
conservation and restoration company. He prepared several types of restoration projects on historical
The Geometric Patterns ofİzzettin Keykavus Tomb
artifacts (castles, mosques, churches, underground cities, baths, etc.) In 2013, he participated in a
scientific research project supported by Anadolu University aiming to research Ottoman artifacts in
Kosovo. He conducting studies on historical artifacts at ASIR PROJE and the Turkish Architecture
Research Center in Ankara.
