Somayyeh Omranifar1 , Shabnam Namdar2, Nader ghaemi3, Yaser Narimanpour Maleki4  
1 PhD student of Architecture, Architecture Department, Islamic Azad University, Tabriz Branch, Tabriz, Iran  
2 Assistant professor, Architecture Department, Islamic Azad University, Tabriz Branch, Tabriz, Iran  
3 Architecture Department, Islamic Azad University, Tabriz Branch, Tabriz, Iran  
4 Department of Architecture, Islamic Azad University, Mashad Branch, Mashad, Iran  
In the present article, the collocation of the Fractal Geometries and Voronoi Algorithm  
has been proved in line with the designing of garden’s digital architecture and it was  
Research Article  
made clear that the designing of the ancient four-segment gardens is considerably  
PII: S238315531800004-7  
matching with the contemporary architecture which has come about subject to the  
effect of computers and software. The present study is a qualitative research of  
descriptive-analytical type and it has been conducted with a glance at the history of  
Iranian gardens’ designs and their delineated geometrical analyses concerning the  
structural philosophy of Voronoi and Fractal algorithms. The primary goal of the  
researcher is showcasing the timelessness in the designing of the Iranian garden so as  
to prove that the prior designing has been in accordance with the novel patterns of the  
contemporary architecture and it can get the paradise manifested like before in a  
corner of the earthy ground in today's machining world. It can be stated as a part of the  
obtained results that the idea of exhibiting water in the garden and the system of water  
transmission to the most distant spots in the garden tries depicting the ancient  
imaginations of the previous architects about Voronoi algorithm considering the use of  
the shortest path which is a perpendicular line drawn towards a dot on a straight line.  
Moreover, the use of four-segment garden’s basic module and its division in an  
internally descending manner displays Fractal geometry in each of Voronoi levels of  
the garden’s plots.  
Received: 10 Sep. 2018  
Revised: 05 Dec. 2018  
Published: 15 Dec. 2018  
Corresponding author’s e-mail:  
Iranian garden, Geometry, Voronoi algorithm, Fractal, Water  
Iranians. The architectural context and garden-  
building methods lost their performance and beauty  
The thing that is currently seen in the Iranian cities  
as buildings and urban spaces and elements has not  
been able to crystalize the fruitful background of  
Iranian architecture and garden-building and it is,  
instead, depicting a sort of distinct identity lack and  
self-alienation. Lack of paying attention to the social  
and cultural values and hasty movement towards  
modernity without the real perception of what is  
being constructed and processed have led to visual,  
cultural, social and identity unrest in the level of  
Iranian cities and architecture [1].  
Iranian garden has always endeavored since  
long ago to stay in accord with a philosophical and  
contextual-functional concept and it has been  
embodied as one of the specifications of Iranian  
architecture. Considerable differences were brought  
about in the context of Iranian architecture and  
garden-building with the beginning of the foreign  
advisors’ trip to Iran from Safavid Era and the  
blending of the Iranian architecture and decorations  
with what has been offered as art and technique to  
of their geometrical ground and their connection  
rings were broken in such way that the  
performance occasionally led to the discarding of  
beauty and the paying of attention to the visual  
beauties resulted sometimes in the creation of  
insensible cases in the gardens’ grounds. Therefore,  
the researchers have been coerced to subtly  
contemplate about the functional and visual  
designing of the Iranian gardens so as to identify the  
meanings latent in them beyond the plan of Iranian  
four-segment garden thereby to figure out the  
missing rings of the function and beauty; it is by the  
perception of the concepts existent behind the  
functional and visual designing of the gardens that  
solutions can be found for logical attachment to the  
contemporary mankind’s psychological needs. In the  
present study, the author proves the basics existent  
in fractal geometry and digital Voronoi algorithms  
within the former geometrical context of the Iranian  
garden and shows that the nature’s flexible spirit has  
been portrayed in the meaning-oriented designs by  
Citation: Omranifar S, Namdar Sh, Ghaemi N,  
Narimanpour Maleki  
(2018). Iranian Garden’s  
Journal of Art and Architecture Studies  
Architectural Designing: Collocation of Voronoi and  
Fractal Algorithms in the Context of Natural Background.  
J. Art Arch. Stud., 7 (2): 28-33.  
ISSN 2383-1553  
J. Art Arch. Stud. 7(2): 28-33, Dec. 15, 2018  
J Art Arch Stud. 7(2): 28-33, 2018  
the imaginative ancient architect in such a way that building of the garden falls in the intersection point  
it has been able to absorb the psychological states [3].  
and satisfy the needs of the today’s mankind in the  
area of the contemporary naturalism.  
System of Geometrical Structure:  
The researcher seminally deals with the  
The geometrical structure of the Iranian garden  
theoretical foundations and the concepts usually is shaped in two substantial forms with one of the  
encountered regarding the Iranian gardens; then, indices of its geometrical structure being the  
the references of the Iranian gardens’ patters are relationship between the interior and exterior spaces  
extracted. Next and after deeply studying the [4]: one is the creation of three stretches (in three  
structural philosophy and theoretical foundations of axes) in parallel and along with the garden and the  
Voronoi and fractal algorithms, the feasibility of other is the consideration of two primary  
adjusting the current theoretical foundations and perpendicular axes and then dividing of the garden  
their structural philosophy to the geometrical plan of into squares that per se have their own regular  
the Iranian garden has been analyzed. All the geometrical divisions (Figure 1). In Iranian garden,  
analyses have been discussed geometrically and, special attention was paid to the geometrical shapes  
considering the functionalist and human-oriented and square that showcased a simple and clear-cut  
grounds of the contemporary architecture and distance between the garden’s components enjoyed a  
garden-building, the existing functional factor has particular importance [5].  
been displayed as a connecting ring.  
The main question of the present study is that  
what is the primary functional factor that can, as a  
connecting ring of the garden’s humanism and  
naturalism concepts, make the geometrical design of  
the garden approach the timeless concept?  
How are such digital and naturalist geometries  
like Voronoi and fractal algorithms matched on the  
grounds of the Iranian garden? The author tries  
answering the above-posited questions in five  
chapters, namely the Iranian garden and formation  
Figure 1. Geometrical structure of the garden [3].  
process, Voronoi and fractal algorithms, Iranian  
garden and collocation of water-transmission  
function with Voronoi algorithm and Iranian garden  
and fractal geometry in the geometry of the plots.  
Voronoi Algorithm:  
Voronoi algorithm is a geometrical algorithm.  
This algorithm receives a series of points as its input.  
These points can be selected randomly and/or be  
certain points on a plane [6]. In this type of  
algorithm, the input points lead to the production of  
a series of regions on the plane and, problem-  
specifically, give the closest answers with the highest  
likelihood for the calculation of the region of a point  
by considering all the line segments between the  
given point and the other points. Then, the bisectors  
of these line segments are drawn and the regional  
bisectors are created around the given point [7]. In  
this case, the algorithm juxtaposes the given regions  
without overlapping. It has been recently proved  
that many of the structures follow this algorithm in  
nature (Figure 2), especially the micro-structures of  
the nature wherein the Voronoi schemes are  
abundantly found [6].  
Iranian Garden and Formation Process:  
The concept of “garden architecture” reflects a  
“sense of place”; garden is defined as a space that  
has the perfect and utmost reflection of the world in  
it. This concept that fosters order and coordination  
might be expressed more tangibly by means of  
numbers, geometry, color and matter [2]. Based on  
the studies that have been carried out on the  
reference patterns of the Iranian garden, the  
primitive sample of the archetypes has been applied  
as the common contact surface between the patterns  
of Iranian gardens and it has also been crystalized in  
the Iranian carpets; it depicts a square-shaped area  
of a relatively vast land which is surrounded inside  
walls. The surface area of this land has been divided  
by two axes (two water streams) perpendicular to  
and intersecting one another with the main stream  
eventually pouring into four ditches and with trees  
finally surrounding the square. The most major  
Omranifar et al., 2018  
Figure 4 illustrates the unique and well-known  
example of Voronoi algorithm has been displayed in  
the cubic water building (the name that has been  
given to the stadium of water sports in Pecan,  
China). Besides having a lot of advantages in terms  
of the architectural form and creation of modern and  
innovative structures, this building is completely  
environment-friendly and it is enumerated, in other  
words, as a green and environment-compliant  
structure. The primary idea of this complex’s  
designing has been the formation of a cube that has  
been brought to existence from the joining of water-  
filled bubbles and it is completely transparent [8].  
Figure 2. Voronoi algorithm in nature [8].  
There are also more complex forms of Voronoi  
algorithms the regions of which are made by soft  
curves in lieu of the broken lines. Furthermore, there  
are also 3D images of this algorithm that receive the  
initial points in a given space and gives the area  
zoning by dividing of the spaces (Figure 3). In  
architecture, various Voronoi algorithms have been  
widely applied [8].  
Fractal is a geometrical shape that is created  
with the repetition of a simple logic in a recursive  
manner. In a normal manner, the shape obtained  
from the fractals can be divided into several parts  
each of which resembling the overall shape. The  
other interpretation existent about the fractals is  
that they have indefinite components and/or they  
possess similar structures making them look  
identical in the various magnifications. The term  
“fractal” was first used by Mandelbert in 1975. It is  
derived from the Latin root “fractus” meaning  
broken [9].  
In the process of a fractal’s production, there are  
at least two primary shapes: one basic and one  
generative. In every stage of the repetition, the  
generative shape is placed in the position of every  
one of the basic shape’s fundamental segment or line  
segment. From the theoretical perspectives, this  
repetition can be continued endlessly. The algorithm  
used for producing the fractal shapes is a basic  
process the job of which is placing one shape  
between two points. This process includes the size  
change, rotation and displacement of the generative  
shape for taking a position between the two  
presumed points or two ends of a line segment.  
Amongst the simplest types of the fractals are the  
Sierpinski triangle (Figure 5) and fractal cube [6].  
In the traditional Iranian architecture, as well,  
the fractal pattern can be seen in Karbandi  
Figure 3. The structure of Voronoi algorithm [8].  
Yazdibandi [roof covering] and ornamental corbel  
vaults. In these patterns, the basic module is iterated  
with a special order to create an integrated whole.  
The fractal pattern has also been inserted in the  
architectural form. This pattern is occasionally  
symbolic and it is sometimes inserted in the wefts  
and warps of the architectural function (Figure 6).  
Figure 4. The structure of Voronoi algorithm  
J Art Arch Stud. 7(2): 28-33, 2018  
input points, an operator point and two result  
Figure 7. Functional formation of the basic four-  
segment garden (source: the author)  
In order to accomplish the objective of water  
display and water transmission to all the garden’s  
spots, preparatory measures should be taken so that  
the water can be transmitted to all the various spots  
of the garden in respect to the land’s slope. In line  
with this goal, some other points have been  
considered as the input points of Voronoi algorithm  
and, considering the water-transmission function  
and water display, the shortest paths have been  
offered according to the philosophy of Voronoi  
algorithm in two states of the land slope.  
Iranian Garden and Collocation of Water-  
Figure 8. Formation of Voronoi algorithm in the  
garden’s geometry (Source: the author)  
Iranian garden is the geometrical embodiment  
of paradise in the context of a scorching desert with  
an extreme inclination towards water exhibition in  
every corner being one of its primary specifications  
[9]. That is because is considered as the symbol of  
dynamicity and life in the desert. The most original  
solution for showcasing it and taking a more  
appropriate advantage thereof is its management in  
the middle of the garden in such a way that the water  
starts flowing from the longest edge of the garden  
and proves presence in the entire garden while  
performing its duty of satiation. Concentration on  
the soil’s natural slope has been the easiest solution  
in the previous gardens.  
According to figure 7, points A and C are  
considered as the input points of Voronoi algorithm.  
Considering the use of square geometry for the easy  
recognition of the distances in the garden, point B is  
the place where the pond should be placed and its  
stretching gives the points E and D. this way, the  
preliminary pattern of the four-segment gardens is  
formed by dividing the land into four areas with two  
As it is seen in figure 8, the first case pertains to  
gardens that have been positioned in respect to the  
land’s natural slope in an ascending or descending  
form. In this type of gardens, water starts flowing  
from its source and is brought from a garden to  
another [5]. Water enters through the point A and  
reaches the point B from which it is distributed  
along the operator ED axis in the lower half of the  
garden. In order to transmit water to the upper  
plots, the upper axis includes water inlet as the  
secondary operator axis and water is subsequently  
distributed in the upper plots.  
The second case shows the method of water  
transmission in the gardens that are usually irrigated  
by wells’ water. In this type of gardens, the  
intersection of the two primary axes is considered as  
the water inflowing point and the land’s slope is  
assumed to be towards the walls on the four sides of  
the garden [5]. In this state, only one operator axis is  
Omranifar et al., 2018  
considered for the garden. The gardens in the the Iranian garden is adjusted to the contemporary  
deserts usually make use of this method. The water digital architectural patterns drawn on Voronoi  
line axis shows the water-transmission system.  
Considering the Voronoi algorithm and  
The stretch of the traditional market’s body  
according to the goal of transmitting water to the within the city limits of Tabriz and the prominent  
most distant spots in the garden, certain points are historical buildings like Dar Al-Fonun School, Darb  
considered on the peripheral walls and vertical lines Baghmishe, Haramkhaneh Complex, municipality  
are presumed from the water’s primary stream building and so forth as well as the valuable business  
principle edifices like the large market of artefacts and the  
recommending the selection of the shortest large bazar of Tabriz’s leather have caused this  
perpendicular route from the point onto the water region to be transformed into one of the tourism  
stream following which the Iranian garden’s water- spots of the city. The adjacency of the unique  
transmission network/system is formed within the tourism potentials to Tabriz County’s trade pole,  
format of four-segment garden. It is by locating organized within the limits of Jomhouri Street, has  
certain points on the wall that squares and caused the vehicles’ access to the aforesaid centers to  
rectangles are geometrically formed in respect to the encounter considerable limitations and factors like  
amount of water required to be transmitted to the excessive congestion of the region, lack of cargo-  
transportation infrastructures and blending of the  
Considering the Voronoi algorithm, the thing industrial and tourism uses and heavy traffic on  
that is effective in the shape obtained from the Shahid Madani Street and adjacent streets have  
algorithm is the number of the points to which water caused the creation of problems in terms of urban  
should be transmitted in respect to water power and tourism in the foresaid region.  
land’s shape. If a garden is square in shape with  
large dimensions and the soil slope is not so much  
Iranian Garden and Fractal in the Plots’  
steep and/or if water is withdrawn from wells for Geometry:  
satiating the garden, smaller Voronoi areas will be  
The most original factor in the formation of  
attained. However, if the surface area is considered fractal geometry is the basic module and the  
similar to the previous example with a rectangular scattering path and method of this module in the  
geometry, the resulting shape will be different and general format. Considering the functional system of  
follows the land’s slope following which the number the square-shaped geometry of the garden, the upper  
of the plots will be reduced.  
section is defined as the basic module in the garden  
and the entirety of the garden’s surface area is  
covered through the arrangement of this module  
based on Voronoi algorithm and the primary  
geometry of the plots is formed in a descending  
(downsizing) trend in the module-dividing state.  
Figure 10 illustrates the plots’ fractal process in  
Iranian gardens.  
Figure 9. Voronoi algorithm based on the basic  
four-segment pattern (Source: the author)  
Figure 9 exhibit the Voronoi algorithm in Figure 10. Formation of the fractal pattern (source:  
collocation with the water-transmission function in the author)  
the garden and proves that the original pattern of  
J Art Arch Stud. 7(2): 28-33, 2018  
According to the pictorial information and their implemented in the adjacency of the water inlet with  
analysis, it was made clear that the four-segment an internal-descending multiplication in the whole  
garden’s geometry that has been applied as the four-segment garden on the ground surface.  
primary geometry in the architectural designing of  
the Iranian gardens follows and is matched with the  
patterns produced in Voronoi and fractal algorithms.  
Competing interests  
The authors declare that there is no competing  
In addition, the water-transmission function and the interest.  
idea of water display is the ring connecting the  
garden’s geometry to Voronoi and fractal algorithms.  
The former architect and garden-builder followed  
the nature to create a beautiful paradise in the heart  
of a scorching desert and getting the imaginary  
nature manifested in the grounds of the available  
land. It is worth mentioning that the naturalism  
ground in Voronoi and fractal algorithms as well as  
the geometrical designing of the gardens have  
created the aforesaid harmony and match.  
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