CC BY
11целесообразным решением задачи повышения эффективности работы КН при их работе на различных режимах.
Литература
1. Кукис В.С. Влияние рециркуляции и нейтрализации отработавших газов на экологические показатели дизеля / В.С. Кукис, Е.А. Омельченко // Materials of IX international research and practice conference «Fundamental and applied science» (October 30-Movember 7, 2014.V. 18. Technical science). Sheffield UK, 2014. - P.26-30.
2. Постановление Главного государственного санитарного врача РФ от 30.05.2003 № 114 (ред. от 03.11.2005) «О
введении в действие ГН 2.1.6.1338-03» (вместе с «Гигиеническими нормативами «Предельно допустимые концентрации (ПДК) загрязняющих веществ в атмосферном воздухе населенных мест. ГН 2.1.6.1338-03», утв. Главным государ-ствен-ным санитарным врачом РФ 21.05.2003) (Зарегистрировано в Минюсте РФ 11.06.2003 № 4679).
3. Мельберт А.А. Основные направления снижения дым-ности и ток-сичности отработавших газов дизелей / А.А. Мельберт, Н.Д. Новоселова, А.А. Унгефук, Т.А. Стопорова // Дымность дизелей: Сб. статей. Барнаул, 2005. - С. 4-9.
MATHEMATICAL REGULARITIES OF DESIGN OF AZERBAIJANI CARPET
PATTERNS
Kazim-zada A.K.
Azerbaijan University of Architecture and Construction, "Information technology and systems" Department Ph.D (engineering sciences)
ABSTRACT
The paper treats information signs of identification of hand-made carpets and carpet patterns demonstrating the presence of "golden section" rule, Fibonacci and Lucas numbers, symmetry. The researches were performed using "visual centres", golden triangles, rectangles, lozenges as well as variants of composition meshes based on them. These parameters are regarded as criteria for correctness of the making of a carpet pattern, arrangement of the patterns and as indicators during identification of the carpet patterns too.
Keywords: carpet, Fibonacci numbers, golden section, visual centres
1. Introduction
A number of teachings on harmony appeared in ancient Greece of classical epoch the most profound trace in the world culture has been left by Pythagorean teaching of all these. The Pythagoreans sought for mathematical substantiation of beauty. They studies human body proportions and established a mathematical canon of beauty in accordance with which Polyclet, the Greek sculptor, made his statue "Canon". The influence of Pythagorean teaching on proportions was felt not only by medieval scientists but also by art and science of the Renaissance epoch. Leonardo da Vinci wrote about the use of proportion in art in his treatise on paintings. "A painter embodies in the form of proportion the same regularities concealed in nature which a scientist learns in the form of numerical law".
Patterns in the ancient hand-made Azerbaijani carpets kept in many museums of the world represent geometrical and curvilinear figures. The patterns symmetrically arranged both in vertical and horizontal import harmony, balance and peace to the design of the carpets. Azerbaijani carpets are included in the UNESCO list of Non-Material Cultural Heritage.
2. Statement of the problem
The essence of the "golden section" is as follows: if there exists a point B on a straight line segment of arbitrary length AC which divides the said segment into unequal parts at which all the segment AC is to greater part BC as the greater part BC itself is to the smaller part AB (fig.1) [1, c.25]. The ratio of these segments (denoted by F) is always equal F=1,618034... The quantity 1/F is denoted by j and equals to 0,618034...
AC BC ^
-= — = o
BC AB (1)
A B C
Fig. 1. Division of straight line segment by the "golden section"
Golden figures -golden triangle, rectangle, lozenge, pentagon, the golden Archimedean spiral constructed on the "golden section" principle are of common knowledge.
The presence of the "golden section" in the works of art cannot be considered separately, without relation to symmetry. It is precisely the presence of these two notions that imparts balance, harmony, emotional perception and effect to the works of art.
The study of hand-made Azerbaijani carpets and carpet patterns is of great interest from this viewpoint.
3. Solution
Carpet patterns represent colours planar patterns. They display: human figures, animals, birds, plants, objects of everyday life, geometrical shapes etc. They can be both symmetrical and asymmetrical. It is also widely known that the majority of patterns of Azerbaijani carpets are geometrical figures [2].
Some informative features have been revealed for patterns and ornaments of Azerbaijani carpets among which the name of pattern, its geometrical dimensions, number of contours, colour gamut, centre of gravity, area, conformity of pattern to the "golden section" rule and symmetry [3, c.290].
The combination of distribution density of a certain pattern with its centre of gravity, area of surface, the presence of the "golden section" and symmetry are the required information parameters for subsequent comparison to other patterns [4, c.153].
According to the type of symmetry, images of carpets and their patterns can be divided into three classes: asymmetrical, symmetrical along OX-axis and symmetrical both along OX-
axis and OY-axis. The studies have shown that the majority of patterns in Azerbaijani carpets comprising both one and several patterns inserted into one another, represent patterns with mirror and rotational (900) symmetry. Examples of these patterns are
given in Fig. 2. ( Fig.2a - unsymmetrical; Fig.2b - symmetrical by OX and OY; Fig.2c - symmetrical by OY; Fig.2d - symmetrical by OX )
a)
Fig. 2. Examples of carpet patterns.
b)
c)
d)
For studying the carpet patterns from the viewpoint of symmetry, we shall consider the following geometrical figures.
Fig.3a, b, c demonstrate three denoted by ABC.
;golden" isosceles triangles
a)
Fig. 3. Examples of "golden" isosceles triangles
b)
c)
According to the properties of the "golden" triangle shown in fig.3a zACB=zBAC =36°, and zABC=108°. In this case the ratio of the sides AC/AB = AC/BC =1.62. The angles of the second triangle demonstrated in fig.3b are equal, respectively to zACB=zBAC =72°, and zABC=36°. The ratio of the sides here is AB/AC = BC/AC = 1.62. For the third golden triangle (fig.3c) zABC= 144°, zACB = zCAB=18°. In this case the ratio of the sides also is AC/AB = AC/BC =1.62.
If the triangles shown in fig.3a, b, c are bilaterally represented at the base AC we shall obtain three figures having the shape of "golden" lozenges.
It should be noted that all the three triangles demonstrated in fig.3a, b, c possess vertical mirror symmetry and the lozenges obtained through mirror representation of the triangles possess both vertical and horizontal symmetry.
F = F , + F 2
n n-1 n -2
at the following initial values:
As is known Fibonacci numbers Fn are in close connection with the "golden section". By Fibonacci numbers the following numerical sequence (2) is implied in mathematics:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ... (2)
Fibonacci numbers are given by recurrent relationship:
(3)
(4)
This numerical sequence has a number of peculiarities one of which is that the ratio of a subsequent number of (F ) series to a preceding one (Fn) tends to the value of 1.62.
Tables 1 and 2 given below contains the values of the first 10 initial numerical values of Fibonacci and Lukas series and their ratios. As seen from the table starting from the sixth column the ratios of numbers for Fibonacci numbers are equal to F number (1,62) [1, c.143].
F1 = 1 h F2 = 1
Table 1
Ratio of subsequent Fibonacci numbers to preceding ones
№ 1 2 3 4 5 6 7 8 9 10
F n 1 1 2 3 5 8 13 21 34 55
Fn+1 1 2 3 5 8 13 21 34 55 89
Fn+1 / Fn 1 2 1.5 1.6 1.62 1.62 1.62 1.62 1.62 1.62
The given value (1.62), however, forms in the course of Lukas numbers Ln are given by recurrent relationship :
consideration of ratios of Lukas numbers likewise. By Lukas L = L , + L „ (6)
J n n-1 n -2 v y
numbers the numerical sequence presented below (2) is implied: at the following initial values:
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, ... ( 5 ) L1 = 1 h L2 = 3. ( 7 )
Table 2
Ratio of the subsequent Lucas numbers to the preceding ones
№ 1 2 3 4 5 6 7 8 9 10
L n 1 3 4 7 11 18 29 47 76 123
Ln+1 3 4 7 11 18 29 47 76 123 199
Ln+1 / Ln 1 2 1.5 1.66 1.62 1.62 1.62 1.62 1.62 1.62
1 2 3 4 5 6 7 Fig. 4. Graphs of ratio values of the succeeding Fibonacci and Lukas numbers
Fig 4. displays graphs of ratio values of the succeeding Fibonacci and Lukas numbers given in tables 1 and 2. The studies of hand-made Azerbaijani carpets of the XVIII-XX centuries have disclosed that the majority of the carpet products are of 1:1, 1:2, 1:3, 2:3, 3:5, 5:8, 1:3, 3:4, 4:7 in dimensions or close to them which corresponds to Fibonacci and Lukas numbers.
It is likely to have depended not only on the size of rooms for which the carpets products were intended. Of importance was human nature as well, aspiration for harmony and balance.
On the basic of golden geometrical figures one can construct lots of variants of composition nets created in accordance with the "golden section" rule and Fibonacci numbers. Examples of variants of these nets are given in Fig.5 [5].
a) b) c) d)
Fig. 5. Variants of basic nets created according to the "golden section" rule
e)
In the epoch of Renaissance artists had found that any painting had certain points which involuntarily arrested our attention, so called "visual centres". This discovery was termed "golden section" of a painting by the artists of that period. That is why for attracting attention to the main element of a painting and the dimensions of the given plane are of no importance at
all. As an example, is used the painting "Bacchus and Ariadne" of Tiziano (fig.7). it is necessary to match this element with one of visual centres. Here it is absolutely unimportant of what format the painting is - horizontal or vertical. There are only four such points and they are located at the distance of 3/8 and 5/8 from the respective edges of the plane (fig. 6)
Fig. 6. Points of "visual centres"
Fig.7 Example of the use of "visual centres" in the works of the outstanding artists
Examples of the use of "golden" figures by outstanding painters shown on Fig. 8. After carrying out harmonic analysis of a picture of Raphael Santi, the prominent painter of the Renaissance epoch, the researches have discovered that the composition plan of this picture was based on golden isosceles triangle (Fig. 8a).
The outstanding Italian painter of the Renaissance epoch Pietro della Francesca also was an outstanding mathematician who made a considerable contribution in polyhedron theory. One of brilliant examples of his artistic style using golden polyhedrons is the painting named "Baptism of Christ". Fig. 8 b,c demonstrates harmonic analysis of this painting.
a) b) c)
Fig. 8. Examples of the use of "golden" figures in the works of the outstanding artists
One of examples of a painting based on the "golden section" principle is Russian artist K.K.Vasiliyev's painting "At the Windows". The artist placed the girl's face in the "golden" point of the painting located at the intersection of two golden lineshorizontal and vertical the crossing of which passes through, the girl's eye (Fig. 9 a)
Studying composition structures of paintings, art critics paid attention to the wide use of the law of golden section in landscape
paintings. An example of the use of this law is I.I.Shishkin's painting "Ship Grove". The pine which is brightly lit by the sun (standing in the foreground) divides the painting along the horizontal by golden section. The sunlit hillock on the right of the pine divides the painting along the vertical by golden section (Fig. 9 b) [1, c.195].
By matching the mentioned nets with carpet wares, their patterns and orna-ments one can obtain a mathematical expression of beauty and harmony of the works of art created by skilled Azerbaijani craftsmen. Examples of matching these nets on the images of carpet wares are given in Fig. 10 a, b, c, d, e.
As a rule, a carpet comprises several tens and even hundreds of patterns. Depending on the quantity of carpet patterns we
have used some or other composition nets. The net demonstrated in Fig. 10 a, b we have used in the process of studying symmetry of few patterns on the carpet plane. The nets demonstrated in Fig. 10 c, d, e we have used in studying symmetry of numerous carpet patterns on the carpet plane.
a) b) c)
Fig. 10. Examples of matching nets on the images of the carpets
d)
e)
As seen in fig. 10 a, b, c, d, e the location of central patterns in Azerbaijani carpets conforms to the "visual centres" rule. Owing to the use of the composition nets made up of "golden" geometrical figures symmetry, harmony, balance in the pattern location can be clearly displayed from the viewpoint of mathematics.
4. Conclusion
The results of analysis and studies of more than 620 handmade Azerbaijani carpets of the XVIII-XX centuries have shown that some of the carpets were characterized by symmetry of patterns along OX-axis, other carpets had symmetry of patterns along OY-axis while there were the carpets having the said symmetry both along OX and OY axis. Numerical values obtained in study of the carpets are presents in table 2.
Table 3
Numerical values of the presence of symmetry and "visual centres" in the carpets
Name of informative features Quantity %
Location of the central pattern in "visual centres" 406 65.5
Symmetry along OX-axis 452 73
Symmetry along Oy -axis 520 84
Symmetry along OX and Oy axis 412 66.5
It is distinctly displayed in the presented figures, that the location of the carpet patterns is symmetrical along OX and OY axis, along the triangle lines as well as along bisecting line. Such location complies with the principles of golden proportion and it should be said that in percent the location of patterns in the middle zone of the carpets complying with the principles of
golden proportion is more as compared to the ratio of geometrical dimension (length/width) of the carpets.
We have also studied elements of decoration of the carpets. The studies have shown that the majority of the patterns satisfy the golden section rule and Fibonacci serious. As for example, an element of decoration demonstrated in fig. 11.
Fig. 6. Element of decoration
The ratio of geometrical dimensions of this pattern is 8:5 which represents Fibonacci numbers and golden section.
References
1. Стахов А.П. Код да Винчи и ряды Фибоначчи: Универсальный Код Природы. СПб.: Питер, 2007, 316 с.
2. Керимов Лятиф. Азербайджанский ковер. Баку-Ленинград: Изд.АН Аз.ССР, 1961, 210 с.
3. Abdullayeva G.G., Kazim-zada A.K., Recognition and Identification of Plane Color Images in the Case of Carpet
Designs. ISSN 0146-4116, Automatic Control and Computer Sciences, 2008, Vol. 42, No. 6, pp. 288-294. © Allerton Press, Inc., 2008.
4. Кязим-заде А.К. Определение центра тяжести криволинейных плоских фигур на примере ковровых узоров. Известия НАНА, том XXVIH, № 3, 2008 г. с 152-157.
5. www.zhamkov.com
DEVELOPING A HACCP PLAN FOR QUALITY AND SAFETY OF MINERAL
WATER
Yaheliuk S.
Lutsk national technical university, associate Professor, PhD, Senior Lecturer of department of commodity
Peredriy O.
Lutsk national technical university, associate Professor, PhDm Senior Lecturer of department of commodity
Tkachuk V.
Lutsk national technical university, associate Professor, PhD, Senior Lecturer of department of commodity
ABSTRACT
The outcomes of studies on the system of quality control of food products, including mineral water, are given in this contribution. As a part of the study comprehensive commodity product description with the ingredients and packaging have been conducted, risks connected with technological operations have been determined and critical control points of the mineral water production process have been detected. The HACCP plan, which permits to increase the quality, safety and competitive ability of mineral water, has been designed.
Keywords: quality, mineral water, HACCP plan, the consumer characteristics, quality, safety, process, risks.
Introduction
It is impossible to provide quality and safety of domestic goods without implementation of systems that control the quality, especially HACCP (Hazard Analysis and Critical Control Point, a system of risk, danger factors and critical control point analysis), on the food industry enterprises. The struggle for safety and quality of food takes place all over the world, including main trade partners of Ukraine, countries of CIS and Europe, that set mandatory condition of work and organization of HACCP systems and follow the transfer of production. It is important to put in remembrance that setting up HACCP and other systems ensure the competitive ability of manufacturing facilities in the food industry and help entrepreneurs make their business more successful. It boosts effectiveness of food industry and saves expenses. HACCP is also called an admission pass to international markets where it is prestigious and advantageous to work. Furthermore, setting up a HACCP assists attraction of investments and growth of enterprise value. Currently, there are a lot of Ukrainian enterprises on the markets of the EU and
CIS that are in compliance with standards of HACCP system. Nowadays HACCP or similar systems that provide security during the fabrication of food are established on 709 enterprises, 54 of such systems are on the stage of establishing and 138 are being developed [1].
Reference Analysis and Problem Statement
According to international requirements for food quality, the quality control only is insufficient. It cannot guarantee the safety from all detected deviations, and limits the ability to determine random distortion of chemical residues and contamination levels. The new global food industry introduces new quality management system to solve such problems. HACCP is a program that aims to eliminate the health risks associated with consumption of food. The principles of this system are included in the legislation of several countries, and its presence on the manufacturer's enterprise is in most cases a prerequisite for entering into contracts for the supply of food. Plenty of researches in the sphere of the use of HACCP system are conducted in order to assure quality and safety of food. For
