Proportions

Achaemenid art is royal Courtly art which has been influenced by subordinate nations arts. The different nations like Mesopotamian, Egyptian and Anatolian. As it is clear that Achaemenid art as a magnificent court art was a modest imitation of the another nations art But they did not ever blindly imitate others' art,but they transformed what they took from others and changed into Iranian art;and becouse of that, the coordination and harmony that was seen in the Achaemenid style,  was the localization of the artistic element of other nations.This paper trying to discuss, analyze and compare of human proportions of statues and reliefs as one of the most important artificial evidence of Achaemenid dynasty. Proportion also is visible in nature. Anatomical proportion is a mathematical concept which that important in visual arts. The formula of a golden division in Proportion is also visible at the nature and human artifacts such as statues and monuments. In this study, the influences of the Achaemenid art from subordinate nation's arts have been considered as a source of proportions in this period. The results indicate that human proportions in Achaemenid art has been Influenced by art and artistic proportions rules of Mesopotamia, Greece and specially Egypt. It is noted that highest point of Achaemenid art in terms of use principles Proportion in architecture and sculpture is very similar to the Egyptian art.

Comparative analysis of the concepts of colors and
geometric shapes P. Florensky, E. Trubetskoy, and Vasily
Kandinsky showed their unity is symbolic and empirical
perception. Use in the analysis of ancient temples in X–ХV
centuries of the Christian symbolism of geometric forms
and light, allowed to identify a common set of proportional
laws, which is «life-giving pillar» is visualized by the light
from the windows of the drum and shown in the plans
geometric solution of the problem of «squaring the circle».
Сравнительный анализ концепций цветов и геометрических форм П. Флоренского, Е. Трубецкого и В. Кандинского выявил единство их символического и эмпирического восприятия. Использование в анализе древ-нерусских храмов X–ХV вв. христианской символики геометрических форм и света позволило выявить единые для них пропорциональные закономерности, в основе которых находится «Животворящий столп» визуализирован-ный светом из окон барабана и проявленный в планах геометрическим решением задачи «квадратура круга».

The plan's proportions of building interiors Old Russian temples of X-XV centuries were been
solved with the problem of “the quadrature of circle”. When ancient builders of temples were
going to begin to measure interior dimensions, they could draw by means of pegs and ropes
symbolical proportions 8:9, 10:13, 2:3, 5:8 of width to length of temples by means of simple
solutions of a task a quadrature. This geometry allows identifying the structural code in the
center of cathedrals with several naves and churches with the narthex. We may see the
proportion as 10:13 at that structural соde of interiors at plans of Old Russian temples XXV
centuries. Half circles from the geometric constructions, that we may see in temple’s plans,
were build the depth of altars in the temples with the of interior proportions 10:13, 2:3, 5:8 or in
structural codes in the center of cathedrals with several naves. In geometrical creation of
problems of “the quadrature of circle”, which has been imposed on interiors of plans of Old
Russian churches, we can see a symbolical view of “the earth”, “the sky on the earth” and “the
sky invisible”. And also we can observe symbolical partitioning in plans of these cathedrals: the
“world visible” and the “world invisible”, which are divided by an iconostasis.
Простые геометрические решения задачи «квадратуры круга», а точнее её обратной
задачи «кругатуры квадрата», которые можно произвести на строительной площадке при
помощи колышков и шнуров, позволяют объяснить пропорционирование внутреннего
пространства в планах древнерусских храмов X–XV веков. Геометрическим начертанием
равных по площади круга и квадрата выявляются системы размерения как общих
габаритов, которые в древнерусских храмах, чаще всего, имеют пропорции ширины к
длине такие, как 8:9, 10:13, 2:3, 5:8, так и основных символических делений на «мир
видимый» до иконостаса и «мир невидимый» в алтаре, а также символических мест в
храмах, называемых «небо на земле», «земля» и «небо невидимое». Эта же геометрия
позволяет выявить в планах многонефных соборов и церквей с притворами структурное
ядро, состоящее из девяти ячеек, посередине которого находится центральный
подкупольный квадрат. Ширина ядра храма относится к его длине как 10:13, в
большинстве храмов.

The mathematician John Dee, by making architecture one of his 21 derivative mathematical arts, followed a well-known Renaissance tradition that strove to disengage architecture from its medieval classification within the mechanical arts and to include it instead among the liberal arts. Mathematics had been the most obvious choice for association, not only because of the practical application of arithmetic, geometry, and perspective in the delineation and execution of architectural projects, but also because it elevated the art of building into a philosophical exercise, since proportional correspondences between the architectural members were thought to express Nature's universal laws. Within the realm of the eighteenth century's emphasis on science and method, and in the wake of important technological advances in the manufacture of instruments for measurement, the relationship between architecture and mathematics was reevaluated.

Solutions of the problem of “squaring of circle” made it possible to explain “inaccuracies” in the
whole-measure proportional analysis of ancient Russian churches of the Tenth–Fifteenth
centuries. This geometry determined the most common width to length ratios in three-nave
churches: 8:9, 10:13, 2:3, 5:8, and justified the proportions of the main symbolic divisions:
“visible world” to the iconostasis; “invisible world” in the altar; symbolic temple spaces called
“heaven on the earth”, “earth” and “invisible heaven”. The structural core with proportions of
10:13 was identified in plans of multi-nave cathedrals and churches with vestibules. The geometric analysis allowed to confirm the existence of “The Life Giving Pillar” in the centre of
the dome space, as confirmed by theological works, chronicles and images of ancient Russian
icons. The double diameter of its horizontal section represents the geometric record of the
Dogma of the Holy Trinity and is equal to the internal width of three central naves, both in threenave
and multi-nave churches. The dome square as the symbol of earth in 75% study churches
is constructed to this circle as the symbol of heaven, equal in area. The height to the central
point of the vault is proportioned in relation to the width of “The Life Giving Pillar” as 1:3, 1:4, 1:5
or in relation to the temple’s square side as 5:8. The practical method of layout of dimensions
and main symbolic divisions in plans of cathedral cores was reconstructed in close connection
with the Christian theology. The geometric layout of two symbolic Throne variants (as the Holy
Sepulcher or the Throne of the Lord of Heaven and Earth) was revealed in temple cores.
Решения задачи «квадратуры круга» позволили объяснить возникающие «неточности»
при цельномерном пропорциональном анализе древнерусских храмов X–XV веков. Эта
геометрия определила чаще всего встречающиеся отношения ширины к длине в
трёхнефных церквях: 8:9, 10:13, 2:3, 5:8, и обосновала пропорции основных
символических делений: «мир видимый» до иконостаса; «мир невидимый» в алтаре;
символические места в храмах, называемые «небо на земле», «земля» и «небо
невидимое». В планах многонефных соборов и церквей с притворами выявлено
структурное ядро с пропорциями 10:13. Геометрический анализ позволил подтвердить
существование в центральном подкупольном пространстве «Животворящего столпа», о
чём имеются богословские упоминания, летописные источники и изображения на
древнерусских иконах, удвоенный диаметр горизонтального сечения которого является
геометрической записью догмата о Троице, ему равна внутренняя ширина трёх
центральных нефов как в трёхнефных, так и в многонефных соборах. К этому кругу,
символу «неба», выстраивается равный ему по площади подкупольный квадрат как
символ земли в 75% обследованных храмов. Высота до центральной точки свода
центральной главы пропорционируется относительно ширины «Животворящего столпа» в
соотношениях 1:3, 1:4, 1:5, или относительно стороны квадрата храма в соотношении 5:8.
Реконструирован практический способ разбивки габаритов и основных символических
делений в планах ядер храмов в тесной связи с христианским богословием. Выявлены
геометрические схемы расположения в ядрах храмов двух символических вариантов
Престолов: как Гроб Господень или Трон «Царя небу и земли».

The reform of the Polish education system initiated in 1998 is a new challenge for teachers of mathematics, especially at the early teaching level and the primary school level. Since that time mathematical content has been bound up with other items of education. It was necessary to prepare a different way of teacher training – so as to prepare teachers to go through the new content of the subject “math”. There was a chance to: • Create a new approach to teaching early geometry. • Create new connections between arithmetic and geometry, keeping the essence of arithmetical and geometrical cognition.