The fresco "The School of Athens" by Renaissance artist Raphael depicts philosophers in a grand architectural setting. It includes over 50 figures in a variety of poses and locations within the fresco. The archways are painted in a semicircular shape. The fresco is one of Raphael's most famous works and an excellent example of Renaissance artistic techniques and ideals.
Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
Inscribe Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
SHARIGUIN PROBLEMS IN PLANE GEOMETRY
This volume contains over 600 problems in plane geometry and consists of two parts. The first part contains rather simple problems to be solved in classes and at home. The second part also contains hints and detailed solutions. Over 200 new problems have been added to the 1982 edition, the simpler problems in the first addition having been eliminated, and a number of new sections- (circles and tangents, polygons, combinations of figures, etc.) having been introduced, The general structure of the book has been changed somewhat to
accord with the new, more detailed, classification of the problems. As a result, all the problems in this volume have been rearranged.
Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
Inscribe Circles in Triangles using Geometric ConstructionColin
Inscribe circles in triangles using geometric construction, with PC geometry software such as Dr Geo on Linux. This slide show presents some innovative constructions. The slide show and included instructions are public domain. Basic use of compass and straightedge is advised as a prerequisite topic.
SHARIGUIN PROBLEMS IN PLANE GEOMETRY
This volume contains over 600 problems in plane geometry and consists of two parts. The first part contains rather simple problems to be solved in classes and at home. The second part also contains hints and detailed solutions. Over 200 new problems have been added to the 1982 edition, the simpler problems in the first addition having been eliminated, and a number of new sections- (circles and tangents, polygons, combinations of figures, etc.) having been introduced, The general structure of the book has been changed somewhat to
accord with the new, more detailed, classification of the problems. As a result, all the problems in this volume have been rearranged.
One of the best PPT on HERONS' FORMULA You will get here.Contains all most all information about Heron, its formula.Formulas of some other shapes also.Area of triangles and its derivation.
How to calculate the area of a triangleChloeDaniel2
The space that is occupied by a flat shape or the surface of an object. The standard unit of measurement is mostly either ㎡ or cm2. We are going to discuss the Area of triangles here
One of the best PPT on HERONS' FORMULA You will get here.Contains all most all information about Heron, its formula.Formulas of some other shapes also.Area of triangles and its derivation.
How to calculate the area of a triangleChloeDaniel2
The space that is occupied by a flat shape or the surface of an object. The standard unit of measurement is mostly either ㎡ or cm2. We are going to discuss the Area of triangles here
Right foundation at the right stage is the most important factor in the success of any student in exam and in life the APEX IIT / PMT foundation program is aimed at students studying in class IX, who aspire to prepare for engineering / medical entrance in future. The program keeps the school curriculum as base and further upgrades the students’ knowledge to meet the requirements of competitive exams. The program has been design in a way so as to develop orientation of the students as well as to motivate him to excel in competitive exams.
Four Year Classroom Program is the ideal program for students who wish to start early in their quest for a seat at the IITs. This program helps the student not only to excel in IIT-JEE but also in Olympiads & KVPY by building a strong foundation, enhance their IQ & analytical ability and develop parallel thinking processes from a very early stage in their academic career. Students joining this program will have more time to clear their fundamentals and practice extensively for IIT-JEE, their ultimate goal!
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Basic phrases for greeting and assisting costumers
Italija vokietija
1. “The School of Athens” is the fresco made by the Italian Renaissance artist Raphael (1483–1520).
The archs are in a semi-circle form.
1) In a coordinate system draw a circle, when the circle‘s radius is equal to 4, and the center is in
O( -2; 3) point. Calculate the circumference of the given circle and its area.
Write down the circumference of the circle.
2) In a coordinate system draw the given semi-circles:
𝑥2
+ 𝑦2
= 9, when 𝑦 ≥ 0 and 𝑥2
+ (𝑦 − 3)2
= 9, when 𝑦 ≤ 3.
Calculate the coordinates of their collision points.
The answers:
1) 𝐶 = 8𝜋, 𝑆 = 16𝜋. (𝑥 + 2)2
+ (𝑦 − 3)2
= 16.
2)
(
3√3
2
;
3
2
) ir (−
3√3
2
;
3
2
)
2. Albrecht Dürer (1471-1528 m.) – the most popular German Renaissance artist,
copper carver and art theoretic.
“Melancholia” is one of the most popular masterpieces created by the artist.
„Melancholia“
In A. Dürer’s “Melancholia” you can see a
magical square. In every line, column, and
diagonal you can sum up every number and
get the same answer.
1) Fill in the magical square.
2) When was “Melancholia” created?
The answer is hidden in the mystical
square. Two numbers one next to the
other in the same line.
16 13
10 11
6 12
4 14
Answers:
1) 2) 1514 year
3. Jacek Yerka (Polish surrealist)
“The Tower of Subconsciousness“ and “The Garden“
Tasks:
Once upon a time a master asked his servants to scoop some water from a magical well and fill a
reservoir with it. Three servants offered their service to complete this mission. Each of them took a
bucket and started scooping water from the magical well. The surface of the part of the tower which
had to be filled was of a shape of a perfect hexagon, which sides were each 20 m, and the height of
the reservoir was 15 m.
1. Which spatial geometrical shape meets the part of the tower which has to be filled?
2. How many litres are needed to fill the reservoir fully? Give the answer using an integral
number.
3. How many buckets would fill the reservoir fully if it is known that the diameter of a bucket‘s
surface is 26 cm, the diameter of the bucket‘s top is 36 cm, and the height of the bucket is 40
cm. Consider the value of Pi as ≈3,14. Give the answer using an integral number.
4. The first servant alone would fill the reservoir in 70 h, the second in 72 h, and all three
together in 25 h. How much time would it take for the third servant to fill the reservoir if he
was doing it all by himself? Give the answer in units worth.
Answers:
1. Vertical hexagonal prism.
2. 𝑉 = 15588000 𝑙
3. 519600 buckets
4. 85 h.
4. Osman Hamdi Bey (Turkish artist)
„Girl reading the Quran“
Tasks:
1. The length of the a side of the rhombus highlighted in the painting is 0,5 m, and the shorter
diagonal is 60 cm. long. Find the area of the rhombus.
2. What would the area of the rhombus be, if the area of four hexagonal tiles was deducted
from it? Here MN is the median of ABC.
3. What percentage of the area of the rhombus meets the area of triangle BNM? Give the answer
in exactness of 0,01%.
4. In what percentage is the area of all hexagonal tiles bigger than the area of all equilateral
triangles which are in the rhombus?
Answers:
1. 𝑆𝑟𝑜𝑚𝑏ℎ𝑢𝑠 = 2400 𝑐𝑚2
= 0,24 𝑚2
.
2. 𝑆 = 2400 − 1350√3 𝑐𝑚2
3. 4,06 %
4. 200%
5. Math in Arts
Greek Amphora
A classic Greek amphora is composed of two parts – a sphere A, from which
the bottom segment is cut off, and a neck formed by a sphere B rotating
around the vertical axes. The sphere B has the same diameter as the sphere A,
and its center is positioned on the opposite side of the square formed on the
diagonal of the square inscribed into the circle of the bottom’s sphere plane
projection (look at the picture below).
Problem
With the diameter of the bottom sphere of the
amphora be equal to 40 cm, Find the area of the
amphora’s projection on the vertical plane.
Solution
After noticing that segments x and y have the same
area, we can conclude that the amphora’s
projection area is twice as large as the inscribed
square.
Let d be the diameter of the sphere.
r = d/2
a = r / √2
S = 2 (2a)2
= 2 (r √2)2
= 4 r2
= 4 (d/2)2
= d2
=
= 402
cm2
= 1600 cm2
6. Perspective in Painting
Artists use perspective to
distinguish close objects
from remote objects. This
is done by reducing an
object’s visible size
proportionally to the
distance from the object.
Let’s consider a
masterpiece from Xanthos
Hadjisoteriou, an
acclaimed Greek Cypriot
painter.
Problem
Assuming all houses on this
picture have windows of
the same size, find the real
height of the belfry in the
far-left corner if the height
of the woman on the front
is 160 cm.
Solution
Let’s assume that the view
point in this picture is at
the bottom center of this
picture. This is not exactly
true, but will work for our
approximations.
Let’s draw vectors from the
view point to all objects we
want to take into account.
The table below shows
horizontal and vertical sizes
of each vector taken from
PowerPoint, where the
vectors were drawn, and
the calculated length of
each vector.
7. Vector # Horizontal size (“) Vertical size (“) Length (‘)
1 2.2 2 2.973
2 2.25 3.06 3.798
3 0.88 2.94 3.069
4 0.95 3.46 3.588
5 0.12 5.64 5.641
6 0.18 5.97 5.973
7 2.75 5.76 6.383
8 2.78 7.06 7.588
1. Knowing that sizes of windows residing on vectors 3-4 and 5-6 are the same, define the proportion
between the visible size and distance from the view point:
a. Window 3-4 visible height = .52”
b. Window 5-6 visible height = .33”
c. Proportion between distance and visible height is:
<length of vector-5> / <length of vector-3>
:
<window 3-4 visible height> / <window 5-6 visible height>
or
5.641 / 3.069 : .52 / .33, which means that
“Objects that are 1.838 times further seem 1.576 times smaller”
2. The belfry is 6.383/2.973= 2.147 times further than the woman. Hence, it seems 1.576/1.838*2.147=
1.841 times smaller.
3. Woman’s visible height: 3.06” – 2” = 1.06”
4. Belfry’s visible height: 7.06” – 5.76” = 1.30”
5. Belfry’s real height: 1.30/1.06*1.841=2.258 is larger than woman’s or is:
160*2.258=3m 61cm
Conclusion
Since in the real life a belfry is much higher, either the assumption about the view point is incorrect, or the
artist diverges from true proportions to produce special impressions.