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Файл:M&Ms. Movies and Maths. Учебно-методическое пособие
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Unit 6
The GR
Lead-in
A. Look at the pictures below. Describe every picture. What do they have in common?
In pairs, make assumptions about the mathematical concepts that can be behind all of
these pictures.
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Viewing 1
A. Watch a short video clip about the golden ratio and Fibonacci numbers – Fibonacci
Sequence in Nature – https://www.youtube.com/watch?v=nt2OlMAJj6o. Which of the
objects above are shown in the clip?
B. Read the following expressions. Which one was not mentioned in the video clip?
DNA spiral a pine-cone
the fruitlets of a pine-apple a bud
animal bodies rose petal
sand dollar starfish
the nautilus shell the golden rectangle
the human body blueprint braces to indicate proportions
C. Watch the video clip again. Answer the following questions.
● If you break down the perfect rectangle into smaller squares, what pattern can
be seen?
● What do the spirals that we observe in nature, for example, in ocean waves,
reveal?
● What is the greatest example of Fibonacci sequence, in the opinion of the
speaker in the video clip?
Vocabulary
A. Watch the video clip again, if necessary. What do these words/phrases mean in
the context?
a DNA spiral a pine-cone
a bud a petal
a sequence to obtain
a blueprint braces
to indicate proportions the divine proportion
the golden rectangle a curve
to reveal to break down
a pattern indefinitely
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B. Read the following sentences. Match the words in bold with the correct definitions.
1. The mathematical definition of sequence goes as follows: a sequence is
an enumerated collection of objects in which repetitions are allowed.
2. In the second experiment they obtained clear results.
3. Let’s break down this diagram into smaller parts.
4. A pattern is beginning to emerge from our analysis of the experiment data.
5. His biography revealed what kind of person he was.
6. This path curves round to the right.
7. There is a mathematical explanation why we are so attracted to pine-cones.
8. This sequence can continue indefinitely.
9. In the springtime all the trees are in bud.
10. Can you show the blueprint for your equipment?
a._____________________
adv.
in a manner not clearly determined
b. _____________________
v.
to allow something that has been kept hidden to be
seen
c. _____________________
v.
to separate into simpler elements
d. _____________________
n.
the seed-producing structure of a pine tree
e. _____________________
v.
to gain possession of; acquire; get
f. _____________________
n.
a swelling on a plant stem consisting of overlapping
leaves or petals
g. _____________________
n.
an arrangement of two or more things following one
after another
h. _____________________
n.
a plan indicating how someone will build, design or
achieve something
i. _____________________
n.
a particular way in which something is done,
organized or arranged; a plan or a diagram
j. _____________________
v.
to move in a line that has no straight parts; to bend
C. Fill in the gaps with the words or phrases from the exercise above.
1. This project can _____________ funding only if you show how the institute can
benefit from it.
2. The _____________ of patterns will vary from one specific problem to another.
3. They’ve got the _____________, they’ve already started building.
4. The speed of a planet produces a line that is known as a rotation _____________.
5. The measures taken can last _____________.
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D. Which examples of the golden ratio from the video clip are the most appealing to
you?
Grammar
A. Study the following example from the video clip.
Example: We begin to see Fibonacci spiral. The spiral in and of itself is insignificant,
its importance is revealed in where we find it.
Why is the definite article used in the second sentence?
B. Study the following rules on the use of the definite article.
THE is used:
1) when we know which object is meant because either we have already mentioned it,
or we say which one is meant, or it’s clear from the situation:
Can you pass me the book which is next to your computer?
Could you close the door? (there’s only one door)
2) when there is the only one(s) around (something unique), e.g. the planets, the stars,
the peoples: the Japanese, the Earth, the Thirty Years’ War, the arms trade, the international market, the
North Pole, the police
3) with expressions referring to our physical environment – the world around us and
its climate: the sea, the mountains, the rain, the wind, the weather, the sunshine, the fog, the night, the
future, the past, the climate, the ground, the sky
My wife likes the seaside, but I prefer the mountains.
4) superlatives: This is the most beautiful flower I’ve ever seen.
5) with adjectives to form the meaning of group of people having one characteristic in
common: the rich, the old, the sick, the unemployed, the injured, the blind, the deaf, etc.
Do you think the rich should pay more taxes?
The man over there is collecting money for the blind.
6) to talk about things in general, more common with scientific instruments and
inventions, musical instruments, animal and plant species:
Life would be quieter without the telephone.
The tiger is in danger of becoming extinct.
7) with mountain groups, island groups: the Altay mountains, the Alps, the Bahamas, the Kuriles;
- oceans, seas, rivers, canals, deserts: the Black sea, the Mediterranean; the Nile, the Suez Canal; the Sahara
- regions: the Middle East, the Far East, the north of Russia (but: northern Russia);
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- most hotels: the Hilton, the Station Hotel;
- most cinemas and theaters: the National Theater, the Palace Theater; the Odeon;
- museums and art galleries: the British Museum;
- newspapers: the Herald, the Times
C. Read the sentences below. Match the sentence and the rule from exercise B that it
follows.
1) Take for example the sunflower.
2) Currents move through the ocean and the tide rolls on to the shore.
3) But the greatest example of all is directly above our heads.
D. Which is correct: A/AN or THE? Fill in the gaps.
1. The Golden Ratio is one of ____ most attractive topics for an Arts student.
2. She could hear ____wind whistling through the trees outside.
3. How can we combine economic growth and respect for ___ environment?
4. It has been found that the shells of mollusks grow annually following ___ golden
ratio.
5. He told he’d bought a house somewhere near ___ Mediterranean Sea.
E. All the articles have been taken out of the following sentences. Supply them where
necessary.
1) Different parts are all related to each other and whole. Ideal correlation of parts
is based on golden ratio principle.
2) Cardan, Bombelli and others included problems in their texts on finding golden
ratio using quadratic equations.
3) To calculate most aesthetically pleasing rectangle, you simply multiply length
of short side by golden ratio approximation of 1.618.
Speaking
A. Look at the following pictures of various art products. In pairs or groups, describe
them and say which of them exhibit the golden ratio. Place the pictures in the
chronological order.
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1. The Parthenon
2. “Living Still Life” by Salvador Dali
3. “Composition in Red, Yellow and
Blue” by Piet Mondrian
4. The Chartre Cathedral
5. Stradivarius Violin
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B. Look at the following rectangles. Find the one that we can see in the pictures and
constructions from exercise A.
C. Conduct an experiment. Ask four people in the group which rectangle from exercise
B they find aesthetically pleasing. Compare the results of your survey with the
following test result:
In the 1860s, a psychologist named Gustav Fechner conducted experiments to
explore this question. He presented subjects with an array of varying rectangles, and
asked them which was their favorite. The results showed that 76% of all choices
focused on the three rectangles with ratios of 1.75:1, 1.62:1, and 1.50:1. The winner
was the “Golden Rectangle” (D - with ratio of 1.618:1).
Reading
A. Scan through the text. Which of the artistic creations from the speaking exercise
above are mentioned in it?
The Golden Ratio
Mathematicians since Euclid (300 BC) studied the golden ratio, also known as
the divine proportion, golden mean, or golden section. It is a number often encountered
when taking the ratios of distances in simple geometric figures such as the pentagon,
pentagram, decagon, dodecahedron. It is denoted , or sometimes .
The term “golden section” seems to first have been used by Martin Ohm in
the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002,
p. 6). The symbol ("phi") was apparently first used by Mark Barr at the beginning of
the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC),
who a number of art historians claim made extensive use of the golden ratio in his
works (Livio 2002, pp. 5-6). Similarly, the alternate notation is an abbreviation of
the Greek tome, meaning “to cut”.
The golden ratio is the ratio between two segments such that the smaller (bc)
segment is to the larger segment (ab) as the larger segment (ab) is to the sum of the two
segments (ac), or bc/ab = ab/ac ≈ 1.618.
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a b c
The golden ratio is found throughout nature, art, and architecture. Pinecones,
seashells, and the human body all exhibit the golden ratio. Piet Mondrian and Leonardo
da Vinci commonly incorporated the golden ratio into their paintings. Stradivari
utilized the golden ratio in the construction of his violins. The Parthenon, the Great
pyramid of Giza, Stonehenge, and the Chartres Cathedral all exhibit the golden ratio.
While many manifestations of the golden ratio in early art and architecture were
likely caused by processes not involving knowledge of the golden ratio, it may be that
these manifestations result from a more fundamental subconscious preference for
the aesthetic resulting from the ratio. A substantial body of research comparing
individual preferences for rectangles of various proportions supports a preference
based on the golden ratio.
[Abridged from http://mathworld.wolfram.com/GoldenRatio.html and
“Universal Principles of Design” by William Lidwell, Kritina Holden, Jill Butler]
B. Read the text again. Look through the following statements. Find the ones that are
not true according to the text.
1. The golden ratio is often encountered while studying simple geometrical figures.
2. The term “golden section” was introduced in the 19th century.
3. The number that represents the golden ratio approximates 1,618.
4. Claims of the significance of the golden ratio appearing in art tend to be
exaggerated.
5. Many great artists, sculptors and architects utilized the golden ratio in their works.
C. In pairs, compose one question to the group mates based on the ideas from the text.
Ask this question to 4-5 students. What does the survey reveal?
D. Using the vocabulary from the reading selection in exercise A, translate the
following text into English.
Золотое сечение
Золотое сечение – это универсальное проявление структурной гармонии.
Оно встречается в природе, науке, искусстве – во всем, с чем может
соприкоснуться человек. Однажды познакомившись с золотым правилом,
человечество больше ему не изменяло.
Наиболее емкое (exact) определение золотого сечения гласит, что меньшая
часть относится к большей, как большая ко всему целому. Приблизительная его
величина – 1,6180339887. В округленном процентном значении (rounded value)
пропорции частей целого будут соотноситься как 62% к 38%. Это соотношение
действует в формах пространства и времени. Древние видели в золотом сечении
отражение космического порядка, а Иоганн Кеплер называл его одним из
сокровищ геометрии. Современная наука рассматривает золотое сечение как
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«ассиметричную симметрию», называя его в широком смысле универсальным
правилом, отражающим структуру и порядок нашего мироустройства.
[Abridged from http://russian7.ru/2014/09/zolotoe-sechenie-kak-ehto-rabotaet/]
Vocabulary 2
A. Work with your partner. Read the word in bold and three definitions to it. Decide
which one is NOT correct.
WORD DEFINITIONS
1. to exhibit
to show
to display to the
public
to tire out
2. to incorporate
to form a united
whole
to be included into a
corporation
to include
3. to utilize
to use in an
effective way
to make practical use
of smth
to use something
continuously
4. a manifestation
an indication of
something existing
a demonstration
exposure
5. to result from
to make something
happen
to be caused by
to happen as a result
of
6. substantial
of considerable
size
important
subsequent
B. Read the sentences below. Fill in the gaps with the words in bold from exercise A.
You may need to use more than one word for one gap.
1. His difficulties at work _____________ the lack of attention.
2. The changes in weather patterns are obvious _____________ of global warming.
3. The vitamins come in a form that is easily _________ by the body.
4. Suggestions from the survey have been _______________ into the final design.
5. He __________ great self-control considering her rudeness.
Viewing 2
A. In the following video selection Keith Devlin, professor of mathematics at Stanford,
speaks about the golden ratio – http://www.youtube.com/watch?v=4oyyXC5IzEE.
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Watch a part of his lecture (6:40-11:35) and say which aspects of the Golden Ratio
topic he covers:
● the history of research;
● the connection between art and mathematics;
● math concept of the golden ratio.
B. Watch the lecture again and arrange the following utterances in the order of
occurrence.
1. Newton and Leibniz invented calculus
2. Quotients of numbers couldn’t measure all lengths
3. The golden rectangle
4. The decimal part repeats
5. The ratio of two lengths
6. These are clearly cultural numbers
7. The ratio of the whole to the larger part equals the ratio of the larger part to the
smaller
8. Mathematicians finally fused together those two conceptions of numbers
(arithmetic and geometry)
9. The rectangle whose aspects ratio the human cognitive system finds the most
pleasing
10. This was in Euclid’s “Elements”
11. Solve it using the formula for the quadratic
C. Read the sentences. Select the answer, a or b, which best matches the meaning of
the word in bold.
1. The decimal numeral system has ten as its base.
a. using powers of ten
b. expressed as ten
2. The prize was split between the two outstanding students.
a. to have two parts
b. to divide into two or more parts
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