Angličtina pro matematiky IV

COURSE MATERIALS AND HOMEWORK X.

 

 

Golden ratio
From Wikipedia, the free encyclopedia
The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b.
 
a) Answer these questions.
 
  1. What does it mean when two quantities are in the golden ratio?
  2. How many synonyms are there of the golden ratio? How would you translate it into Czech (Slovak)?
  3. How do you distinguish (in notation) the golden ratio and its reciprocal?
  4. Why was the golden ratio interesting for architects?
  5. Why were mathematicians interested in it?
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887. Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean. Other terms encountered include extreme and mean ratio, medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut, golden number, and mean of Phidias. In this article the golden ratio is denoted by the Greek lowercase letter phi ( ) , while its reciprocal, or , is denoted by the uppercase variant Phi ( ).
The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:
This equation has one positive solution in the set of algebraic irrational numbers:
At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.
 
b) Study the drawing of the golden rectangle and try to write instructions for its construction.
 
 
Construction of a golden rectangle:
1. ………………………………………………………………….
2. ……………………………………………………………………….
3. ……………………………………………………………………………………………………….
 
 
A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship
 
Geometry
 
c) Try to explain the meaning of these words.
 
 diagonal        regular icosahedron              orthogonal        Fibonacci sequence
 
apothem         geometric progression          arithmetic progression      perimeter
 
irrational number      pyramid                  tangent           coincidental relationship

 

Golden ratio – listening
 
Pre-listening
 
What is the golden ratio?
Where can you find the golden ratio?
 
Listening – fill in the missing information
 
  1. The golden ratio can be found in ............................................................................
  2. Each number in the Fibonacci sequence is ..............................
  3. When you divide one number by the number before it, you obtain ......................
  4. Da Vinci and Le Corbusier used the golden ratio .................................................
  5. t/s ratio is always ..............................to the golden ratio.
  6. The golden ratio applies to an ............................ human body.
  7. List some examples of the golden ratio in human body.
a)      .............................................................
b)      ............................................................
c)      .............................................................
d)      ............................................................
e)      ...........................................................
             List some examples of the golden ratio on human face.
a)      ................................................................
b)      ...............................................................
c)      .............................................................       
 
  1. The study of American physicist revealed the golden ratio in .....................................
  2. The cochlea serves to ..................................................................
  3. DNA consists of two intertwined .................................................................
 
 
 
 
 
 
 
Golden ratio – listening
 
Pre-listening
 
What is the golden ratio? 1.618
Where can you find the golden ratio?
 
Listening – fill in the missing information
 
  1. The golden ratio can be found in ..........pyramids, flower, snails, body, art
  2. Each number in Fibonacci sequence is .......the sum of preceding two numbers...
  3. When you divide one number by the number before it, you obtain ..numbers close to one another
  4. Da Vinci and Le Corbusier used golden ratio ........in their designs..
  5. t/s ratio is always .........equivalent.....................to the golden ratio.
  6. Golden ratio applies to an ..........idealized.................. human body.
  7. List some examples of golden ratio in human body.
f)       ..........navel to foot...................................................
g)      ..........wrist elbow..................................................
h)      ...........shoulder line top of the head..................................................
i)        .........navel shoulder top of the head...................................................
j)        ..........navel knee, knee end of the foot.................................................
             List some examples of golden ratio on human face.
d)      .............teeth...................................................
e)      .............length ..of the face, width of the face................................................
f)        .............pupils, eyebrows, nostrils................................................       
 
  1. The study of American physicist revealed the golden ratio in ........lungs...........
  2. The cochlea serves to .............transmit sound vibrations.............
  3. DNA consists of two intertwined ..........perpendicular helixes.........
       
       
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