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Melisa Sude Yağmur

Asking our questions about the project

RATIO AND PROPORTION IN MY DREAMS

WHAT DO YOU SEE??

Hi, my name is Melisa. I want to be a painter. I like observing my enviroment and pitcurizing what I see. I think one painting can be expected as a successful magnum opus if there is a good balance between ratio and proportion in itself.

For example, in the drawings like human face there is certain ratio and it is essential to draw figures in that ratios.

I want to mention about one more example. In the painting, the objects which get closer seem as if their size increases but this does not mean that their size increses, this means that they get closer. Therefore, getting this ratio in the perspective is very important. Otherwise, the ratio of objects jumble and there arises a painting which disturb people.  

DO YOU KNOW THESE?

GOLDEN RATIO

Definitions of Golden Ratio
1) Numeric definition

Here is a 'Fibonacci series'.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..

If we take the ratio of two successive numbers in this series and divide each by the number before it, we will find the following series of numbers.

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...

The ratio seems to be settling down to a particular value, which we call the golden ratio(Phi=1.618..).



2) Geometric definition

We can notice if we have a 1 by 1 square and add a square with side lengths equal to the length longer rectangle side, then what remains is another golden rectangle. This could go on forever. We can get bigger and bigger golden rectangles, adding off these big squares.

Step 1 Start with a square 1 by 1
Step 2 Find the longer side
Step 3 Add another square of that side to whole thing



Here is the list we can get adding the square;
1 x 1, 2 x 1, 3 x 2, 5 x 3, 8 x 5, 13 x 8, 21 x 13, 34 x 21.
with each addition coming ever closer to multiplying by Phi.

start 1 by 1, add 1 by 1 => Now, it is 2 by 1, add 2 by 2



Now, it is 3 by 2, add 3 by 3 => Now, it is 5 by 3, add 5 by 5










Now, it is 8 by 5.










3) Algebraic and Geometric definition

We can realize that Phi + 1 = Phi * Phi.


















Start with a golden rectangle with a short side one unit long.
Since the long side of a golden rectangle equals the short side multiplied by Phi, the long side of the new rectangle is 1*Phi = Phi.

If we swing the long side to make a new golden rectangle, the short side of the new rectangle is Phi and the long side is Phi * Phi.

We also know from simple geometry that the new long side equals the sum of the two sides of the original rectangle, or Phi + 1. (figure in page4)

Since these two expressions describe the same thing, they are equivalent, and so
Phi + 1 = Phi * Phi.



II. Some Golden Geometry

1) The Golden Rectangle








A Golden Rectangle is a rectangle with proportions that are two consecutive numbers from the Fibonacci sequence.

The Golden Rectangle has been said to be one of the most visually satisfying of all
geometric forms. We can find many examples in art masterpieces such as in edifices of ancient Greece.





2) The Golden Triangle



If we rotate the shorter side through the base angle until it touches one of the legs, and then, from the endpoint, we draw a segment down to the opposite base vertex, the original isosceles triangle is split into two golden triangles. Aslo, we can find that the ratio of the area of the taller triangle to that of the smaller triangle is also 1.618. (=Phi)








If the golden rectangle is split into two triangles, they are called golden triangles suing the Pythagorean theorem, we can find the hypotenuse of the triangle.

3) The Golden Spiral









The Golden Spiral above is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with the golden rectangle.
If you take one point, and then a second point one-quarter of a turn away from it, the second point is Phi times farther from the center than the first point. The spiral increases by a factor of Phi.





This shape is found in many shells, particularly the nautilus.








4) Penrose Tilings





The British physicist and mathematician, Roger Penrose, has developed an aperiodic tiling which incorporates the golden section. The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B).



When a plane is tiled according to Penrose's directions, the ratio of tile A to tile B is the Golden Ratio.








In addition to the unusual symmetry, Penrose tilings reveal a pattern of overlapping decagons. Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean.



5) Pentagon and Pentagram





We can see there are lots of lines divided in the golden ratio. Such lines appear in the pentagon and the relationship between its sides and the diagonals.



We can get an approximate pentagon and pentagram using the Fibonacci numbers as lengths of lines. In above figure, there are the Fibonacci numbers; 2, 3, 5, 8. The ratio of these three pairs of consecutive Fibonacci numbers is roughly equal to the golden ratio.

RATIO AND PROPORTION

From a project

A proportion is a name we give to a statement that two ratios are equal. It can be written in two ways:

• two equal fractions,

  or,

• using a colon, a:b = c:d

When two ratios are equal, then the cross products of the ratios are equal.
That is, for the proportion, a:b = c:d , a x d = b x
A proportion is simply a statement that two ratios are equal. It can be written in two ways: as two equal fractions a/b = c/d; or using a colon, a:b = c:d. The following proportion is read as "twenty is to twenty-five as four is to five."

In problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. To find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

Here, 20 and 5 are the extremes, and 25 and 4 are the means. Since the cross products are both equal to one hundred, we know that these ratios are equal and that this is a true proportion.

We can also use cross products to find a missing term in a proportion. Here's an example. In a horror movie featuring a giant beetle, the beetle appeared to be 50 feet long. However, a model was used for the beetle that was really only 20 inches long. A 30-inch tall model building was also used in the movie. How tall did the building seem in the movie?

First, write the proportion, using a letter to stand for the missing term. We find the cross products by multiplying 20 times x, and 50 times 30. Then divide to find x. Study this step closely, because this is a technique we will use often in algebra. We are trying to get our unknown number, x, on the left side of the equation, all by itself. Since x is multiplied by 20, we can use the "inverse" of multiplying, which is dividing, to get rid of the 20. We can divide both sides of the equation by the same number, without changing the meaning of the equation. When we divide both sides by 20, we find that the building will appear to be 75 feet tall.

Note that we're using the inverse of multiplying by 20-that is, dividing by 20, to get x alone on one side.

 

A GREAT DAY: )

Selma in the kıtchen

Selmaaa...
When Selma wakes up, she heards her mothers cheerful voice. She gets closer to her and says that your aunt and your cousin come to us today. She is very happy because she likes spending time with her cousin, Halenur, very much. As soon as hearing this news, she takes her mother's recipe book and goes to kitchen to make her delicious chocolate cake. In the book there is a note which says that this cake serve around 10 people. There will 5 people so she thinks that I should mix HALF of the all ingredients written in the book. She makes the cake by putting 1/2 of the ingredients and the cake is perfect.

Her aunt and her cousin come. After they spend some time together, her aunt gaves her money to go to cinema with her cousin.
They take money and go to cinema. Selma asks the price of one ticket and learns that it is 10 tl. She have to multiply 10 by 2 to find the cost of TWO tickets, one for Halenur and one for Selma. She pays 20 tl and gets their tickets. After watching film, they decides to wander a shopping center. Selma buys her course book from a bookstore. They come back to home.

After her aunt and her aunt go to their home, Selma handles the book and looks at the number of pages of the book. It has five hundred pages. Selma have to finish this book in FIVE days. She computes how much she should read each day so that she can finish the book on time by dividing 500 to 5. She gets 100 as a result. After reading 100 pages, she sleeps...