There are some mind blowing facts and games related to mathematics that even mathematics cannot explain. I just like to call those facts “The Abracadabra of Mathematics”.
Let’s take a look at one of those games or tricks. Assume that you are in a classroom with a population of at least 25 students and I am the instructor. I give a blank paper to each of you to write a digit , any number between 0–9, inclusive. After you are done writing your digit on your paper and fold it, I will collect the papers. Of course I don’t have any idea of what you write on the paper. However, I guarantee that I will know the most written number in the classroom.
Now, my claim is that most of the students chose the number 7. If a student comes and unfolds all of the papers and checks the papers, he would say, “You are correct! But how?” Unfortunately, I don’t have any explanation for this, though it is always true. Most people always choose number 7. I might have played this game more than 100 times, and I have never been wrong.
I learned this game from one of my favorite professors, Ali Nesin, 10 years ago. To play this game, there are some conditions. First, you need at least 25 people. Otherwise, it will be risky. You might think that this is about probability but actually it is not. Since there are 10 digits, the probability of each digit that is chosen is 1/10 for each student. So, the mathematical explanation is now working here. I think that this can also be explained with physiology or sociology.
There is also something which is extremely interesting that mathematics cannot explain. Here, we have 4 different rectangles. If we ask people which rectangle is more beautiful, 70%-80% would choose the green one.
We cannot also explain this situation using only mathematics because we don’t have a definition for beauty in mathematics, and this fact is mathematically incomprehensible. However, marketers used this information very effectively. When they realized that most people prefer a specific design.
After many years, we still couldn’t find an answer why people choose number 7, but an academician, Adrian Bejan found an answer as to why people choose the green rectangle. The professor discovered that “ the human eye is capable of interpreting an image featuring the golden ratio faster than any other.” So the green rectangle has the golden ratio and it seems more beautifully than the other rectangles.
You might’ve heard of Euclid. I have written some articles about him before. He had a book called “Elements”. I definitely recommend that you buy that book. In his book, “Elements”, Euclid defined the golden ratio as such:
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
In other words, Euclid was saying: There is a point on a segment such that we can call it the golden point which divides the line perfectly. His claim was assertive but also true.
Mathematically speaking, if you have a segment |AB|, and there exists a point C between the points A and B where we can get some ratios such as |AB|/|AC| and |AC|/|BC|. Then if this two ratios are equal, they are going to be golden ratio, 1.618…., φ (phi).Phi (/faɪ/; uppercase Φ, lowercase φ.
I am pretty sure that, this special ratio made you very very curious and you are wondering how Euclid got the value of the golden ratio? Let’s try to find it together.
Assume that: the length of the |AC| = x and the length of the |CB| = y. Then
That means if we can find the value of x/y we can find the value of φ and this will lead us to a quadratic equation.If we do cross multiply - (x+y) times y, and x times y - we get: x² = xy + y²Then we can combine all of the variables on one side and this time we get: x² - xy + y² = 0Reminder: our purpose is finding x/y. So if we divide all of the terms by y², then we get:
If we can define (x/y)= φ, we get:φ² — φ — 1 =0.Here, we need to remember the quadratic formula.The Quadratic Formula:Let a, b, and c be real number. The solutions of ax²+bx+c = 0 are
The quadratic formula tells us the product of the roots of our equation (c/a) is equal to -1. Thus, one the product of the roots is negative and the other one is positive. However, by the definition of the golden ratio, φ, cannot be negative. So we are going to choose the positive root. Now we can solve the equation.
So far we worked on a segment. We still need to show why people choose the green rectangle above and why Euclid called it the golden rectangle.
When we go back to our segment |AB| with the point C, we can fold it from the golden point C and get a right angle. So we can build a rectangle now. This rectangle is going to be the golden rectangle because the length of the sides will be x and y and we already showed that x/y is equal to the golden ratio, φ.
The golden rectangle has a property that no other rectangle has. This is a special rectangle such that, if you cut a square part from it, the leftover rectangle will be also a golden rectangle. I can leave this one as an exercise for you!
So far so good. Now, we can try something different such as find a golden triangle if there exists.
First of all, we need to decide what kind of triangle do we need to work on. Remember, when we remove a square part from a golden rectangle, we still have a golden rectangle. We need the same property for triangles. I think it is obvious that an equilateral triangle cannot be golden triangle because if you cut an equilateral triangle from an equilateral triangle the leftover part will not be an equilateral triangle.
However, I have good news, we can work on isosceles triangle! The steps will be clear. We will have an isosceles triangle and then we will cut another isosceles triangle from our original isosceles triangle and we will check if the leftover triangle is going to be similar to the original one or not. If yes, we will try to call it a golden triangle. I said we will try, because the next step will be finding the ratio of sides is equal to the golden ratio.
Let’s start with an isosceles triangle ABC with the base angles equal to 2α. Then we draw a line from the point B to the side |AC| to get two different isosceles triangles ABD and BCD. Here we get something interesting because the base angles of the triangle BCD are also 2α and the base angles of ABD are α because the sum of two interior angles will give us the exterior angle. So the angles of the triangle ABC is α, 2α, 2α. This gives us 5α = 180 and α = 36.
Now we get a very special triangle with top angle 36 and base angles 72.
For the second step, we need to check the ratio of the lengths of the triangle. If we say |AB| = |AC| = x and |BC| = y, then we get:|AD| = |BD| = y and |CD| = x-y.And our purpose is trying to find x/y = φ or not. By similarity, we get:
As you can see, we get the same quadratic equation at the end. So the triangle with the angles 36–72–72 deserves to be golden triangle. By the way, if you keep digging you will see that 108–36–36 is also a golden triangle. This information will be very useful when we deal with a pentagon.
Let’s do an example. If you have a pentagon, what is the ratio between the length of its diagonal and the length of its side?If you draw a diagonal from any edge we will get a triangle which is a golden triangle because the angles will be 108, 36, and 36.So, if the length of one side of the pentagon is 1, then the length of the diagonal will be φ.
We solve a challenge question without doing any math. Without any knowledge of the golden ratio, we have to deal with a lot of lines, quadratic equations, etc…
Let’s do another example. If we draw all the diagonals of a pentagon we will get something unique. There will be smaller pentagon in the middle and every triangle you see is going to be a golden triangle.
Let’s do another example. If we draw all the diagonals of a pentagon we will get something unique. There will be smaller pentagon in the middle and all the triangles you see is going to be a golden triangle.Here is the question: what is the ratio between the area of the small pentagon and the area of the big one?We can easily do it because if we call one side of the small pentagon 1, the length of the other side of the small triangles will be φ. Then the same ratio will tell us the base length of the triangle will be φ². Now the similarity will give us the ratio. The similarity ratio is 1/φ², then the ratio of the areas will be 1/φ⁴.
There is also another unique property that φ has. If we go back, we will remember the quadratic equation of φ.
φ² — φ — 1 = 0, so this will give us:φ² = φ +1
φ is the only number which its square is equal to the sum of itself and 1. So there is no real number such that when you add 1 that you get square of that real number. And interestingly enough, we can have something like this:
φ² = 1φ + 1φ³ = 2φ + 1φ⁴ = 3φ + 2φ⁵ = 5φ + 3φ⁶ = 8φ + 5...
Here the constant numbers are interesting. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …. These are not random numbers. These numbers are coming from the Fibonacci sequence where each term is the sum of the previous two terms.
The relation between Fibonacci sequence and the golden ratio is unique. The ratio of 2 consecutive numbers from the sequence is getting the golden ratio after a while. You are getting every digit of golden ratio if you deal with big number from the sequence.
For instance 5/3= 1.666…8/5= 1.613/8 = 1.61…21/13 = 1.618…
If you keep doing this you will get a new digit of a φ.
This information is very useful because we can also find sin 18 or cos 36 easily without a calculator. This is also exercise for you!