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Flower Card

Do you know Hanafuda?

According to WIKIPEDIA,
Hanafuda are playing cards of Japanese origin,
used to play a number of games.
The name literally translates as "flower cards".
The name refers to the set of cards
and can be used to play any number of variations of games.


It's really fun.

I was interested in their unique illustrations.
And their card size are made by golden ratio.

I feel they have another golden ratio n their illustrations, probably.

Gold, silver, bronze 5

I found the golden ratio in my favorite bike!
Please see this picture.


Yes, Sachs Madass!
Do you remember this?
In this picture, we can see the golden tetragon.
And if we draw the diagonal line of it, we can get Madass" body line.

In addition, if we rotate the diagonal line by golden angle,
we can get another body line.

As I said in my presentation in today's lecture,
I found it by chance.
So I was really surprised.
It is glad to find the golden ratio in what I like unawares.

gold, silver, bronze 3

Today I show you another expression.
The golden, silver and bronze ratio can be expressed with fraction.

Of course, lambda is the same role.
If n=1, lambda expresses the golden number.
If n=2, lambda expresses the silver number.
If n=3, lambda expresses the bronze number.

In fact, this fraction relates the Fibonacci sequence.
In Fibonacci sequence, each number is the sum of the previous two numbers.

This tool is called "yupana".
This tool's detail is below (by wikipedia).

A yupana (Quechua for "counting tool") is a calculator that was used by the Incas. Researchers believe that calculations were based on Fibonacci numbers to minimize the number of necessary grains per field.

Gold, silver, bronze 2

My last blog, I showed the special number.
Why are they special?

In fact, these ratio means the golden, silver and bronze ratio.
Yes, they can be expressed similar equations.

By the way, I find the thesis published by Kyusyu University.
In it, these similar equations are put together.
Of course, lambda is positive answer.
In the thesis, lambda called "n-th noble metal number".
If n=1, it expresses the golden ratio.
If n=2, it expresses the silver ratio.
If n=3, it expresses the bronze ratio.

Like this, we can get some irrational ratios.
If n=4, ?...something cheaper than copper???
If n=-1, ?...Roentgenium???
We should find the new element in periodic table.

Oh, but n=-1 means the golden ratio!

Gold, silver, bronze 1

I showed you some irrational ratio which relates the beauty.
We call them the gold, silver and bronze ratio.

Today, I will show one of the relationships between them and math.
The number which expresses these ratio can be defined by equation below.

On sequentially, these are equations which relates gold, silver and bronze ratio.
Positive answers of these three equations shows the special number.
Why these are the special number?
Because... .
Let's talk about this the next time.

By the way the first picture is the golden crystal.
Does the golden ratio exist in this crystal?
I will check it.

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