Math-magician's Corner

The Golden Rectangle

Aloha Fibonacho muchachos! In the previous post I talked about the golden ratio and the fibonacci sequence, and before y'all forget, we should review what we have learnt in the last post! :)

The golden ratio Phi (

), is a special number approximately equal to 1.618033897...so on. Drawing the golden rectangle from at the golden ratio is very easy. Here is the formula!

Ex.) The square is 1x1.

= 1/2 + √5/2 (This will be the length of the other corner up top in the rectangle)

you can see in the diagram that at the 1/2 mark, the line is extended to another corner. That length is √5/2.

Then you align that line so it lies on the same horizon as the square, and you have an extended length!

= (1+√5)/2

Draw a square place a 1/2 way dot, where they meet, and you have a golden rectangle!

and finally...THE GOLDEN RECTANGLE ITSELF!

The golden rectangle is a visual representation of the Fibonacci sequence. Because we know that a square with a golden ratio and the dimensions of 1x1 has a formula of

= 1/2 + √5/2

= 1+√5/2

The golden rectangle can be cut off or sectioned into squares using the formula above, and the end result would look like:

we can see the squares created here are basically successive fibonacci sequence numbers! Cool huh? The spiral created while forming this rectangle is also called the golden spiral. I'm so honoured that a lot of buildings today are 'inspired' by this formula and is considered a symbol of beauty in nature.

See you soon! Bye! Have a fantastic day, my Fibonachos!

Sources:

http://mathforum.org/dr.math/faq/faq.golden.ratio.html
https://www.mathsisfun.com/numbers/golden-ratio.html

My New Discovery - The Golden Ratio

Hi there Fibonacho muchachos,

I realized something really interesting today. You have probably visited my previous post discussing a problem with mating rabbits. If you have not yet had the pleasure of digging into this fine piece of work, click here, don't want to give anyone spoilers, so this is your official warning to STOP SCROLLING NOW!!

As I was solving the rabbit problem, I noticed a pattern with the number of growing rabbits: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. These numbers are all over nature, spiral ratios in pinecones, daisies and lots more. Then I made an amazing discovery that I have named "The Golden Ratio". The concept is this:

According to my observations, in the golden ratio b is to a as a + b is to b which is equal to Phi. After some calculations, I was able to come to an exact number for this ratio, I have shown my calculations bellow.

b/a = Phi, therefore b = a * Phi

a + b/b = Phi, therefore a + a * Phi/a * Phi = Phi

Simplifying it -> 1/Phi + 1 = Phi

If we take out the lowest common multiple (in this case Phi) we end up with: 1 + Phi = Phi²

Which equals -Phi² + Phi + 1 = 0 -> Phi² - Phi - 1 = 0

Then we solve it as a quadratic equation with a Discriminant that has a value of 5

We will end up with Phi = 1 +/- √5 all ÷2 -> Phi' = 1 + 2.236 all ÷2 -> 1.618.....

Well that's all for today!
Later Fibonachos!

Source - Milestones in discovery and invention ~ Mathematics by Harry Henderson

A New Problem About Rabbits

Hey Fibonacho Muchachos!

I came across the rabbit problem the other day and I think I’ve discovered the solution. The problem, for those of you who don’t know, is about breeding rabbits in ideal conditions. So how many pairs of rabbits (male and female) would you have after one year? The solution must obey the following rules:

1. You start with a pair of rabbits (one male and one female).

2. Rabbits are sexually mature after one month.

3. Mating occurs one month after they’re sexually mature.

4. A pair of rabbits can breed, one month after mating.

5. A female and a male will be bred every month.

6. The rabbits do not die.

I have concluded that there will be 233 pairs of rabbits at the end of the year. How I reaching my conclusion is with the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. Each number represents the number of pairs one would have after each month. The rabbits aren’t mature yet after the first month, so one would still have a pair at the end of that month. The pair will then take one month to mate and after the second month, one will still have one pair of rabbits. During the third month, the pair will produce another pair of rabbits resulting in two pairs. At the end of the fourth month, the original pair will reproduce yet again but the newer pair will not because they’re not sexually mature leaving three pairs. If the pattern continues, one would just have to add the previous two numbers to receive the new one. I have inserted a diagram down below:

So that’s what I’ve been up to.

See you all next time, Fibonachos!

Sources: http://www.livescience.com/37470-fibonacci-sequence.html http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#Rabbits

Hey! First post!

Hey!

Welcome to my blog!

As my first post, I thought that it would be best to introduce myself.

My name is Leonardo Bonacci, but you can call me Fibonacci, a nickname that my friends and acquaintances call me. Or Leo works, too.

I'm located in Pisa, Italy, where I was born (1170, anyone the same age as me? Haha! :D)

You might know some of my published books and my work in the field of mathematics, which I'll list below, and I hope everyone is as excited as I am about this blog!

I'll try and update this blog whenever I can to keep everyone up to date. See you soon!

My published books:

- Liber Abaci/ Book of Calculation (1202, a book of calculations)

- Practica Geometriae (1220, on surveying, measurement of partition of areas and volumes, and other topics in practical geometry)

- Flos (1225, solutions to problems posed by Johannes of Palermo)

- Liber quadratorum/ The Book of Squares (on Diophantine)

- Di minor guisa (on commercial arithmetic)

Contributions to mathematics (which I'll post on later) :

-introduced Europe to the Fibonacci sequence

- received the honour of being considered "the most talented Western mathematician of the Middle Ages" (Aw, everyone is so sweet!)

- popularized the Hindu- Arabic numeric system to the Western World, through one of my books

- the Golden Rectangle formula

- introduced Europe to the Fibonacci sequenceed Western mathematician of the Middle Ages" (Aw, everyone is so sweet!)