Random musings

Warning! Math geekiness ahead!

Sunday is Fibonacci Day

Who or what is a Fibonacci you may ask and what has it got to do with math? I'm glad you asked!

Fibonacci was a person who lived during the Middle Ages. According to Wikipedia:

Leonardo Bonacci (c. 1170 – c. 1250)^[2]—known as Fibonacci (Italian: [fiboˈnattʃi]), and also Leonardo of Pisa, Leonardo Pisano, Leonardo Pisano Bigollo, Leonardo Fibonacci—was an Italian mathematician, considered as "the most talented Western mathematician of the Middle Ages.".^[3][4]

He is best known for creating a sequence of numbers, known as the fibonacci numbers or the fibonacci sequence. This sequence is 0, 1, 1, 2, 3, 5…where each number is the sum of the previous 2 numbers starting with 0 and 1 as the seeds.

Fibonacci also introduced the Arabic numerals 0-9 and place notation to Europe.

See the Wikipedia article about his life.

Fibonacci numbers

This sequence of numbers was used to describe a mathematical problem in Fibonacci's book. According to Wikipedia:

In the West, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci.^[5] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

• At the end of the first month, they mate, but there is still only 1 pair.
• At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
• At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
• At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.^[15]

There are many uses for these numbers.

They are used to approximate the golden ratio. According to Wikipedia:

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

where the Greek letter phi (φ) represents the golden ratio. Its value is:

The golden ratio is also called the golden section (Latin: sectio aurea) or golden mean.^[1][2][3] Other names include extreme and mean ratio,^[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,^[5] and golden number.^[6][7][8]

Here is a photo to explain this:

The ratio of any two adjacent fibonacci numbers is an approximation of the golden ratio, and as the numbers get larger the ratio gets closer and closer to the golden ratio, i.e. F(n)/F(n-1)

Also according to Wikipedia:

Fibonacci sequences appear in biological settings,^[9] in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,^[10] the flowering of artichoke, an uncurling fern and the arrangement of a pine cone,^[11] and the family tree of honeybees.^[55] 

I've also seen them used as a betting sequence in blackjack.

 According to Wikipedia:

Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.

Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.

(This is computer science stuff).

Here is the link to the Wikipedia article on fibonacci numbers.

Interesting days

Today - https://www.daysoftheyear.com/days/hello-day/ Did you know that today was almost Ahoy-hoy Day?

Tomorrow - https://www.daysoftheyear.com/days/go-for-a-ride-day/ It's supposed to rain tomorrow

Next Friday - https://www.daysoftheyear.com/days/french-toast-day/ I think Denny's after work sounds good

December 21 - https://www.daysoftheyear.com/days/crossword-puzzle-day/ Grab the Sunday NY Times today