PATTERNS OR SEQUENCES

PURPOSE

• To appreciate and investigate a numerical pattern

• To look for evidence of mathematical patterns in nature

Order is the key

• What do these situations have in common?

• 1.- There are people waiting for their turn to buy some fruit

• 2.- You are looking for your name on a list because you want to know the mark you got on your last exam

• 3.- You are trying to follow instructions, being careful to do every step of the process in order

KEY WORDS

• The situations given before are related to a concept known as SEQUENCESEQUENCE

• A sequence is an ordered list of things (objects, events, numbers,...)

• Like a set it contains members (elements or TERMSTERMS

GUESS THE NEXT TERM OR ELEMENT OF THE SERIES GIVEN BELOW

• A) O, T, T, F,....?

• B) 1, 3, 6, 10,...? (HINT: TRIANGLE NUMBERS)

• C) 1, 4, 9, 16,...?

• D) -5, 4, -3, 2,...?

• E) 1, 1, 2, 3, 5,... (FIBONACCI SEQUENCE)

WHO WAS FIBONACCI?

» The “Greatest European mathematician” of the middle ages, his full name was Leonardo of Pisa

• He was born in Pisa about 1175 AD

• He was one of the first people to introduce the Hindu-Arabic number system into Europe

• He discovered Fibonacci sequence after an investigation on the reproduction of rabbits

• The number sequence was known to Indian Mathematician as early as the 6th century, but Fibonacci introduced it to the west

The rabbit problem

• Suppose a newly-born pair of rabbits, one male, one female, are put in a field.

• Rabbits are abble 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.

• Suppose that our rabbits never die and that the female always produces a new pair (one male, one female) every month from the second month on

• How many pairs will there be in one year?

• The numbers of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Fibonacci pattern in nature

» In the head and petals of sunflowers

» Pinecones show the Fibonacci Spiral

In music

– A piano keyboard has 8 white keys, 5 black keys in groups of 2 and 3 these 13 keys comprise one octave

• The number of petals on a flower are often Fibonacci numbers

• This important pattern can be found in pineapples, bananas, cauliflowers

The Golden Ratio

• The Golden ratio is an irrational mathematical constant, approximately equals to 1.6180339887

• A golden rectangle is a rectangle where the ratio of its length to width is the golden number

Relation between Fibonacci Sequence and the Golden ratio

• Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584,...

• If you calculate the ratios...

• 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.6666...

• 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.615384..

• 34/21 = 1.61904... 55/34 = 1.617647...

• Aha! Notice that as we continue down the sequence, the ratios seem to be converging upon one number

• If we continue to look at the ratios as the numbers in the sequence get larger and larger the ratio will become the same number, and this number is THE GOLDEN RATIO 1.6180339887...

Golden ratio in nature

» Nautilus Shell

» Butterfly

In human body

» Ear

» Fingers

In art

In design, architecture, publicity,...