Return to site

Things 2 8

broken image


The Fibonacci Sequence is the series of numbers: Super photocut 2 0 5.

Things even supports GTD, so if you're familiar and comfortable with GTD concepts, you'll feel right at home. Version 2.8.1: Full compatibility with OS X 10.11 El Capitan. Support for OS X 10.11 El Capitan's new Split View mode - click and hold the green zoom button to move Things into the left or right pane. There are a number of things that come in groups of 8. These include batteries in a pack, crayons in a standard pack, 8 sides of an octagon, as well as hot dogs in a pack. The world has turned Upside Down. Stranger Things 2 is now streaming on Netflix. Watch Stranger Things, Only on Netflix: https://www.netflix.com/title/800572. 'Things To Do® is the premiere organization for young professionals to socialize and network. With a membership of over 175,000, we produce a wide variety of original, interactive social, educational and cultural events that enrich your social and cultural experiences.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ..

The next number is found by adding up the two numbers before it:

  • the 2 is found by adding the two numbers before it (1+1),
  • the 3 is found by adding the two numbers before it (1+2),
  • the 5 is (2+3),
  • and so on!

Example: the next number in the sequence above is 21+34 = 55

It is that simple!

Here is a longer list:

Things

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ..

Can you figure out the next few numbers?

Makes A Spiral

Things 2 1998

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ..

Can you figure out the next few numbers?

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.


This spiral is found in nature!
See: Nature, The Golden Ratio,and Fibonacci

The Rule

The Fibonacci Sequence can be written as a 'Rule' (see Sequences and Series).

First, the terms are numbered from 0 onwards like this:

n =01234567891011121314..
xn =01123581321345589144233377..

So term number 6 is called x6 (which equals 8).

Example: the 8th term is
the 7th term plus the 6th term:


x8 = x7 + x6

So we can write the rule:

The Rule is xn = xn−1 + xn−2

where:

Things To Do 2/8/20

  • xn is term number 'n'
  • xn−1 is the previous term (n−1)
  • xn−2 is the term before that (n−2)

Example: term 9 is calculated like this:

= x8 + x7
= 34

Golden Ratio

And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio 'φ' which is approximately 1.618034..

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

B
2
1.5
3
1.666666666..
5
1.6
8
1.625
..
..
144
1.618055556..
233
1.618025751..
..
..

We don't have to start with 2 and 3, here I randomly chose 192 and 16 (and got the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ..):

A
B / A
16
0.08333333..
208
13
224
1.07692308..
432
1.92857143..
..
..
11984
1.61771058..
19392
1.61815754..
..
..

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

xn = φn − (1−φ)n√5

The answer comes out as a whole number, exactly equal to the addition of the previous two terms.

Example: x6

x6 = (1.618034..)6 − (1−1.618034..)6√5

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033 , a more accurate calculation would be closer to 8.

Try n=12 and see what you get.

You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):

Example: 8 × φ = 8 × 1.618034.. = 12.94427.. = 13 (rounded)

Some Interesting Things

Here is the Fibonacci sequence again:

n =0123456789101112131415..
xn =01123581321345589144233377610..

There is an interesting pattern:

  • Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ..)
  • Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, ..)
  • Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, ..)

And so on (every nth number is a multiple of xn).

1/89 = 0.011235955056179775..

Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?

In a way they all are, except multiple digit numbers (13, 21, etc) overlap, like this:

0.0
0.01
0.001
0.0002
0.00003
0.000005
0.0000008
0.00000013
0.000000021
.. etc ..

0.011235955056179775.. = 1/89

Stranger Things 2 80s Music

Terms Below Zero

The sequence works below zero also, like this:

n =..−6−5−4−3−2−10123456..
xn =..−85−32−110112358..

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- .. pattern. It can be written like this:

x−n = (−1)n+1xn

Which says that term '−n' is equal to (−1)n+1 times term 'n', and the value (−1)n+1 neatly makes the correct +1, −1, +1, −1, .. pattern.

History

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

About Fibonacci The Man

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

'Fibonacci' was his nickname, which roughly means 'Son of Bonacci'.

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits '1, 1, 2, 3' which is part of the sequence. So next Nov 23 let everyone know!

Enter what you're reading or your whole library. It's an easy, library-quality catalog.

LibraryThing connects you to people who read what you do.

Check it out

  • Take the tour.
  • Sign up. It's more fun than the tour.
  • See what bloggers and the media say about LibraryThing.

What's Good?

  • Catalog your books from Amazon, the Library of Congress and 4,941 other libraries.
  • Catalog your movies and music too.
  • LibraryThing is entirely free.
  • Find new books to read.
  • Talk about what you love with other committed bibliophiles.
  • Track and lend your books.
  • Snag a book from over 2000 early-release books every month.
  • Available in many languages: (others)

Recent Activity

El Conde de Montemolin: historia de la vida pública y privada de D. Carlos Luis de Borbón y de Braganza, primogénito de D. Carlos María Isidro by Leopoldo Augusto de Centurion
(read review)
Aristotle and Dante discover the secrets of the universe by Benjamin Alire Sáenz
(read review)
LLETRA ENCICLICA EN L'ESCAIENÇA DEL PRIMER CENTENARI DE LES APARICIONS DE LOURDES by PIUS XII
(read review)
The Other Mrs. by Mary Kubica
(read review)
(read review)
Boston Scream Murder (A Deputy Donut Mystery) by Ginger Bolton
(read review)
Tomie dePaola's The Quicksand Book by Tomie dePaola
(read review)
The Art of Deception: A Daughter of Sherlock Holmes Mystery (The Daughter of Sherlock Holmes Mysteries, 4) by Leonard Goldberg

'Of course, LibraryThing is even more useful if you post your book collection, and the process is wonderfully easy.' (link)

'LibraryThing is an impressive cataloging app that feels like del.icio.us
for books' (link)

'There are also suggestions of related books to read; it's a
virtual feast of information.' All Things Considered

'Many social connections thrive at the site. Although members can keep all details of their online catalog private, most choose to display their libraries..' (link)

'LibraryThing can also connect likeminded readers — a sort of MySpace for bookworms.'

'Now, with LibraryThing.com, we can peek at thousands of libraries.'

'Not surprisingly, librarians love LibraryThing.'
Library Journal (March 15, 2007)

About | Contact | Privacy/Terms | Help/FAQs | Blog | Store | APIs | TinyCat | Legacy Libraries | Early Reviewers | Common Knowledge | 151,489,159 books! | Top bar: Always visible
Copyright LibraryThing and/or members of LibraryThing, authors, publishers, libraries, cover designers, Amazon, Bol, Bruna, etc. | static: /




broken image