Fibonacci and the Mandelbrot

The Mandelbrot set shown here has been written about often in maths books, appears in magazines and posters, greeting cards and wrapping paper and in lots of places on the Net. A detail from the Mandelbrot set picture is shown here. It is also a link to a page on how the Fibonacci numbers occur in the Mandelbrot Set (at Boston University Mathematics Department). 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 1

The Mathematics of the Fibonacci series

Take a look at the Fibonacci Numbers List or, better, open another window in your Browser, then you can refer to this page and the list together. The i m line means there is a Things to do investigation at the end of the section. Patterns in the Fibonacci Numbers Factors of Fibonacci Numbers i m Benford's Law and Initial Digits The Fibonacci Numbers in Pascal's Triangle i m o Why do the Diagonals sum to Fibonacci numbers o Another arrangement of Pascal's Triangle o Fibonacci's Rabbit...

Easier Fibonacci puzzles

All these puzzles except one which have the Fibonacci numbers as their answers. So now you have the puzzle and the answer - so what's left Just the explanation of why the Fibonacci numbers are the answer -that's the real puzzle Puzzles on this page have fairly straight-forward and simple explanations as to why their solution invovles the Fibonacci numbers . Puzzles on the next page are harder to explain but they still have the Fibonacci Numbers as their solutions. So does a simple explanation...

Binets formula for noninteger values of n

This section is optional and at an advanced level i.e. post 16 years education. Take me back to the Fibonacci Home page now or learn about square roots of negative numbers in what follows Well now we've tried negative values for n, why not try fractional or other non-whole values for n This doesn't make sense in terms of numbers in a series there is a 2nd and a 3rd term and even perhaps a -2nd term but what can we possibly mean by a 2 5th term for instance However, this could give us some...

Leaf arrangements

Fibonacci Sequence Plants

Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their stems. If we look down on a plant, the leaves are often arranged so that leaves above do not hide leaves below. This means that each gets a good share of the sunlight and catches the most rain to channel down to the roots as it runs down the leaf to the stem. The computer generated ray-traced picture here is created by my brother, Brian, and here's another, based on an African violet type of plant,...

Honeybees Fibonacci numbers and Family trees

Fibonacci Numbers Bee

There are over 30,000 species of bees and in most of them the bees live solitary lives. The one most of us know best is the honeybee and it, unusually, lives in a colony called a hive and they have an unusual Family Tree. In fact, there are many unusual features of honeybees and in this section we will show how the Fibonacci numbers count a honeybee's ancestors in this section a bee will mean a honeybee . First, some unusual facts about honeybees such as not all of them have two parents In a...