Contents of this Page Uva
The i line means there is a Things to do investigation at the end of the section. From 2-dimensional flat shapes, we turn to 3-dimensional ones solids . JDice Shapes We need symmetry in dice if they are to be fair, but is the cube the only possible shape No, there are 5 and only 5 fair dice shapes J Coordinates and other statistics of the 5 Platonic Solids m The Tetrahedron J The Cube or Hexahedron m The Octahedron m The Dodecahedron Some other relationships between these shapes M Golden...
Phi and another Isosceles triangle
If we copy the BCD triangle from the red diagram above the 36 -72 -72 triangle , and put another triangle on the side as we see in this green diagram, we are again using P Phi as above and get a similar shape - another isosceles triangle - but a flat triangle. The red triangle of the pentagon has angles 72 , 72 and 36 , this green one has 36 , 36 , and 72 . Again the ratio of the shorter to longer sides is Phi, but the two equal sides here are the shorter ones they were the longer ones in the...
Fibonacci paper
If we take a sheet of paper and fold a corner over to make a square at the top and then cut off that square, then we have a new smaller piece of paper.
i 0
Remember that the rows and columns of Pascal's triangle in this formula begin at 0 For example, in month 8, there are 4 levels and the number on each level is When k is bigger than 4, the column number exceeds the row number in Pascal's Triangle and all those entries are 0. SUM is F 8 21 col 0 1 2 3 4 5 6 7 8 9 SUM is F 8 21 col 0 1 2 3 4 5 6 7 8 9 The general pattern for month n and level generation k is Level k is Pascal's triangle row n-k and column k-1 For month n we sum all the generations...
Quasicrystals
Penrose found that there are two simple shapes that you can use to fill a space as large as you like and which have five-fold axes of symmetry. The shapes are built from 6 flat faces which are , that is, shapes with all sides of equal length like a square and which has oppopsite sides paralled again like a square , but which does not have all its angles equal - so they are diamond shaped rhombs, rhombuses . The Penrose tiling shown on the Flat Phi page is also made from two rhombuses and fills...
The Fibonacci base system
What if we labelled the columns with the Fibonacci numbers instead of powers of 10 We follow the usual conventions of larger column sizes being on the LEFT We represent number representations in this system by putting Fib after the representation eg 8 5 3 2 1 ten 1 0 0 1 0Fib 8 2 This time it is not clear what digits we should use in the columns. For instance, there are many ways to represent the value ten in this system as well as in the example above 5 3 2 1110Fib 3X3 1 301Fib 10 X 1 AFib...
Fibonacci Numbers and Branching Plants
One plant in particular shows the Fibonacci numbers in the number of growing points that it has. Suppose that when a plant puts out a new shoot, that shoot has to grow two months before it is strong enough to support branching. If it branches every month after that at the growing point, we get the picture shown here. A plant that grows very much like this is the sneezewort Achillea ptarmica. 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 More
Bipyramids as dice
Putting both of the above shapes together, we get a dice which is two n-gon-al pyramids, joined at their bases the n-gons to form a double pyramid or bi-pyramid. The picture shows a 12-sided dice formed from two 6-sided pyramids joined at their hexagonal bases. Perhaps we should call it a bi-hexahedral dice. If we used pentagons then the bi-pyramidal dice would be 10-sided. It would be useful for generating random numbers up to 10. By using two of them, say a red one for tens digits and a green...
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...
Why is the Golden section the best arrangement
The line means there is a Things to do investigation at the end of the section. J Packing Why exact fractions are fruitless The rational answer is the irrationals J Links and References On the first page on the Fibonacci Numbers and Nature we saw that the Fibonacci numbers appeared in idealised rabbit, cow and bee populations, and in the arrangements of petals round a flower, leaves round branches and seeds on seed-heads and pinecones and in everyday fruit and vegetables. We explained why they...
The Fibonacci Rectangles and Shell Spirals
We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21, if we start with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 1 1 . We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long and then another touching both the 2-square and the 3-square which has sides of 5 units . We can continue adding squares around the picture, each new square having a side which is...
A quote from Coxeter on Phyllotaxis
Finally, note that, although the Fibonacci numbers and golden section seem to appear in many situations in nature, they are not the only such numbers. H S M Coxeter, in his Introduction to Geometry 1961, Wiley, page 172 - see the references at the foot of this page - has the following important quote it should be frankly admitted that in some plants the numbers do not belong to the sequence of f's Fibonacci numbers but to the sequence of gs Lucas numbers or even to the still more anomalous...
Fibonacci Fingers
2 hands each of which has 5 fingers, each of which has Is this just a coincidence or not However, if you measure the lengths of the bones in your finger best seen by slightly bending the finger does it look as if the ratio of the longest bone in a finger to the middle bone is Phi What about the ratio of the middle bone to the shortest bone at the end of the finger - Phi again Can you find any ratios in the lengths of the fingers that looks like Phi ---or does it look as if it could be any...
Leaf arrangements
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,...
Harder 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 The Fibonaci puzzles are split into two sections those with fairly straight-forward and simple explanations as to why the answer is the Fibonacci numbers are on the Easier Fibonacci Puzzles page. This page contains the second set where it is not so simple to explain why...
Pine cones
Pine cones show the Fibonacci Spirals clearly. Here is a picture of a pinecone seen from its base sorry the quality is a bit poor and another with the spirals emphasised red in one direction and green in the other. Click on the images to enlarge them. How many red spirals are there Collect some pine cones for yourself and count the spirals in both directions. A tip Soak the cones in water so that they close up and counting the spirals is easier. a. Does the number of spirals differ for each...
Honeybees Fibonacci numbers and Family trees
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...