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Ron's Project Page

Seeds Spirals

I've wanted a dynamic demonstration of why the golden section appears in nature in so many places.
My Fibonacci Numbers and Nature page gives more details.
This demo lets you play with the angle between successive seeds in a seed-head and it's made with Geometer's Sketchpad Demonstration to show that the golden section value gives the best packing.
THIS IS NOW DONE!

The first picture is what happens if we have 0.403 of a turn (which is about 145 degrees) between successive seeds as they are "grown" from the active "meristem" (that tiny tip which is the actual growing point of the seed-head).

The second angle is just a bit above 90 degrees (as the number of seeds placed increases, to the seeds wrap slightly beyond the quarter turn each time and we see it curling to the right).
This is about 2 fifths of a turn between seeds - still no good as the seeds are too closely packed along an "arm" and there is much space wasted between the arms.
This final picture is better - at 0.387 of a turn between the seeds. The spacing makes much better use of the available seed-head and there is little wasted space between the seeds - the only evidence of any "arms" being towards the outer part of the seed-head.

Egyptian Fractions

There are many interesting ways to represent a fraction as a sum of unit fractions - which are just reciprocals of the positive integers. For instance:
```          2/5  = 1/3 + 1/15
```

Here is my first attempt -- especially for Jon Basden!

Other ideas

Looking at some stuff on Representations of Integers
Fibonacci number representations
The basic information is already on my Fibonacci Bases page.

The columns are headed not with base 10 powers but with the Fibonacci numbers:

```        ...  13   8   5   3   2   1
```
An interesting new(ish) problem still beiong actively researched was reported in a couple of papers at the 8th International Fibonacci Numbers conference two weeks ago in Rochester, New York state:
How many ways are there to represent each integer n as a sum of Fibonacci numbers?. This is denoted as R(n). For instance, R(1)=1, R(2)=1 but R(3)=2 since the first number with more than one representation is 3 which can be 100 or else 11. To distinguish these from base 10 (or binary or any other base for that matter) we write Fib after a Fibonacci representation:
310 = 100Fib = 11Fib
Base Phi representations of integers
Again this is in basic form on my Phigits and Base Phi page. This is just a fun way of representing integers which more advanced maths students may like to play with.
Pascal Triangle Representations
Every number appears in Pascal's Triangle in column 2 which is just 0, 1, 2, 3, 4, ... !!

Each number is also a sum of Pascal's Triangle numbers. A simple solution is to sum the first n ones from column 1 !!

So an interesting question is...

What is a unique way to find a sum of Pascal Triangle numbers whose sum is any given integer n?
I know of at least one family of methods (from D E Knuth's Fundamental Algorithms, Volume 1) but.... are there more??
... OR ... Developing specific teaching resources based on my Fibonacci and Golden Section site Please email me if you think one of these ideas is more worthwhile than the others.
R.Knott
Contact me on R.Knott@surrey.ac.uk
Tuesday, 06-Sep-2016 14:10:26 EDT