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To: "MWForum" <mwforum@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
Subject: Mathematics and Islamic Art
From: "Mike Sandy" <mjsandy@xxxxxxxxxxxxxx>
Date: Sun, 28 Jan 2007 19:14:53 -0000

The title comes from an article in Mathematics in School
which is a journal of the Mathematical Association (UK) aimed 
particulary at school teachers.
The idea of the article is to generate diagrams linked
to a sequence of numbers, in particular the Fibonacci sequence.
The numbers in this sequence are generated by the rule that each 
number is the sum of the two preceding numbers:
   f(n)=f(n-1)+f(n-2)
except for the first two which are 0, 1.
 Giving 0, 1, 1, 2, 3, 5, ......
A diagram can be formed from this series by letting the numbers
represent the lengths of segments and the angle between the segments
have some fixed(though arbitrary) value. In turtle terms: fd :d rt :angle,
where :d
takes the successive values 1, 1, 2, 3, &.(or some fixed multiple
of these values).
The numbers increase, so this will generate a spiral - of not much interest.

If the numbers are reduced using a remainder with respect to a fixed integer
then interesting patterns are formed. 
Sections of the sequence repeat and can form a linear or cyclic pattern.
 
Different diagrams are formed by: using different angles; taking every 2nd,
3rd, &c.
element; changing the starting values of the sequence e.g. 3, 1.

In the attached program, I have not used the strict definition of the fibonacci
sequence but f(n)=(remainder f(n-1) n) + remainder f(n-2) n, where n is
value given by
MODULUS (0 being replaced by n);
POINTS is the length of the sequence used. If the ends of the cyclic diagrams
are 
not joined then increase POINTS.
DILATION values alter the size of the diagram. 
JUMP indicates whether the values are taken every 1, 2, 3, &c. at a time.
Start by clicking on DIAGRAM.

I have used 2 colours for the plot. Comment out lines in proc. DIAGRAM for
single
colour.

Good hunting.
Mike


  

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