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Friday, February 18, 2011

Creating a fractal with Excel or Calc

If you have Excel 2010 then do the following, otherwise apply the equivalent code / options for your spreadsheet.

Create a new workbook
Make sure that zeros are displayed as whitespace:
- \File\Options
- Select 'Advanced' on left sidebar
- ( Look for 'Display options for this worksheet: Sheet1' )
- When found deselect 'Show a zero in cells that have a zero value'
Set the column-width to the first 130 rows or so to about 3.0

Now apply some values to the following four cells.
A1: p ( any prime ), use a small prime, for example 3.
C3: 1
A4: =A3+1 ( not necessary, but shows interesting properties )
In C4 we implement the Pascal recurrence modulo a prime
C4: =MOD(C3+B3;*A*1) ( replace * with dollar sign conflict with MathJax )
Now you have to do some copying and pasting:
- you have to copy this cell at least 128 cells to the right
- copy the contents of row 4 at least 127 times

If you have done this correct you will see patterns like this:
Mod 2, zoom 30%

Mod 3, zoom 30%

Mod 7, zoom 30%
It will be interesting to see what happens if you try mod p to a power, i.e. 4, 8 and 16, or mod 9.

It should be possible to control graphics at the pixel level with Mathematica, I might give it a try.

It was OU course M381 ( which I am on at the moment ) that gave me the idea to look at the Pascal triangle modulo a prime because one of the topics in M381 is the Fibonacci series modulo a prime. The Pascal Triangle and the Fibonacci series are definitely among my favourite mathematical objects. If I had to choose I would say the PT of course, because the Fibonacci series is neatly contained within it. What is not? One might ask.

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)