Do the exquisite patterns of diatoms
have a mathematical basis?

Text by Brian Darnton (UK)

 

It has been said that microscopists share a limited number of interests beyond their field and that prominent among these are mathematics and music. With the latest developments in the study of inheritance and gene technology, perhaps it’s a good idea to take a look at those little plants over which numerous microscopists have laboured countless hours.

There are few employed in the mounting of these fascinating objects nowadays compared with Victorian times, but we can identify Klaus Kemp at www.diatoms.co.uk as one of the few commercial mounters in the UK working to such a standard.

Some time ago now, although I am mainly interested in the aesthetic and ecological aspects of diatoms, I decided to tackle the mysteries of fractals, their occurrence in nature and the pretty patterns that they could exhibit on the computer. I bought a book on the subject (ref. 2) which contained lots of computer programmes. As promised by the distinguished author, I very soon produced a complex of squares something like an old Islamic illuminated text. The result impressed me immensely. It was not long before I generated the idea that perhaps some of our most beautiful natural patterns such as those on the diatom frustule just might conform to some sort of mathematical order and that the iteration, possible on a computer program might just be able simulate these .

Of course they would not be true fractals and nor would they be real diatoms, but it might just be a fun project for a Christmas holiday.
The early efforts using the BBC BASIC language and then the AMOS AMIGA BASIC language were alarmingly successful. Since those days we are now left with home computers that are rarely equipped with the BASIC language. Only QBASIC is commonly available under the file “OLD MS DOS” on the Windows 95 Companion disk, although it can be found more commonly on WINDOWS 3.1. It is normally run on the MS DOS MODE.
In order to draw a simple circle in the Cartesian system, most programmers use the general formula X=SINE A and Y=COSINE A ,when A increases from 0 to PI TIMES 2. Of course its only a tiny circle, so a multiplier must be used. By manipulation of the co-ordinates, ellipses, circles and other shapes can be formed. By expanding one axis and reducing the other in consecutive scans, a set of dots begins to unfold. At one point a Navicula shape emerges and then a double ellipse becomes almost the shape of Biddulphia antedeluviana. A three dimensional format would perhaps become very complicated, but even in this simple form there are some interesting comparisons with real diatoms.

A. The external shape can be very similar.
B. The orientation of the dots was very similar to some species.
C. The spacing between the lines of dots is similar to some diatoms.
D. If the value of the increments ( C ) is changed from fractions of pi then asymmetrical arrays can be produced with alternating lines.


LITERATURE.
1) Barber and Haworth.
Freshwater Biological Association (FBA) Publication No. 44, “The Diatom Frustule” ISBN 0-900386-428. Gives the whole range of shapes of the diatom.


2) Lauwerier H., Penguin, ”Fractals, Images of Chaos”, ISBN 0-14-014411-0. A basic book of fractals.

3) Darnton. B., “Turkey, fractals and suchlike nonsense”, Balsam Post, April 1994, Issue No. 23, ISSN 0961-043X. (Balsam Post is the quarterly magazine of the Postal Microscopical Society, UK.)

Comments to the author Brian Darnton are welcomed.

 


PROGRAMMES AND THEIR IMAGES

If you have trouble with writing or running the 3 programmes, send me an e-mail request and I will send the files to you as attachments. The files will only work on Qbasic and require translation into other basic languages. Do try manipulating the co-ordinates and stepping values and see what shapes and orientations you can create. Note that the 'G loop' is only inserted to slow down development in order to enjoy the image creation: It is not essential. I have pared the language down to a minimum for rapid results. The images are low resolution digital photographs from the screen.

 

1. NAVICULOID.

05 REM: NAV
10 SCREEN 1:CLS:Pi=3.141593:COLOR(0)
20 M=100:V=0:Z=0:C=Pi/40:N=0
30 M=M-3:V=V+3:N=N+3
40 X=(SIN(Z)*M+150):Y=(COS(Z)*N+100)
50 Z=Z+C
60 PSET(X,Y)
70 IF N>=20 THEN END
80 FOR G=1 TO 1000:NEXT G
90 IF Z>=PI*2 THEN GOTO 30
100 IFZ<= PI*2 THEN GOTO 40
110 END

 

2. CENTRIC.

05 REM:CENTRIC
10 SCREEN 1:CLS
20 PI=1.141593
30 M=95:V=10:Z=0:C=Pi/100
40 M=M-3:V=V+3:Z=0
50 IF M < 85 THEN C=Pi/100
60 X=(SIN(Z)*M+150):Y=(COS(Z)*M+100)
70 Z=Z+C
80 COLOR(0)
90 PSET(X,Y)
100 IF M<=0 THEN END
110 IF Z>=Pi*2 THEN GOTO 40
110 IF Z<=PI*2 THEN GOTO 60

 

3. QUAD.

05 REM:QUAD
10 SCREEN 1:CLS:COLOR(0):PI=1.141593
20 M=100:V=0:Z=0:C=Pi/40:N=0
30 M=M-3:V=V+3:Z=0:N=N+3
40 X=(SIN(Z)N+150):Y=(COS(Z)*M+100)
50 Z=Z+C
60 PSET(X,Y)
70 IF M<=0 THEN END
80 FOR G=1 TO 1000:NEXT G
90 IF Z>=Pi*2 THEN GOTO 30
100 IF Z<=Pi*2 THEN GOTO 40
110 END


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