Mathematical and Physical Journal
for High Schools
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Problem I. 188. (May 2008)

I. 188. Aristid Lindenmayer (1925-1989) was a Hungarian-born biologist and mathematician living in the Netherlands. He modelled the growth of plants with algorithms and studied line fractals that can be described by textual formulae (``generative grammar''). We now call these L-systems and they can be considered as predecessors of turtle graphics (symbols encode the direction and motion of the pen).

In order to draw a fractal, we need to know an axiom, a rule and a turning angle. Complexity of figures depends on the number of recursive steps.

The Koch curve is a classical example of a line fractal.

You should make the five L-system fractals on the given figures by using the Inkscape program (a freely downloadable vector-graphics editor). Use your imagination to create the sixth figure. (The function to generate an L-system is a bit hidden in the program: see Effects/Display/L-system.)

Possible axioms and rules in Inkscape:

F, A, B, C, D, E Line drawing
G, H, I, J, K, L Movement
+ Turn to the left
- Turn to the right
[ Save the turtle's state into the stack
] Restore the turtle's state from the stack
X, Y Rules (but not for drawing)

The following example illustrates the usage of the stack when drawing a tree:

Axiom: F
Formula: F=F[+F][-F]
Angle: 25

Figures can be easily coloured by Inkscape.

The parameters to generate the given 5 figures (axiom, rule, angle and number of recursions) should be submitted together with a short documentation (i188.txt, i188.doc, ...), further one of your own figures and its parameter set (i188.svg, i188.png, ...).

(10 pont)

Deadline expired on June 16, 2008.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás

A feladatra sok jó megoldás érkezett. Néhány ábra előállítható többféle axiómával és szabállyal is. Lindenmayer munkásságáról sok cikket olvashatunk. Most egyet ajánlok figyelmetekbe.

Ábra Axióma Szabály Szög Helyettesítések száma
1. F F=F+F-F-F+F 90 4
2. F+F+F F=F+F-F-F+F 120 3
2. -F+F+F F=F[+F+F]F Bal: 120 Jobb: 90 3
3. F-F-F-F F=FF-F-F-F-F-F+F 90 2
3. F+F+F+F F=FF-F[-F-F]+F 90 2
4. F F=F[+F]F[-F]F 26 3
5. F F=FF-[-F+F+F]+[+F-F-F] 15 3

A szabadon választott 6. ábrának nagyon szép és látványos rajzok érkeztek. Többen a kész ábra színezésével is foglakoztak.

Példaként álljon itt Horváth 135 Lóránd rajza:

Statistics:

7 students sent a solution.
10 points:Fábián András, Földes Imre, Horváth 135 Loránd, Pap 987 Dávid, Póta Kristóf, Véges Márton.
6 points:1 student.

Problems in Information Technology of KöMaL, May 2008