Barnsley Farn

Englische Anleitung zur Computererzeugung des fraktalen Farns

# Basiswissen

Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer practically mandatory. Many different computer models of Barnsley's fern are popular with contemporary mathematicians. As long as the math is programmed correctly using Barnsley's matrix of constants, the same fern shape will be produced.

The first point drawn is at the origin (x0 = 0, y0 = 0) and then the new points xn and yn are iteratively computed by randomly applying one of the following four coordinate transformations:

* stands for 'times'

x = the old x-coordinate

xn = the next to be x-coordinate

y = the old x-coordinate

yn = the next to be x-coordinate

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# ƒ1

xn = 0

yn = 0.16 * y

This coordinate transformation is chosen 1% of the time and just maps any point to a point in the first line segment at the base of the stem. This part of the figure is the first to be completed in during the course of iterations. The asteriks * stands for "times" (multiplication).

# ƒ2

xn = 0.85 * x + 0.04 * y

yn = −0.04 * x + 0.85 * y + 1.6

This coordinate transformation is chosen 85% of the time and maps any point inside the leaflet represented by the red triangle to a point inside the opposite, smaller leaflet represented by the blue triangle in the figure.

# ƒ3

xn = 0.2 * x − 0.26 * y

yn = 0.23 * x + 0.22 * y + 1.6

This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (it flips it).

# ƒ4

xn = −0.15 * x + 0.28 * y

yn = 0.26 * x + 0.24 * y + 0.44

This coordinate transformation is chosen 7% of the time and maps any point inside the leaflet (or pinna) represented by the blue triangle to a point inside the alternating corresponding triangle across the stem (without flipping it).

The first coordinate transformation draws the stem. The second generates successive copies of the stem and bottom fronds to make the complete fern. The third draws the bottom frond on the left. The fourth draws the bottom frond on the right. The recursive nature of the IFS guarantees that the whole is a larger replica of each frond. Note that the complete fern is within the range −2.1820 < x < 2.6558 and 0 ≤ y < 9.9983.

Zitiervorlage Gunter Heim (2016): Barnsley Farn, in: Rhetos Lern-Lexikon der Physik. URL: www.rhetos.de/physik/lexikon/barnsley_farn_mathematisch.htm (Stand: 22 Jun. 2024)

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