How is the Koch snowflake based on the Koch curve?
It is based on the Koch curve, which appeared in a 1904 paper titled “On a Continuous Curve Without Tangents, Constructible from Elementary Geometry” by the Swedish mathematician Helge von Koch . The Koch snowflake can be built up iteratively, in a sequence of stages.
Is the perimeter of the Koch snowflake fractal infinite?
As Sal says on this video the perimeter of this koch snowflake is infinite. One really intriguing question popped out of my mind. Are not all irrational numbers like pi based on some simple recursive formula as fractals do. So we could be able to make a clear definition to irrational numbers by fractals.
How are Koch snowflakes used in a turtle graphic?
Koch snowflakes and Koch antisnowflakes of the same size may be used to tile the plane. A turtle graphic is the curve that is generated if an automaton is programmed with a sequence. If the Thue–Morse sequence members are used in order to select program states: the resulting curve converges to the Koch snowflake.
How is the Koch snowflake calculated in Lindenmayer?
A turtle graphic is the curve that is generated if an automaton is programmed with a sequence. If the Thue–Morse sequence members are used in order to select program states: the resulting curve converges to the Koch snowflake. The Koch curve can be expressed by the following rewrite system ( Lindenmayer system ):
When did Helge von Koch create the Koch curve?
It is based on the Koch curve, which appeared in a 1904 paper titled “On a continuous curve without tangents, constructible from elementary geometry” by the Swedish mathematician Helge von Koch.
How to draw the Koch curve in Python?
In order draw the Koch curve, we’ll use Python’s turtle library. So before we can define any function, we have to import the library into our program, and create a turtle object that will draw lines for us. We’ll also specify a window, line width, background color, etc:
How is the Koch curve iterated system colored?
Applying these to the Koch curve iterated system produces the following fractal with four-fold rotational symmetry. It is colored using pixel counting (the color depends on how many times a particular pixel is plotted during the drawing of the fractal using the random chaos game algorithm). Here is the fractal obtained using the dihedral group D 4.
