Abstract
The external rays of the Mandelbrot set are a valuable graphic tool in order to study this set. They are drawn using computer programs starting from the Böttcher coordinate. However, the drawing of an external ray cannot be completed because it reaches a point from which the drawing tool cannot continue drawing. This point is influenced by the resolution of the standard for floating-point computation used by the drawing program. The IEEE 754 Standard for Floating-Point Arithmetic is the most widely used standard for floating-point computation, and we analyze the possibilities of the quadruple 128 bits format of the current IEEE 754-2008 Standard in order to draw external rays. When the drawing is not possible, due to a lack of resolution of this standard, we introduce a method to draw external rays based on the escape lines and Bézier curves.
Highlights
As is well known, the Mandelbrot set can be defined by M = {c ∈ C : fc∘k(0) ∞ as k → ∞}, where fc∘k(0) is the k-iteration of the parameter-dependent quadratic function fc(z) = z2 + c (z and c complex) from the initial value z0 = 0.In the 1980s, Douady and Hubbard published the external arguments theory of the Mandelbrot set [1, 2]
The field lines are the external rays of Douady and Hubbard, and the numbers associated with the external rays are the external arguments of Douady and Hubbard
When we draw an external ray by means of a computer program using the Bottcher coordinate, we observe that the drawing of the ray is interrupted when it comes close to the landing point; that is, the drawing of an external ray has an end point
Summary
As we will see, the resolution of the quadruple format is not sufficient in some of the cases, and the same occurs with a hypothetical octuple 256 bits format (not defined yet)
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