Editing Mandelbrot set (section)

== Geometry ==<!--[[Douady rabbit]] links directly here.-->
For every rational number <math>\tfrac{p}{q}</math>, where ''p'' and ''q'' are [[coprime]], a hyperbolic component of period ''q'' bifurcates from the main cardioid at a point on the edge of the cardioid corresponding to an [[internal angle]] of <math>\tfrac{2\pi p}{q}</math>.<ref name="guild">{{cite web |title=Number Sequences in the Mandelbrot Set |url=https://www.youtube.com/watch?v=oNxPSP2tQEk | archive-url=https://ghostarchive.org/varchive/youtube/20211030/oNxPSP2tQEk| archive-date=2021-10-30|website=youtube.com |publisher=The Mathemagicians' Guild |date=4 June 2020}}{{cbignore}}</ref> The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the '''''p''/''q''-limb'''. Computer experiments suggest that the [[diameter]] of the limb tends to zero like <math>\tfrac{1}{q^2}</math>. The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like <math>\tfrac{1}{q}</math>.{{Citation needed|date=July 2023}}

A period-''q'' limb will have <math>q-1</math> "antennae" at the top of its limb. The period of a given bulb is determined by counting these antennas. The numerator of the rotation number, ''p'', is found by numbering each antenna counterclockwise from the limb from 1 to <math>q-1</math> and finding which antenna is the shortest.<ref name="guild" />

=== Pi in the Mandelbrot set ===
There are intriguing experiments in the Mandelbrot set that lead to the occurrence of the number <math>\pi</math>. For a parameter <math>c = -\tfrac{3}{4}+ i\varepsilon</math> with <math>\varepsilon>0</math>, verifying that <math>c</math> is not in the Mandelbrot set means iterating the sequence <math>z \mapsto z^2 + c</math> starting with <math>z=0</math>, until the sequence leaves the disk around <math>0</math> of any radius <math>R>2</math>. This is motivated by the (still open) question whether the vertical line at real part <math>-3/4</math> intersects the Mandelbrot set at points away from the real line. It turns out that the necessary number of iterations, multiplied by <math>\varepsilon</math>, converges to pi. For example, for ''<math>\varepsilon</math>'' = 0.0000001, and <math>R=2</math>, the number of iterations is 31415928 and the product is 3.1415928.<ref>{{cite book |first=Gary William |last=Flake |title=The Computational Beauty of Nature |year=1998 |page=125 |publisher=MIT Press |isbn=978-0-262-56127-3 }}</ref> This experiment was performed independently by many people in the early 1990s, if not before; for instance by David Boll.

Analogous observations have also been made at the parameters <math>c=-5/4</math> and <math>c=1/4</math> (with a necessary modification in the latter case). In 2001, Aaron Klebanoff published a (non-conceptual) proof for this phenomenon at <math>c=1/4</math><ref>{{cite journal |last=Klebanoff |first=Aaron D. |title=π in the Mandelbrot Set |journal=Fractals |volume=9 |issue=4 |pages=393–402 |year=2001 |doi=10.1142/S0218348X01000828 }}</ref>

In 2023, Paul Siewert developed, in his Bachelor thesis, a conceptual proof also for the value <math>c=1/4</math>, explaining why the number pi occurs (geometrically as half the circumference of the unit circle).<ref>Paul Siewert, Pi in the Mandelbrot set. Bachelor Thesis, Universität Göttingen, 2023</ref>

In 2025, the three high school students Thies Brockmöller, Oscar Scherz, and Nedim Srkalovic extended the theory and the conceptual proof to all the infinitely bifurcation points in the Mandelbrot set.<ref>{{cite arXiv | eprint=2505.07138 | last1=Brockmoeller | first1=Thies | last2=Scherz | first2=Oscar | last3=Srkalovic | first3=Nedim | title=Pi in the Mandelbrot set everywhere | date=2025 | class=math.DS }}</ref>

=== Fibonacci sequence in the Mandelbrot set ===
The Mandelbrot Set features a fundamental cardioid shape adorned with numerous bulbs directly attached to it.<ref name=":1">{{Cite journal |last=Devaney |first=Robert L. |date=April 1999 |title=The Mandelbrot Set, the Farey Tree, and the Fibonacci Sequence |url=http://dx.doi.org/10.2307/2589552 |journal=The American Mathematical Monthly |volume=106 |issue=4 |pages=289–302 |doi=10.2307/2589552 |jstor=2589552 |issn=0002-9890}}</ref> Understanding the arrangement of these bulbs requires a detailed examination of the Mandelbrot Set's boundary. As one zooms into specific portions with a geometric perspective, precise deducible information about the location within the boundary and the corresponding dynamical behavior for parameters drawn from associated bulbs emerges.<ref name=":2">{{Cite web |last=Devaney |first=Robert L. |date=January 7, 2019 |title=Illuminating the Mandelbrot set |url=https://math.bu.edu/people/bob/papers/mar-athan.pdf}}</ref>

The iteration of the quadratic polynomial <math>f_c(z) = z^2 + c</math>, where <math>c</math>&nbsp;is a parameter drawn from one of the bulbs attached to the main cardioid within the Mandelbrot Set, gives rise to maps featuring attracting cycles of a specified period <math>q</math>&nbsp;and a rotation number <math>p/q</math>. In this context, the attracting cycle of &nbsp;exhibits rotational motion around a central fixed point, completing an average of <math>p/q</math>&nbsp;revolutions at each iteration.<ref name=":2" /><ref>{{Cite web |last=Allaway |first=Emily |date=May 2016 |title=The Mandelbrot Set and the Farey Tree |url=https://sites.math.washington.edu/~morrow/336_16/2016papers/emily.pdf}}</ref>

The bulbs within the Mandelbrot Set are distinguishable by both their attracting cycles and the geometric features of their structure. Each bulb is characterized by an antenna attached to it, emanating from a junction point and displaying a certain number of spokes indicative of its period. For instance, the <math>2/5</math> bulb is identified by its attracting cycle with a rotation number of <math>2/5</math>. Its distinctive antenna-like structure comprises a junction point from which five spokes emanate. Among these spokes, called the principal spoke is directly attached to the <math>2/5</math> bulb, and the 'smallest' non-principal spoke is positioned approximately <math>2/5</math> of a turn counterclockwise from the principal spoke, providing a distinctive identification as a <math>2/5</math>-bulb.<ref name=":3">{{Cite web |last=Devaney |first=Robert L. |date=December 29, 1997 |title=The Mandelbrot Set and the Farey Tree |url=https://math.bu.edu/people/bob/papers/farey.pdf}}</ref> This raises the question: how does one discern which among these spokes is the 'smallest'?<ref name=":1" /><ref name=":3" /> In the theory of [[external ray]]s developed by [[Adrien Douady|Douady]] and [[John H. Hubbard|Hubbard]],<ref>{{Cite web |last1=Douady, A. |last2=Hubbard, J |date=1982 |title=Iteration des Polynomials Quadratiques Complexes |url=https://pi.math.cornell.edu/~hubbard/CR.pdf}}</ref> there are precisely two external rays landing at the root point of a satellite hyperbolic component of the Mandelbrot Set. Each of these rays possesses an external angle that undergoes doubling under the angle doubling map <math>\theta\mapsto</math> <math>2\theta</math>. According to this theorem, when two rays land at the same point, no other rays between them can intersect. Thus, the 'size' of this region is measured by determining the length of the arc between the two angles.<ref name=":2" />

If the root point of the main cardioid is the cusp at <math>c=1/4</math>, then the main cardioid is the <math>0/1</math>-bulb. The root point of any other bulb is just the point where this bulb is attached to the main cardioid. This prompts the inquiry: which is the largest bulb between the root points of the <math>0/1</math> and <math>1/2</math>-bulbs? It is clearly the <math>1/3</math>-bulb. And note that <math>1/3</math> is obtained from the previous two fractions by [[Farey sequence|Farey addition]], i.e., adding the numerators and adding the denominators

<math>\frac{0}{1}</math> <math>\oplus</math> <math>\frac{1}{2}</math><math>=</math><math>\frac{1}{3}</math>

Similarly, the largest bulb between the <math>1/3</math> and <math>1/2</math>-bulbs is the <math>2/5</math>-bulb, again given by Farey addition.

<math>\frac{1}{3}</math> <math>\oplus</math> <math>\frac{1}{2}</math><math>=</math><math>\frac{2}{5}</math>

The largest bulb between the <math>2/5</math> and <math>1/2</math>-bulb is the <math>3/7</math>-bulb, while the largest bulb between the <math>2/5</math> and <math>1/3</math>-bulbs is the <math>3/8</math>-bulb, and so on.<ref name=":2" /><ref>{{Cite web |title=The Mandelbrot Set Explorer Welcome Page |url=http://math.bu.edu/DYSYS/explorer/ |access-date=2024-02-17 |website=math.bu.edu}}</ref> The arrangement of bulbs within the Mandelbrot set follows a remarkable pattern governed by the [[Farey tree]], a structure encompassing all rationals between <math>0</math> and <math>1</math>. This ordering positions the bulbs along the boundary of the main cardioid precisely according to the [[rational number]]s in the [[unit interval]].<ref name=":3" />
[[File:Fibonacci sequence within the Mandelbrot set.png|left|thumb|Fibonacci sequence within the Mandelbrot set]]
Starting with the <math>1/3</math> bulb at the top and progressing towards the <math>1/2</math> circle, the sequence unfolds systematically: the largest bulb between <math>1/2</math> and <math>1/3</math> is <math>2/5</math>, between <math>1/3</math> and <math>2/5</math> is <math>3/8</math>, and so forth.<ref>{{Cite web |title=Maths Town |url=https://www.patreon.com/mathstown |access-date=2024-02-17 |website=Patreon}}</ref> Intriguingly, the denominators of the periods of circular bulbs at sequential scales in the Mandelbrot Set conform to the [[Fibonacci sequence|Fibonacci number sequence]], the sequence that is made by adding the previous two terms – 1, 2, 3, 5, 8, 13, 21...<ref>{{Cite journal |last1=Fang |first1=Fang |last2=Aschheim |first2=Raymond |last3=Irwin |first3=Klee |date=December 2019 |title=The Unexpected Fractal Signatures in Fibonacci Chains |journal=Fractal and Fractional |language=en |volume=3 |issue=4 |pages=49 |doi=10.3390/fractalfract3040049 |doi-access=free |issn=2504-3110|arxiv=1609.01159 }}</ref><ref>{{Cite web |title=7 The Fibonacci Sequence |url=https://math.bu.edu/DYSYS/FRACGEOM2/node7.html#SECTION00070000000000000000 |access-date=2024-02-17 |website=math.bu.edu}}</ref>

The Fibonacci sequence manifests in the number of spiral arms at a unique spot on the Mandelbrot set, mirrored both at the top and bottom. This distinctive location demands the highest number of iterations of &nbsp;for a detailed fractal visual, with intricate details repeating as one zooms in.<ref>{{Cite web |title=fibomandel angle 0.51 |url=https://www.desmos.com/calculator/oasdhfehoc |access-date=2024-02-17 |website=Desmos |language=en}}</ref>

===Image gallery of a zoom sequence===
The boundary of the Mandelbrot set shows more intricate detail the closer one looks or [[magnification|magnifies]] the image. The following is an example of an image sequence zooming to a selected ''c'' value.

The magnification of the last image relative to the first one is about 10<sup>10</sup> to 1. Relating to an ordinary [[computer monitor]], it represents a section of a Mandelbrot set with a diameter of 4 million kilometers.
{{clear}}

<gallery mode="packed">
Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment.
Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley"<ref name=":7">{{Cite book |last=Lisle |first=Jason |url=https://books.google.com/books?id=h-czEAAAQBAJ |title=Fractals: The Secret Code of Creation |date=2021-07-01 |publisher=New Leaf Publishing Group |isbn=978-1-61458-780-4 |pages=28 |language=en}}</ref>
Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right
Mandel zoom 03 seehorse.jpg|"Seahorse" upside down
</gallery>

The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes"<ref>{{Cite book |last=Devaney |first=Robert L. |url=https://books.google.com/books?id=GUpaDwAAQBAJ |title=A First Course In Chaotic Dynamical Systems: Theory And Experiment |date=2018-05-04 |publisher=CRC Press |isbn=978-0-429-97203-4 |pages=259 |language=en}}</ref> each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a [[Misiurewicz point]]. Between the "upper part of the body" and the "tail", there is a distorted copy of the Mandelbrot set, called a "satellite".

<gallery mode="packed" heights="180">
File:Mandel zoom 04 seehorse tail.jpg|The central endpoint of the "seahorse tail" is also a [[Misiurewicz point]].
File:Mandel zoom 05 tail part.jpg|Part of the "tail" – there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a [[Simply connected space|simply connected]] set, which means there are no islands and no loop roads around a hole.
File:Mandel zoom 06 double hook.jpg|Satellite. The two "seahorse tails" (also called ''dendritic structures'')<ref>{{Cite book |last=Kappraff |first=Jay |url=https://books.google.com/books?id=vAfBrK678_kC |title=Beyond Measure: A Guided Tour Through Nature, Myth, and Number |date=2002 |publisher=World Scientific |isbn=978-981-02-4702-7 |pages=437 |language=en}}</ref> are the beginning of a series of concentric crowns with the satellite in the center.
File:Mandel zoom 07 satellite.jpg|Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".
File:Mandel zoom 08 satellite antenna.jpg|"Antenna" of the satellite. There are several satellites of second order.
File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley"<ref name=":7" /> of the satellite. All the structures from the start reappear.
File:Mandel zoom 10 satellite seehorse valley.jpg|Double-spirals and "seahorses" – unlike the second image from the start, they have appendices consisting of structures like "seahorse tails"; this demonstrates the typical linking of ''n'' + 1 different structures in the environment of satellites of the order ''n'', here for the simplest case ''n'' = 1.
File:Mandel zoom 11 satellite double spiral.jpg|Double-spirals with satellites of second order – analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna".
File:Mandel zoom 12 satellite spirally wheel with julia islands.jpg|In the outer part of the appendices, islands of structures may be recognized; they have a shape like [[Julia set]]s ''J<sub>c</sub>''; the largest of them may be found in the center of the "double-hook" on the right side.
File:Mandel zoom 13 satellite seehorse tail with julia island.jpg|Part of the "double-hook".
File:Mandel zoom 14 satellite julia island.jpg|Islands.
File:Mandel zoom 15 one island.jpg|A detail of one island.
File:Mandel zoom 16 spiral island.jpg|Detail of the spiral.
</gallery>
The islands in the third-to-last step seem to consist of infinitely many parts, as is the case for the corresponding Julia set <math>J_c</math>. They are connected by tiny structures, so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value of ''<math>c
</math>'' for the corresponding ''<math>J_c</math>'' is not the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th step.

===Inner structure===
While the Mandelbrot set is typically rendered showing outside boundary detail, structure within the bounded set can also be revealed.<ref>{{Cite journal |last=Hooper |first=Kenneth J. |date=1991-01-01 |title=A note on some internal structures of the Mandelbrot Set |url=https://www.sciencedirect.com/science/article/abs/pii/009784939190082S |journal=Computers & Graphics |volume=15 |issue=2 |pages=295–297 |doi=10.1016/0097-8493(91)90082-S |issn=0097-8493}}</ref><ref>{{Cite web |last=Cunningham |first=Adam |date=December 20, 2013 |title=Displaying the Internal Structure of the Mandelbrot Set |url=https://www.acsu.buffalo.edu/~adamcunn/downloads/MandelbrotSet.pdf}}</ref><ref>{{Cite journal |last=Youvan |first=Douglas C |date=2024 |title=Shades Within: Exploring the Mandelbrot Set Through Grayscale Variations |url=https://rgdoi.net/10.13140/RG.2.2.24445.74727 |journal=Pre-print |doi=10.13140/RG.2.2.24445.74727}}</ref> For example, while calculating whether or not a given c value is bound or unbound, while it remains bound, the maximum value that this number reaches can be compared to the c value at that location. If the sum of squares method is used, the calculated number would be max:(real^2 + imaginary^2) − c:(real^2 + imaginary^2).{{Citation needed|date=March 2025}} The magnitude of this calculation can be rendered as a value on a gradient.

This produces results like the following, gradients with distinct edges and contours as the boundaries are approached. The animations serve to highlight the gradient boundaries.

<gallery mode=packed heights=160>
File:Mandelbrot full gradient.gif|Animated gradient structure inside the Mandelbrot set
File:Mandelbrot inner gradient.gif|Animated gradient structure inside the Mandelbrot set, detail
File:Mandelbrot gradient iterations.gif|Rendering of progressive iterations from 285 to approximately 200,000 with corresponding bounded gradients animated
File:Mandelbrot gradient iterations thumb.gif|Thumbnail for gradient in progressive iterations
</gallery>