Mandelbrot set

The Mandelbrot set (/ˈmændəlbroʊt, -brɒt/)^[1]^[2] is a two-dimensional set that is defined in the complex plane as the complex numbers $c$ for which the function $f_{c}(z)=z^{2}+c$ does not diverge to infinity when iterated starting at $z=0$ , i.e., for which the sequence $f_{c}(0)$ , $f_{c}(f_{c}(0))$ , etc., remains bounded in absolute value.^[3]

This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups.^[4] Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York.^[5]

Images of the Mandelbrot set exhibit an infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications;^[6]^[7] mathematically, the boundary of the Mandelbrot set is a fractal curve.^[8] The "style" of this recursive detail depends on the region of the set boundary being examined.^[9] Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point $c$ , whether the sequence $f_{c}(0),f_{c}(f_{c}(0)),\dotsc$ goes to infinity.^[10]^{[close paraphrasing]} Treating the real and imaginary parts of $c$ as image coordinates on the complex plane, pixels may then be colored according to how soon the sequence $|f_{c}(0)|,|f_{c}(f_{c}(0))|,\dotsc$ crosses an arbitrarily chosen threshold (the threshold must be at least 2, as −2 is the complex number with the largest magnitude within the set, but otherwise the threshold is arbitrary).^[10]^{[close paraphrasing]} If $c$ is held constant and the initial value of $z$ is varied instead, the corresponding Julia set for the point $c$ is obtained.^[11]

The Mandelbrot set is well-known,^[12] even outside mathematics,^[13] for how it exhibits complex fractal structures when visualized and magnified, despite having a relatively simple definition.^[14]^[15]^[16]

^ "Mandelbrot set". Lexico UK English Dictionary. Oxford University Press. Archived from the original on 31 January 2022.
^ "Mandelbrot set". Merriam-Webster.com Dictionary. Merriam-Webster. Retrieved 30 January 2022.
^ Cooper, S. B.; Löwe, Benedikt; Sorbi, Andrea (28 November 2007). New Computational Paradigms: Changing Conceptions of What is Computable. Springer Science & Business Media. p. 450. ISBN 978-0-387-68546-5.
^ Cite error: The named reference :0 was invoked but never defined (see the help page).
^ Nakos, George (20 May 2024). Elementary Linear Algebra with Applications: MATLAB®, Mathematica® and MaplesoftTM. Walter de Gruyter GmbH & Co KG. p. 322. ISBN 978-3-11-133185-0.
^ Addison, Paul S. (1 January 1997). Fractals and Chaos: An illustrated course. CRC Press. p. 110. ISBN 978-0-8493-8443-1.
^ Briggs, John (1992). Fractals: The Patterns of Chaos : a New Aesthetic of Art, Science, and Nature. Simon and Schuster. p. 77. ISBN 978-0-671-74217-1.
^ Hewson, Stephen Fletcher (2009). A Mathematical Bridge: An Intuitive Journey in Higher Mathematics. World Scientific. p. 155. ISBN 978-981-283-407-2.
^ Peitgen, Heinz-Otto; Richter, Peter H. (1 December 2013). The Beauty of Fractals: Images of Complex Dynamical Systems. Springer Science & Business Media. p. 166. ISBN 978-3-642-61717-1. the Mandelbrot set is very diverse in its different regions
^ ^a ^b Hunt, John (1 October 2023). Advanced Guide to Python 3 Programming. Springer Nature. p. 117. ISBN 978-3-031-40336-1.
^ Campuzano, Juan Carlos Ponce (20 November 2020). "Complex Analysis". Complex Analysis — The Mandelbrot Set. Archived from the original on 16 October 2024. Retrieved 5 March 2025.
^ Oberguggenberger, Michael; Ostermann, Alexander (24 October 2018). Analysis for Computer Scientists: Foundations, Methods, and Algorithms. Springer. p. 131. ISBN 978-3-319-91155-7.
^ "Mandelbrot Set". cometcloud.sci.utah.edu. Retrieved 22 March 2025.
^ Peitgen, Heinz-Otto; Jürgens, Hartmut; Saupe, Dietmar (6 December 2012). Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set. Springer Science & Business Media. p. 415. ISBN 978-1-4612-4406-6.
^ Gulick, Denny; Ford, Jeff (10 May 2024). Encounters with Chaos and Fractals. CRC Press. pp. §7.2. ISBN 978-1-003-83578-3.
^ Bialynicki-Birula, Iwo; Bialynicka-Birula, Iwona (21 October 2004). Modeling Reality: How Computers Mirror Life. OUP Oxford. p. 80. ISBN 978-0-19-853100-5.

[1] "Mandelbrot set". Lexico UK English Dictionary. Oxford University Press. Archived from the original on 31 January 2022.

[2] "Mandelbrot set". Merriam-Webster.com Dictionary. Merriam-Webster. Retrieved 30 January 2022.

[3] Cooper, S. B.; Löwe, Benedikt; Sorbi, Andrea (28 November 2007). New Computational Paradigms: Changing Conceptions of What is Computable. Springer Science & Business Media. p. 450. ISBN 978-0-387-68546-5.

[:0-4] Cite error: The named reference :0 was invoked but never defined (see the help page).

[5] Nakos, George (20 May 2024). Elementary Linear Algebra with Applications: MATLAB®, Mathematica® and MaplesoftTM. Walter de Gruyter GmbH & Co KG. p. 322. ISBN 978-3-11-133185-0.

[6] Addison, Paul S. (1 January 1997). Fractals and Chaos: An illustrated course. CRC Press. p. 110. ISBN 978-0-8493-8443-1.

[7] Briggs, John (1992). Fractals: The Patterns of Chaos : a New Aesthetic of Art, Science, and Nature. Simon and Schuster. p. 77. ISBN 978-0-671-74217-1.

[8] Hewson, Stephen Fletcher (2009). A Mathematical Bridge: An Intuitive Journey in Higher Mathematics. World Scientific. p. 155. ISBN 978-981-283-407-2.

[9] Peitgen, Heinz-Otto; Richter, Peter H. (1 December 2013). The Beauty of Fractals: Images of Complex Dynamical Systems. Springer Science & Business Media. p. 166. ISBN 978-3-642-61717-1. the Mandelbrot set is very diverse in its different regions

[:4-10] Hunt, John (1 October 2023). Advanced Guide to Python 3 Programming. Springer Nature. p. 117. ISBN 978-3-031-40336-1.

[11] Campuzano, Juan Carlos Ponce (20 November 2020). "Complex Analysis". Complex Analysis — The Mandelbrot Set. Archived from the original on 16 October 2024. Retrieved 5 March 2025.

[12] Oberguggenberger, Michael; Ostermann, Alexander (24 October 2018). Analysis for Computer Scientists: Foundations, Methods, and Algorithms. Springer. p. 131. ISBN 978-3-319-91155-7.

[13] "Mandelbrot Set". cometcloud.sci.utah.edu. Retrieved 22 March 2025.

[14] Peitgen, Heinz-Otto; Jürgens, Hartmut; Saupe, Dietmar (6 December 2012). Fractals for the Classroom: Part Two: Complex Systems and Mandelbrot Set. Springer Science & Business Media. p. 415. ISBN 978-1-4612-4406-6.

[15] Gulick, Denny; Ford, Jeff (10 May 2024). Encounters with Chaos and Fractals. CRC Press. pp. §7.2. ISBN 978-1-003-83578-3.

[16] Bialynicki-Birula, Iwo; Bialynicka-Birula, Iwona (21 October 2004). Modeling Reality: How Computers Mirror Life. OUP Oxford. p. 80. ISBN 978-0-19-853100-5.

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