Lihat sumber Himpunan Mandelbrot

{{short description|Fraktal dinamai matematikawan Benoit Mandelbrot}}
{{Use dmy dates|date=September 2020}}
[[Berkas:Mandel zoom 00 mandelbrot set.jpg|322px|thumb|Set Mandelbrot (hitam) dalam lingkungan yang terus diwarnai|alt=]]<!-- Urutan \, disisipkan dalam item MATEMATIKA untuk memastikan konsistensi representasi.
  -- Tolong jangan dihapus -->{{Periksa terjemahan|en|Mandelbrot set}}[[Berkas:Progressive infinite iterations of the 'Nautilus' section of the Mandelbrot Set.ogv|thumb|Iterasi progresif tak terbatas dari bagian "Nautilus" dari Set Mandelbrot yang dirender menggunakan webGL]]
[[Berkas:Animation of the growth of the Mandelbrot set as you iterate towards infinity.gif|thumb|Animasi mandelbrot berdasarkan jumlah iterasi statis per piksel]]
[[Berkas:Mandelbrot set image.png|thumb|Detail Himpunan Mandelbrot]]
'''Himpunan Mandelbrot''' adalah [[set (matematika)|himpunan]] dari [[bilangan kompleks]] <math>c</math> yang digunakan sebagai fungsi <math>f_c(z)=z^2+c</math> tidak [[menyimpang (teori stabilitas)|menyimpang]] ketika [[iterasi]] dari <math>z=0</math>, yaitu, urutan dari <math>f_c(0)</math>, <math>f_c(f_c(0))</math>, dll, tetap dibatasi dalam nilai absolut.
[[Berkas:Mandelbrot sequence new.gif|thumb|Memperbesar ke himpunan Mandelbrot]]
 
Definisinya dikreditkan ke [[Adrien Douady]] yang menamakannya sebagai penghormatan kepada [[matematikawan]] [[Benoit Mandelbrot]].<ref name="John H. Hubbard 1985">Adrien Douady dan John H. Hubbard, ''Etude dynamique des polynômes complexes'', Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)</ref> Himpunan tersebut terhubung ke sebuah [[himpunan Julia]], dan himpunan Julia terkait menghasilkan bentuk [[fraktal]] yang kompleks serupa.

Gambar set Mandelbrot dapat dibuat dengan mengambil sampel bilangan kompleks dan pengujian, untuk setiap titik sampel <math>c</math>, apakah urutan dark <math>f_c(0), f_c(f_c(0)),\dotsc</math> [[Urutan#Terikat|pergi ke tak terhingga]] (dalam praktik apakah itu meninggalkan beberapa lingkungan nilai yang telah ditentukan sebelumnya dari 0 setelah jumlah iterasi yang telah ditentukan). Bila [[bilangan riil]] dan [[Bilangan imajiner|bagian imajiner]] dari <math>c</math> sebagai [[koordinat gambar]] pada [[bidang kompleks]], piksel kemudian dapat diwarnai sesuai dengan seberapa cepat urutan dari <math>|f_c(0)|, |f_c(f_c(0))|,\dotsc</math> melintasi ambang yang dipilih secara sewenang-wenang, dengan warna khusus (hitam) digunakan untuk nilai <math>c</math> yang urutannya belum melewati ambang setelah jumlah iterasi yang ditentukan sebelumnya (ini diperlukan untuk membedakan dengan jelas gambar set Mandelbrot dari gambar pelengkap). Bila <math>c</math> dipertahankan konstan dan nilai awal nilai <math>z</math> dinotasikan dengan <math>z_0</math> sebagai gantinya, variabel ini memperoleh [[Himpunan Julia#Polinomial kuadrat|nilai himpunan Julia]] untuk setiap titik <math>c</math> di [[parameter spasi]] dari fungsinya.

Gambar dari himpunan Mandelbrot menunjukkan [[Batas (topologi)|batas]] yang rumit dan sangat rumit yang mengungkapkan detail [[rekursi]]f yang semakin halus pada perbesaran yang meningkat. Dengan kata lain, batas himpunan Mandelbrot adalah ''[[kurva fraktal]]''. "Gaya" dari detail berulang ini bergantung pada wilayah himpunan yang sedang diperiksa. Batas himpunan juga menggabungkan versi yang lebih kecil dari bentuk utama, sehingga properti [[fraktal]] dari [[kemiripan diri]] berlaku untuk seluruh himpunan, dan tidak hanya untuk bagian-bagiannya.

Himpunan Mandelbrot telah menjadi populer di luar [[matematika]] baik karena daya tarik estetikanya maupun sebagai contoh struktur kompleks yang timbul dari penerapan aturan sederhana. Ini adalah salah satu contoh paling terkenal dari [[visualisasi matematika]] dan [[keindahan matematika]].

== Sejarah ==
[[Berkas:Mandel.png|322px|right|thumb|Gambar terbitan pertama dari set Mandelbrot, oleh [[Robert W. Brooks]] dan Peter Matelski pada tahun 1978]]
Himpunan Mandelbrot berawal dari [[dinamika kompleks]], bidang yang pertama kali diselidiki oleh [[matematikawan Prancis]] [[Pierre Fatou]] dan [[Gaston Julia]] pada awal abad ke 20. Fraktal ini pertama kali didefinisikan dan digambar pada tahun 1978 oleh [[Robert W. Brooks]] dan Peter Matelski sebagai bagian dari studi [[kelompok Kleinian]]..<ref>Robert Brooks dan Peter Matelski, ''Dinamika subkelompok 2 generator PSL (2,C)'', di {{cite book|url=http://www.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|title=Permukaan Riemann dan Topik Terkait: Prosiding Konferensi Stony Brook 1978|author=Irwin Kra|date=1 Mei 1981|publisher=Princeton University Press|others=[[Bernard Maskit]]|isbn=0-691-08267-7|editor=Irwin Kra|access-date=1 July 2019|archive-url=https://web.archive.org/web/20190728201429/http://www.math.harvard.edu/archive/118r_spring_05/docs/brooksmatelski.pdf|archive-date=28 July 2019|url-status=dead}}</ref> Pada tanggal 1 Maret 1980, di [[IBM]] [[Pusat Penelitian Thomas J. Watson]] di [[Yorktown Heights, New York|Yorktown Heights]], [[New York]], [[Benoit Mandelbrot]] pertama kali melihat visualisasi.<ref name="bf">{{cite web |url=http://sprott.physics.wisc.edu/pubs/paper311.pdf |title=Fraktal Biofilik dan Perjalanan Visual Penghemat Layar Organik |author=R.P. Taylor & J.C. Sprott |accessdate=1 Januari 2009 |year=2008 |work=Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 12, No. 1 |publisher=Society for Chaos Theory in Psychology & Life Sciences }}</ref>

Mandelbrot mempelajari [[parameter space]] dari [[polinomial kuadrat]] dalam sebuah artikel yang muncul pada tahun 1980.<ref>Benoit Mandelbrot, ''Aspek fraktal dari iterasi <math>z\mapsto\lambda z(1-z)</math> untuk kompleks <math>\lambda, z</math>'', ''Sejarah Akademi Ilmu Pengetahuan New York'' '''357''', 249/259</ref> Studi matematis himpunan Mandelbrot benar-benar dimulai dengan karya ahli matematika [[Adrien Douady]] dan [[John H. Hubbard]] (1985),<ref name="John H. Hubbard 1985"/> yang menetapkan banyak properti fundamentalnya dan menamai himpunan tersebut untuk menghormati Mandelbrot atas karyanya yang berpengaruh di [[geometri fraktal]].

Matematikawan [[Heinz-Otto Peitgen]] dan [[Peter Richter]] menjadi terkenal karena mempromosikan set dengan foto, buku (1986),<ref>{{cite book |title=Keindahan Fraktal |last=Peitgen |first=Heinz-Otto |author2=Richter Peter |year=1986 |publisher=Springer-Verlag |location=Heidelberg |isbn=0-387-15851-0 |title-link=The Beauty of Fractals }}</ref> dan pameran tur internasional [[Goethe-Institut]] Jerman (1985).<ref>[[Frontiers of Chaos]], Pameran Goethe-Institut oleh H.O. Peitgen, P. Richter, H. Jürgens, M. Prüfer, D.Saupe. Sejak 1985 ditampilkan di lebih dari 40 negara.</ref><ref>{{cite book |title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London |pages=229 |title-link=Chaos: Making a New Science }}</ref>

Artikel sampul pada Agustus 1985 ''[[Scientific American]]'' memperkenalkan [[algoritma]] kepada khalayak luas untuk menghitung himpunan Mandelbrot. Sampulnya menampilkan gambar yang terletak di [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.909,%22y%22:-0.275},%22zoom%22:10000} −0.909 + −0.275 ''i''] dan diciptakan oleh Peitgen et al.<ref>{{cite magazine |title= Computer Recreations, Agustus 1985; Mikroskop komputer memperbesar untuk melihat objek paling kompleks dalam matematika |last=Dewdney |first=A. K. |year=1985 |magazine=Scientific American |url=https://www.scientificamerican.com/media/inline/blog/File/Dewdney_Mandelbrot.pdf}}</ref><ref>{{cite book |title=Fractals: The Patterns of Chaos |url=https://archive.org/details/fractalspatterns00brig |author=John Briggs |year=1992 |page=[https://archive.org/details/fractalspatterns00brig/page/80 80]}}</ref> Set Mandelbrot menjadi terkenal pada pertengahan 1980 an sebagai komputer [[Demo (pemrograman komputer)|demo grafis]], ketika [[komputer pribadi]] menjadi cukup kuat untuk memplot dan menampilkan set dalam resolusi tinggi.<ref>{{cite magazine |last=Pountain |first=Dick |date=September 1986 |title= Turbocharging Mandelbrot |url=https://archive.org/stream/byte-magazine-1986-09/1986_09_BYTE_11-09_The_68000_Family#page/n370/mode/1up |magazine= [[Byte (magazine)|Byte]] |access-date=11 November 2015 }}</ref>

Karya Douady dan Hubbard bertepatan dengan peningkatan besar minat dalam dinamika kompleks dan [[matematika abstrak]], dan studi tentang himpunan Mandelbrot telah menjadi pusat perhatian bidang ini. Daftar lengkap dari semua orang yang telah berkontribusi pada pemahaman himpunan ini sejak saat itu masih panjang, tetapi akan mencakup [[Mikhail Lyubich]],<ref>{{cite journal
  | author = Lyubich, Mikhail
  | title = Enam Kuliah tentang Dinamika Nyata dan Kompleks
  | version = 
  | date = Mei-Juni 1999
  | url = http://citeseer.ist.psu.edu/cache/papers/cs/28564/http:zSzzSzwww.math.sunysb.eduzSz~mlyubichzSzlectures.pdf/
  | accessdate = 2007-04-04
  }}{{Pranala mati|date=November 2022 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{cite journal
  | last = Lyubich
  | first = Mikhail
  | authorlink = Mikhail Lyubich
  | title = Regular and stochastic dynamics in the real quadratic family
  | journal = Prosiding National Academy of Sciences of the United States of America
  | volume = 95
  | issue =24
  | pages = 14025–14027
  | date=November 1998
  | url = http://www.pnas.org/cgi/reprint/95/24/14025.pdf
  | doi = 10.1073/pnas.95.24.14025
  | accessdate = 2007-04-04
  | pmid = 9826646
  | pmc = 24319 | bibcode =1998PNAS...9514025L
  }}</ref> [[Curtis T. McMullen|Curt McMullen]], [[John Milnor]], [[Mitsuhiro Shishikura]] dan [[Jean-Christophe Yoccoz]].

== Definisi formal ==
Himpunan Mandelbrot adalah himpunan nilai '' c '' di [[bidang kompleks]] yang [[Orbit (dinamika)|orbit]] dari [[Polinomial kuadrat kompleks#Nilai kritikal|nilai kritikal]] z = 0 di bawah [[Iteratal Kuadrat]]

:<math>z_{n+1} = z_n^2 + c</math>

tetap [[Urutan berbatas|dibatasi]].<ref>{{cite web|url=http://math.bu.edu/DYSYS/explorer/def.html|title=Penjelajah Set Mandelbrot: Daftar Istilah Matematika|accessdate=2007-10-07}}</ref> Jadi, bilangan kompleks ''c'' adalah anggota himpunan Mandelbrot bila, saat dimulai dengan ''z''<sub>0</sub> = 0 dan menerapkan iterasi berulang kali, [[nilai absolut]] dari ''z''<sub>''n''</sub> tetap terikat untuk semua ''n'' > 0.

Contohnya, untuk ''c'' = 1, urutannya adalah 0, 1, 2, 5, 26, ..., yang cenderung [[tak terhingga]], jadi 1 bukan merupakan elemen himpunan Mandelbrot. Sebaliknya, untuk ''c''=−1, urutannya adalah 0, −1, 0, −1, 0, ..., yang dibatasi, jadi −1 memang termasuk dalam himpunan.

[[Berkas:Mandelset hires.png|right|thumb|322px|Penggambaran matematikawan dari himpunan Mandelbrot ''M''. Titik ''c'' diberi warna hitam jika termasuk dalam set, dan putih jika bukan. Re [''c''] dan Im [''c''] menunjukkan bagian nyata dan imajiner dari ''c''.]]

Set Mandelbrot juga bisa didefinisikan sebagai [[lokus keterhubungan]] dari keluarga [[polinomial]].

== Representasi grafis ==
Representasi grafis dari himpunan Mandelbrot dan strukturnya di area tepi hanya mungkin menggunakan [[komputer]] menggunakan apa yang disebut [[generator fraktal]]. Setiap piksel sesuai dengan nilai <math>c</math> bidang kompleks. Untuk setiap piksel, komputer menentukan apakah urutan terkait divergen atau tidak. Begitu jumlahnya <math>|z_n|</math> eines Anggota berurutan nilai <math>s=2</math> melebihi, jelas bahwa hasilnya divergen. Nomor <math>n</math> langkah [[iterasi]] sampai saat itu dapat berfungsi sebagai ukuran derajat divergensi. Pixel diberi warna sesuai dengan tabel warna yang telah ditentukan sebelumnya yang sesuai dengan setiap nilai <math>n</math> memberikan warna.

Untuk mencapai batas yang harmonis antara warna-warna yang berurutan dari sudut pandang estetika, batas tersebut digunakan dalam praktiknya <math>s</math> bukan nilai sekecil mungkin <math>s=2</math> dipilih, tetapi nilai secara signifikan lebih besar dari <math>2</math>, jika tidak, lebar garis warna akan berosilasi. Semakin besar nilai ini dipilih, semakin baik batas warna yang sesuai dengan [[garis ekuipotensial]], yang dihasilkan ketika himpunan Mandelbrot diinterpretasikan sebagai [[muatan listri|bermuatan listrik]] [[konduktor (fisika)|konduktor]]. Untuk gradien warna kontinu, Seperti pada rangkaian gambar zoom di atas, evaluasi faktor diperlukan di mana <math>s</math> terlampaui saat nilainya dilampaui untuk pertama kalinya.

Karena jumlah langkah iterasi <math>n</math>, setelah itu batasnya <math>s</math> terlampaui, dapat berukuran berapa pun, kriteria penghentian dalam bentuk jumlah langkah iterasi maksimum harus ditentukan. Nilai <math>c</math>, konsekuensi yang belum melebihi batas <math>s</math> setelahnya menjadi <math>\mathbb M</math> diharapkan. Semakin kecil jarak <math>c</math> zu <math>\mathbb M</math> adalah, semakin besar angka <math> n</math> biasanya setelah <math>s</math> terlampaui. Semakin besar pembesaran tepi <math>\mathbb M</math> ditampilkan, semakin besar jumlah maksimum langkah iterasi yang harus dipilih, dan semakin banyak waktu komputasi yang diperlukan. Dapat dilihat bahwa urutan nilai awal <math>c</math> konvergen, perhitungan urutannya bisa diakhiri lebih awal.

Representasi tepi secara grafis sangat menarik <math>\mathbb M</math> dengan kekayaan bentuknya. Semakin besar perbesaran yang dipilih, struktur yang lebih kompleks dapat ditemukan di sana. Dengan [[program komputer]] yang sesuai, tepi ini dapat ditampilkan seperti pada [[mikroskop]] dengan perbesaran apa pun. Hanya dua kebebasan artistik yang ada adalah pemilihan bagian gambar dan penetapan warna pada tingkat divergensi.

Perbesaran sering kali diperlukan untuk memeriksa struktur yang diinginkan, yang tidak dapat dihitung dengan tipe data yang didukung perangkat keras karena keakuratannya yang terbatas. Oleh karena itu, beberapa program berisi tipe data bilangan panjang [[aritmatika]] dengan presisi yang dapat dipilih secara sewenang-wenang. Ini berarti bahwa (hampir) semua faktor pembesaran dimungkinkan.

== Properti dasar ==
Himpunan Mandelbrot adalah [[himpunan kompak]], karena ia [[himpunan tertutup|tertutup]] dan terkandung dalam [[disk tertutup]] dengan radius 2 di sekitar [[Asal (matematika)|asal]]. Lebih khusus lagi, satu poin <math>c</math> termasuk dalam kumpulan Mandelbrot jika dan hanya jika
:<math>|P_c^n(0)|\leq 2</math> untuk semua <math>n\geq 0.</math>

Dengan kata lain, [[nilai absolut]] dari <math>P_c^n(0)</math> harus tetap pada atau di bawah 2 untuk <math>c</math> berada di himpunan Mandelbrot, seolah-olah nilai absolut itu melebihi 2, urutannya akan lolos hingga tak terbatas.

[[Berkas:Verhulst-Mandelbrot-Bifurcation.jpg|thumb|Korespondensi antara set Mandelbrot dan [[diagram bifurkasi]] dari [[peta logistik]]]]
[[Berkas:Logistic Map Bifurcations Underneath Mandelbrot Set.gif|thumb|Dengan <math>z_{n}</math> iterasi diplot pada sumbu vertikal, himpunan Mandelbrot dapat dilihat bercabang dimana himpunan terbatas]]
[[Persimpangan (teori himpunan)|persimpangan]] dari <math>M</math> dengan sumbu nyata tepatnya adalah interval [−2, 1/4]. Parameter sepanjang interval ini dapat dimasukkan ke dalam korespondensi satu-satu dengan yang ada di [[peta logistik|keluarga logistik]] yang sebenarnya,
:<math>x_{n+1} = r x_n(1-x_n),\quad r\in[1,4].</math>
Korespondensi diberikan oleh

:<math>z = r\left(\frac12 - x\right),
\quad
c = \frac{r}{2}\left(1-\frac{r}{2}\right).</math>

Faktanya, ini memberikan korespondensi antara seluruh [[ruang parameter]] dari keluarga logistik dan yang ada di himpunan Mandelbrot.

Douady dan Hubbard telah menunjukkan bahwa set Mandelbrot adalah [[ruang yang terhubung|terhubung]]. Faktanya, mereka membangun sebuah [[fungsi holomorfik|isomorfisma konformal]] antara komplemen himpunan Mandelbrot dan komplemen [[disk unit tertutup]]. Mandelbrot awalnya menduga bahwa set Mandelbrot adalah [[Terputus (topologi)|terputus]]. Dugaan ini didasarkan pada gambar komputer yang dihasilkan oleh program yang tidak dapat mendeteksi filamen tipis yang menghubungkan berbagai bagian <math>M</math>. Setelah percobaan lebih lanjut, dia merevisi dugaannya, memutuskan itu <math>M</math> harus terhubung. Ada juga bukti [[topologi]] untuk keterhubungan yang ditemukan pada tahun 2001 oleh [[Jeremy Kahn]].<ref>{{Cite web|url=http://www.math.brown.edu/~kahn/mconn.pdf|title=The Mandelbrot Set is Connected: a Topological Proof|last=Kahn|first=Jeremy|date=8 Agustus 2001}}</ref>
[[Berkas:Wakes near the period 1 continent in the Mandelbrot set.png|thumbnail|right|Sinar eksternal bangun di dekat benua periode 1 di himpunan Mandelbrot]]
Rumus dinamis untuk [[teorema penyeragaman|penyeragaman]] dari pelengkap himpunan Mandelbrot, yang muncul dari bukti Douady dan Hubbard tentang keterkaitan <math>M</math>, memunculkan [[sinar eksternal]] dari himpunan Mandelbrot. Sinar ini dapat digunakan untuk mempelajari himpunan Mandelbrot dalam istilah kombinatorial dan membentuk tulang punggung [[Jean-Christophe Yoccoz#pekerjaan Matematika]].<ref>''The Mandelbrot set, theme and variations''. Tan, Lei. Cambridge University Press, 2000. {{isbn|978-0-521-77476-5}}. Section 2.1, "Yoccoz para-puzzles", [https://books.google.com/books?id=-a_DsYXquVkC&pg=PA121 p.&nbsp;121]</ref>

[[Batas (topologi)|batas]] dari himpunan Mandelbrot persis [[lokus percabangan]] dari keluarga kuadrat; yaitu, kumpulan parameter <math>c</math> di mana dinamika berubah secara tiba-tiba di bawah perubahan kecil <math> c. </math> Itu dapat dibangun sebagai himpunan batas dari urutan [[kurva aljabar|bidang kurva aljabar]], yang ''kurva Mandelbrot'', dari tipe umum yang dikenal sebagai [[lemniskat polinomial]]. Kurva Mandelbrot ditentukan oleh pengaturan ''p''<sub>0</sub> = ''z'', ''p''<sub>''n''+1</sub> = ''p''<sub>''n''</sub><sup>2</sup> + ''z'', dan kemudian menafsirkan kumpulan poin {{abs|''p''<sub>''n''</sub>(''z'')}} = 2 dalam bidang kompleks sebagai kurva dalam [[Sistem koordinat kartesius|bidang Cartesian]] derajat 2<sup>''n''+1</sup> in ''x'' and ''y''. Kurva aljabar ini muncul dalam gambar set Mandelbrot yang dihitung menggunakan "algoritme waktu melarikan diri" yang disebutkan di bawah.

== Perilaku deretan angka ==
Berbagai elemen struktural <math>\mathbb M</math> terkait erat dengan perilaku tertentu dari urutan angka, itu <math>\mathbb M</math> yang mendasari. Bergantung pada nilai <math>c</math>, ada empat kemungkinan:

* Bila nilai [[Nilai batas (urutan)|menyatu]] ke titik tetap.
* Ketika menyatu dengan [[siklus batas]] periodik yang terdiri dari dua atau lebih nilai. Ini juga mencakup kasus di mana urutan berperilaku secara berkala sejak awal.
* Karena hal ini tidak pernah berulang, tetapi tetap terbatas. Beberapa nilai menunjukkan perilaku kacau dengan pergantian antara siklus batas hampir periodik dan perilaku acak.
* Terutama menyimpang menuju tak terbatas ([[nilai batas (urutan)|divergensi tertentu]]).

Semua <math>c</math> nilai yang ''tidak'' pasti dimiliki secara berbeda <math>\mathbb M</math>.

Tabel berikut menunjukkan contoh dari empat perilaku batas iterasi nilai <math>z_{n+1}=z_n^2+c</math> dari <math>z_0=0</math>:

{| class="wikitable" style="text-align:left;"
|-
! Parameter!! Nilai <math>z_2, z_3, z_4, z_5, \ldots</math>                                                 
! Titik tetap
|-
| colspan="3" | '''Nilai sumbu riil...'''
|- bgcolor="#FFF" 
| <math>c= -3</math>        ||<math>6,\, 33,\, 1086,\, 1179393,\, \dotsc</math>
| Perbedaan nilai tertentu <math>+\infty</math>
|- bgcolor="#DDF" 
| <math>c= -2</math>        || <math>2,\, 2,\, 2,\, 2,\, \dotsc</math>                                                              
| Konvergensi tetap menuju titik kontinu <math>2</math>
|- bgcolor="#DFF" 
| <math>c= -1{,}75</math>   || <math>\tfrac{21}{16},\, -\tfrac{7}{256},\, -\tfrac{114639}{65536},\, \dotsc</math>                   
| Konvergensi tetap ke siklus tiga urutan <math>-1{,}74697\ldots,\, 1{,}30193\ldots,\, -0{,}54958\ldots</math>
|- bgcolor="#AAA"
| <math>c= -1{,}5</math>    || <math>\tfrac34,\, -\tfrac{15}{16},\, -\tfrac{159}{256},\, -\tfrac{73023}{65536},\, \dotsc</math>             
| Chaotisches Verhalten
|- bgcolor="#FDD" 
| <math>c= -1{,}4</math>   || <math>0{,}56,\, -1{,}0864,\, -0{,}21973504,\, \dotsc</math>
| Konvergensi tetap 32 ke siklus batas
|- bgcolor="#DEF" 
| <math>c= -1{,}25</math>   || <math>\tfrac{5}{16},\, -\tfrac{295}{256},\, \tfrac{5105}{65536},\, \dotsc</math>             
| Konvergensi terhadap siklus urutan bolak-balik <math>-1{,}20710\ldots,\, 0{,}20710\ldots</math>
|- bgcolor="#DEF" 
| <math>c= -1</math>        || <math>-1,\, 0,\, -1,\, 0,\, -1,\, 0,\, \dotsc</math>                                                 
| Konvergensi instan tidak tetap siklus urutan bolak-balik <math>-1,\, 0</math>
|- bgcolor="#DDF" 
| <math>c= -0{,}75</math>   || <math>-\tfrac{3}{16}, \, -\tfrac{183}{256}, \, -\tfrac{15663}{65536}, \, \dotsc</math> 
| Konvergensi agak tetap menuju titik tertentu <math>-\tfrac12</math> 
|- bgcolor="#DDF" 
| <math>c= -0{,}5</math>    || <math>-\tfrac14,\, -\tfrac{7}{16},\, -\tfrac{79}{256},\, -\tfrac{26527}{65536},\, \dotsc</math>                            
| Konvergensi titik tetap <math>\tfrac12-\sqrt\tfrac34</math> 
|- bgcolor="#DDF" 
| <math>c= -0{,}25</math>   || <math>-\tfrac{3}{16},\, -\tfrac{55}{256},\, -\tfrac{13359}{65536},\, \dotsc</math>
| Nilai titik tetap konvergensi <math>\tfrac12-\sqrt\tfrac12</math> 
|- bgcolor="#DDF" 
| <math>c= 0</math>         || <math>0,\, 0,\, 0,\, 0,\, 0,\, 0,\, \dotsc</math>                                                    
| Konvergensi melawan titik tetap <math>0</math>
|- bgcolor="#DDF" 
| <math>c= +0{,}25</math>   || <math>\tfrac{5}{16},\, \tfrac{89}{256},\, \tfrac{24305}{65536},\, \dotsc</math>
| Konvergensi menuju titik tetap <math>\tfrac12</math>
|- bgcolor="#FFF" 
| <math>c= +1</math>        || <math>2, \, 5, \, 26, \, 677, \, 458330, \,\dotsc</math>                                             
| Perbedaan tertentu pada nilai tetap <math>+\infty</math>
|-
| colspan="3" | '''Dalam bidang bilangan kompleks...'''
|- bgcolor="#DEF" 
| <math>c = -i</math>       || <math>-1-i,\, i,\, -1-i,\, i,\, \dotsc</math>                                             
| Konvergensi instan melawan siklus batas bolak-balik <math>-1-i,\, i</math>
|- bgcolor="#DFF" 
| <math>c = -\tfrac18 +  \tfrac34 i</math> ||<math>-\tfrac{43}{64} +\tfrac{9}{16} i </math>
| Konvergensi menuju siklus tiga batas
|-
|}[[Berkas:Mandelbrot Set – Periodicities coloured.png|thumb|Himpunan Mandelbrot diatur dengan kode warna periode panjang siklus batas]]

=== Penilaian geometris ===
Konvergensi terjadi tepat untuk nilai <math>c</math> yang menyusun interior [[kardioid]], "tubuh" dari <math>\mathbb M</math>, serta untuk banyak titik batas mereka yang tak terhitung banyaknya. Siklus batas periodik dapat ditemukan pada (kurang lebih) "rintisan" dari nilai melingkar seperti pada "kepala", di kardioid satelit serta di banyak titik tepi komponen ini. Sebuah [[dugaan (matematika)|dugaan]] fundamental mengatakan bahwa ada siklus batas untuk semua titik dalam himpunan Mandelbrot. Urutannya benar-benar pra-periodik untuk banyak parameter yang tak terhitung banyaknya, yang sering disebut [[titik Misiurewicz|Titik Misiurewicz-Thurston]] (setelah [[Michał Misiurewicz]] dan [[William Thurston]]). Ini termasuk "ujung antena" seperti titik <math>z=-2</math> paling kiri dan titik cabang himpunan Mandelbrot.

Dalam banyak titik lain dari himpunan Mandelbrot yang tak terhitung banyaknya, urutan dapat berperilaku dalam berbagai cara, yang masing-masing menghasilkan sistem dinamis yang sangat berbeda dan sebagian menjadi subjek penelitian intensif. Je menurut definisi kata tersebut, perilaku "kacau" dapat ditemukan.

== Properti lainnya ==
<!-- [0110101010010101010101001001010]
=== Umbi kardioid dan periode utama ===
<!--[[Douady rabbit]] links directly here.
[[File:Mandelbrot Set – Periodicities coloured.png|right|thumb|Periods of hyperbolic components]]

Upon looking at a picture of the Mandelbrot set, one immediately notices the large [[cardioid]]-shaped region in the center. This ''main cardioid''
is the region of parameters <math>c</math> for which <math>P_c</math> has an [[Periodic points of complex quadratic mappings|attracting fixed point]]. It consists of all parameters of the form
:<math> c = \frac\mu2\left(1-\frac\mu2\right)</math>
for some <math>\mu</math> in the [[open unit disk]].

To the left of the main cardioid, attached to it at the point <math>c=-3/4</math>, a circular-shaped '''bulb''' is visible. This bulb consists of those parameters <math>c</math> for which <math>P_c</math> has an [[Periodic points of complex quadratic mappings|attracting cycle of period 2]]. This set of parameters is an actual circle, namely that of radius 1/4 around −1.

There are infinitely many other bulbs tangent to the main cardioid: for every rational number <math>\tfrac{p}{q}</math>, with ''p'' and ''q'' [[coprime]], there is such a bulb that is tangent at the parameter

:<math> c_{\frac{p}{q}} = \frac{e^{2\pi i\frac pq}}2\left(1-\frac{e^{2\pi i\frac pq}}2\right).</math>

[[File:Animated cycle.gif|left|thumb|200px|Attracting cycle in 2/5-bulb plotted over [[Julia set]] (animation)]]This bulb is called the ''<math>\tfrac{p}{q}</math>-bulb'' of the Mandelbrot set. It consists of parameters that have an attracting cycle of period <math>q</math> and combinatorial rotation number <math>\tfrac{p}{q}</math>. More precisely, the <math>q</math> periodic [[Classification of Fatou components|Fatou components]] containing the attracting cycle all touch at a common point (commonly called the ''<math>\alpha</math>-fixed point''). If we label these components <math>U_0,\dots,U_{q-1}</math> in counterclockwise orientation, then <math>P_c</math> maps the component <math>U_j</math> to the component <math>U_{j+p\,(\operatorname{mod} q)}</math>.

[[File:Juliacycles1.png|right|thumb|300px|Attracting cycles and [[Julia set]]s for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs]]

[[File:Mandel rays.jpg|thumb|right|425px|Cycle periods and antennae]]

The change of behavior occurring at <math>c_{\frac{p}{q}}</math> is known as a [[bifurcation theory|bifurcation]]: the attracting fixed point "collides" with a repelling period ''q''-cycle. As we pass through the bifurcation parameter into the <math>\tfrac{p}{q}</math>-bulb, the attracting fixed point turns into a repelling fixed point (the <math>\alpha</math>-fixed point), and the period ''q''-cycle becomes attracting.

{{clear|left}}

===Hyperbolic components===
All the bulbs we encountered in the previous section were interior components of
the Mandelbrot set in which the maps <math>P_c</math> have an attracting periodic cycle. Such components are called ''hyperbolic components''.

It is conjectured that these are the ''only'' interior regions of <math>M</math>. This problem, known as ''density of hyperbolicity'', may be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components.<ref>''Exploring the Mandelbrot set. The Orsay Notes'' by Adrien Douady and John H. Hubbard. page 12</ref><ref>Wolf Jung, March 2002, [http://www.mndynamics.com/papers/thesis.pdf Homeomorphisms on Edges of the Mandelbrot Set by Wolf Jung]</ref>
For ''real'' quadratic polynomials, this question was answered positively in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the [[Bifurcation diagram|Feigenbaum diagram]]. So this result states that such windows exist near every parameter in the diagram.)

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component ''can'' be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

Each of the hyperbolic components has a ''center'', which is a point ''c'' such that the inner Fatou domain for <math>P_c(z)</math> has a super-attracting cycle – that is, that the attraction is infinite (see the image [[:commons:File:Centers8.png|here]]). This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. We therefore have that <math>P_c</math><sup>n</sup><math>(0) = 0</math> for some ''n''. If we call this polynomial <math>Q^{n}(c)</math> (letting it depend on ''c'' instead of ''z''), we have that <math>Q^{n+1}(c) = Q^{n}(c)^{2} + c</math> and that the degree of <math>Q^{n}(c)</math> is <math>2^{n-1}</math>. We can therefore construct the centers of the hyperbolic components by successively solving the equations <math>Q^{n}(c) = 0, n = 1, 2, 3, ...</math>. The number of new centers produced in each step is given by Sloane's {{oeis|A000740}}.

===Local connectivity===
[[File:Cactus model of Mandelbrot set.svg|right|thumb|Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)]]
[[File:Lavaurs-12.png|right|thumb|Thurston model of Mandelbrot set (abstract Mandelbrot set)]]

It is conjectured that the Mandelbrot set is [[locally connected]]. This famous conjecture is known as ''MLC'' (for ''Mandelbrot locally connected''). By the work of [[Adrien Douady]] and [[John H. Hubbard]], this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important ''hyperbolicity conjecture'' mentioned above.

The work of [[Jean-Christophe Yoccoz]] established local connectivity of the Mandelbrot set at all finitely [[Renormalization|renormalizable]] parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies.<ref name="yoccoz">{{cite book
 | last = Hubbard | first = J. H.
 | contribution = Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz
 | contribution-url = http://www.math.cornell.edu/~hubbard/hubbard.pdf
 | location = Houston, TX
 | mr = 1215974
 | pages = 467–511
 | publisher = Publish or Perish
 | title = Topological methods in modern mathematics (Stony Brook, NY, 1991)
 | year = 1993}}. Hubbard cites as his source a 1989 unpublished manuscript of Yoccoz.</ref> Since then, local connectivity has been proved at many other points of <math>M</math>, but the full conjecture is still open.

===Self-similarity===
[[File:Mandelbrot zoom.gif|right|thumb|201px|[[Self-similarity]] in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-''x'' direction. The display center pans from (−1,&nbsp;0) to (−1.31,&nbsp;0) while the view magnifies from 0.5&nbsp;×&nbsp;0.5 to 0.12&nbsp;×&nbsp;0.12 to approximate the [[Feigenbaum constants|Feigenbaum ratio]] <math>\delta</math>.]]
[[File:Mandelzoom1.jpg|left|thumb|280px|Self-similarity around Misiurewicz point −0.1011&nbsp;+&nbsp;0.9563i.]] The Mandelbrot set is [[self-similar]] under magnification in the neighborhoods of the [[Misiurewicz point]]s. It is also conjectured to be self-similar around generalized [[Feigenbaum point]]s (e.g., −1.401155 or −0.1528&nbsp;+&nbsp;1.0397''i''), in the sense of converging to a limit set.<ref>{{cite journal | last1 = Lei | year = 1990 | title = Similarity between the Mandelbrot set and Julia Sets | url = http://projecteuclid.org/euclid.cmp/1104201823| journal = Communications in Mathematical Physics | volume = 134 | issue = 3| pages = 587–617 | doi=10.1007/bf02098448| bibcode = 1990CMaPh.134..587L}}</ref><ref>{{cite book |author=J. Milnor |chapter=Self-Similarity and Hairiness in the Mandelbrot Set |editor=M. C. Tangora |location=New York |pages=211–257 |title=Computers in Geometry and Topology |url=https://books.google.com/books?id=wuVJAQAAIAAJ |year=1989|publisher=Taylor & Francis}})</ref>
[[File:Blue Mandelbrot Zoom.jpg|left|thumb|340px|Quasi-self-similarity in the Mandelbrot set]]The Mandelbrot set in general is not strictly self-similar but it is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales.

The little copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.

===Further results===
The [[Hausdorff dimension]] of the [[boundary (topology)|boundary]] of the Mandelbrot set equals 2 as determined by a result of [[Mitsuhiro Shishikura]].<ref name="shishikura"/> It is not known whether the boundary of the Mandelbrot set has positive planar [[Lebesgue measure]].

In the [[Blum–Shub–Smale machine|Blum–Shub–Smale]] model of [[real computation]], the Mandelbrot set is not computable, but its complement is [[Recursively enumerable set|computably enumerable]]. However, many simple objects (''e.g.'', the graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on [[computable analysis]], which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.[[File:Relationship between Mandelbrot sets and Julia sets.PNG|3220x|left|thumb|A zoom into the Mandelbrot set illustrating a Julia "island" and a similar Julia set.]]

===Relationship with Julia sets===
As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding [[Julia set]]. For instance, a point is in the Mandelbrot set exactly when the corresponding Julia set is connected.

This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has [[Hausdorff dimension]] two, and then transfers this information to the parameter plane.<ref name="shishikura">{{cite journal
 | last = Shishikura | first = Mitsuhiro
 | arxiv = math.DS/9201282
 | doi = 10.2307/121009
 | issue = 2
 | journal = Annals of Mathematics
 | mr = 1626737
 | pages = 225–267
 | series = Second Series
 | title = The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
 | volume = 147
 | year = 1998| jstor = 121009
 }}.</ref> Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters.<ref name="yoccoz"/> [[Adrien Douady]] phrases this principle as:

:{{quote|Plough in the dynamical plane, and harvest in parameter space.}}-->

== Geometri ==
<!--[[Douady rabbit]] links directly here.
[[File:Unrolled main cardioid of Mandelbrot set for periods 8-14.png|thumbnail|right|upright=2.4|Components on main cardioid for periods 8–14 with 7-13 antennae]]

For every rational number <math>\tfrac{p}{q}</math>, where ''p'' and ''q'' are [[relatively prime]], a hyperbolic component of period ''q'' bifurcates from the main cardioid. 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 ''[[Jean-Christophe Yoccoz|Yoccoz-inequality]]'', which states that the size tends to zero like <math>\tfrac{1}{q}</math>.

A period-''q'' limb will have ''q''&nbsp;−&nbsp;1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.

=== Pi in the Mandelbrot set ===
In an attempt to demonstrate that the thickness of the ''p''/''q''-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the series to diverge for ''z'' = −{{sfrac|3|4}} + ''iε'' (−{{sfrac|3|4}} being the location thereof). As the series doesn't diverge for the exact value of ''z'' = −{{sfrac|3|4}}, the number of iterations required increases with a small ''ε''. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better for smaller ''ε''. For example, for ''ε'' = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.<ref>Gary William Flake, ''The Computational Beauty of Nature'', 1998. p.&nbsp;125. {{isbn|978-0-262-56127-3}}.</ref>

=== Fibonacci sequence in the Mandelbrot set ===
It can be shown that the [[Fibonacci Sequence|Fibonacci sequence]] is located within the Mandelbrot Set and that a relation exists between the main cardioid and the [[Farey sequence|Farey Diagram]]. Upon mapping the main cardioid to a disk, one can notice that the amount of antennae that extends from the next largest Hyperbolic component, and that is located between the two previously selected components, follows suit with the Fibonacci sequence. The amount of antennae also correlates with the Farey Diagram and the denominator amounts within the corresponding fractional values, of which relate to the distance around the disk. Both portions of these fractional values themselves can be summed together after <math>\frac{1}{3}</math> to produce the location of the next Hyperbolic component within the sequence. Thus, the Fibonacci sequence of 1, 2, 3, 5, 8, 13, and 21 can be found within the Mandelbrot set.

=== Galeri gambar dari urutan zoom ===
The Mandelbrot set shows more intricate detail the closer one looks or [[magnification|magnifies]] the image, usually called "zooming in". The following example of an image sequence zooming to a selected ''c'' value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules.

The magnification of the last image relative to the first one is about 10<sup>10</sup> to 1. Relating to an ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border would show an astronomical number of different fractal structures.
{{Clear}}

<gallery mode="packed">
File:Mandel zoom 00 mandelbrot set.jpg|Start. Mandelbrot set with continuously colored environment.
File:Mandel zoom 01 head and shoulder.jpg|Gap between the "head" and the "body", also called the "seahorse valley"
File:Mandel zoom 02 seehorse valley.jpg|Double-spirals on the left, "seahorses" on the right
File:Mandel zoom 03 seehorse.jpg|"Seahorse" upside down
</gallery>

The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes" each and one "spoke" connecting to the main cardioid. These two groups can be attributed by some kind of 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 so-called [[Misiurewicz point]]. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized.

<gallery mode="packed">
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" are the beginning of a series of concentric crowns with the satellite in the center. [https://mandelbrot-svelte.netlify.com/#{%22pos%22:{%22x%22:-0.743904874255535,%22y%22:-0.1317119067802009},%22zoom%22:7502494.442311305} Open this location in an interactive viewer.]
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. Several satellites of second order may be recognized.
File:Mandel zoom 09 satellite head and shoulder.jpg|The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.
File:Mandel zoom 10 satellite seehorse valley.jpg|Double-spirals and "seahorses" – unlike the 2nd 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|Detail of one island
File:Mandel zoom 16 spiral island.jpg|Detail of the spiral. [https://guciek.github.io/web_mandelbrot.html#-0.743643887035763;0.13182590421259918;4.3e-14;7000 Open this location in an interactive viewer.]
</gallery>
The islands in the third-to-last step seem to consist of infinitely many parts like [[Cantor set]]s, as is{{Clarify|date=May 2010}} actually the case for the corresponding Julia set ''J<sub>c</sub>''. However, 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 ''c'' for the corresponding ''J<sub>c</sub>'' is not that of 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 zoom step.

===3D images of Mandelbrot and Julia sets===
{{More citations needed section|talk=3D images|date=January 2020}} Once enough citations have been added to this sectionb then the preceding template can be removed

In addition to creating two dimensional images of the Mandelbrot set, various techniques can be used to render Mandelbrot and Julia sets as 3D [[Heightmap]] images, where each pixel in a 2D image is given a height value, and the resulting image is rendered as a 3D graphic.

The simplest approach to 3D rendering uses the iteration value for each pixel as a height value. This produces images with distinct "steps" in the height value.

[[File: Mandelbrot set 3D integer iterations.jpg|thumb|none|Mandelbrot set rendered in 3D using integer iterations]]

If instead the fractional iteration value (also known as the potential function<ref>[[Julia set#The potential function and the real iteration number|potential function]])</ref>) is used to calculate the height value for each point, steps in the resulting image are avoided. However, images rendered in 3D using fractional iteration data still look rather bumpy and visually noisy.

[[File: Mandelbrot set 3D fractional iterations.jpg|thumb|none| Mandelbrot set rendered in 3D using fractional iteration values]]

An alternative approach is to use Distance Estimate<ref>{{cite book |title=The Science of Fractal Images |last=Peitgen |first=Heinz-Otto |author2=Saupe Dietmar |year=1988 |publisher=Springer-Verlag |location=New York |isbn=0-387-96608-0 |pages=121, 196–197 |title-link=The Beauty of Fractals }}</ref> (DE) data for each point to calculate a height value. Non-linear mapping of distance estimate value using an exponential function can produce visually pleasing images. Images plotted using DE data are often visually striking, and more importantly, the 3D shape makes it easy to visualize the thin "tendrils" that connect points of the set. Color plates 29 and 30 on page 121 of "The Science of Fractal Images" show a 2D and 3D image plotted using External Distance Estimates.

[[File: Mandelbrot set 3D Distance Estimates.jpg |thumb|none| Mandelbrot set rendered in 3D using Distance Estimates]]

Below is a 3D version of the "Image gallery of a zoom sequence" gallery above, rendered as height maps using Distance Estimate data, and using similar cropping and coloring.

<gallery mode="packed">
File:Mandel zoom 00 mandelbrot set 3D.jpg|[[:File: Mandel zoom 00 mandelbrot set.jpg|Zoom 00]]. Start. Mandelbrot set with continuously colored environment.
File:Mandel zoom 01 head and shoulder 3D.jpg|[[:File: Mandel zoom 01 head and shoulder.jpg|Zoom 01]]. Gap between the "head" and the "body", also called the "seahorse valley"
File:Mandel zoom 02 seehorse valley 3D.jpg|[[:File: Mandel zoom 02 seehorse valley.jpg|Zoom 02]]. Double-spirals on the left, "seahorses" on the right
File:Mandel zoom 03 seehorse 3D.jpg|[[:File: Mandel zoom 03 seehorse.jpg|Zoom 03]]. "Seahorse" upside down
</gallery>

<gallery mode="packed">
File:Mandel zoom 04 seehorse tail 3D.jpg|[[:File: Mandel zoom 04 seehorse tail.jpg|Zoom 04]]. A "seahorse tail".
File:Mandel zoom 05 tail part 3D.jpg|[[:File: Mandel zoom 05 tail part.jpg|Zoom 05]]. Part of the "tail".
File:Mandel zoom 06 double hook 3D.jpg|[[:File: Mandel zoom 06 double hook.jpg|Zoom 06]]. Satellite with twin "Seahorse tails."
File:Mandel zoom 07 satellite 3D.jpg|[[:File: Mandel zoom 07 satellite.jpg|Zoom 07]]. Satellite closeup.
File:Mandel zoom 08 satellite antenna 3D.jpg|[[:File: Mandel zoom 08 satellite antenna.jpg|Zoom 08]]. "Antenna" of the satellite. Several satellites of second order may be recognized.
File:Mandel zoom 09 satellite head and shoulder 3D.jpg|[[:File: Mandel zoom 09 satellite head and shoulder.jpg|Zoom 09]]. The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.
File:Mandel zoom 10 satellite seehorse valley 3D.jpg|[[:File: Mandel zoom 10 satellite seehorse valley.jpg|Zoom 10]]. Double-spirals and "seahorses"
File:Mandel zoom 11 satellite double spiral 3D.jpg|[[:File: Mandel zoom 11 satellite double spiral.jpg|Zoom 11]]. Double-spirals with satellites of second order.
File:Mandel zoom 12 satellite spirally wheel with julia islands 3D.jpg|[[:File: Mandel zoom 12 satellite spirally wheel with julia islands.jpg|Zoom 12]]. 
File:Mandel zoom 13 satellite seehorse tail with julia island 3D.jpg|[[:File: Mandel zoom 13 satellite seehorse tail with julia island.jpg|Zoom 13]]. Part of the "double-hook"
File:Mandel zoom 14 satellite julia island 3D.jpg|[[:File: Mandel zoom 14 satellite julia island.jpg|Zoom 14]]. Islands
File:Mandel zoom 15 one island 3D.jpg|[[:File: Mandel zoom 15 one island.jpg|Zoom 15]]. Detail of one island
File:Mandel zoom 16 spiral island 3D.jpg|[[:File: Mandel zoom 16 spiral island.jpg|Zoom 16]]. Detail of the spiral. 
</gallery>

The image below is similar to "zoom 5", above, but is an attempt to create a 3D version of the image "Map 44" from page 85 of the book "The Beauty of Fractals"<ref>{{cite book |title=The Beauty of Fractals |last=Peitgen |first=Heinz-Otto |author2=Richter Peter |year=1986 |publisher=Springer-Verlag |location=Heidelberg |isbn=0-387-15851-0 |pages=[https://archive.org/details/beautyoffractals0000peit/page/85 85] |title-link=The Beauty of Fractals }}</ref> using a visually similar color scheme that shows the details of the plot in 3D.

[[File:A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals".jpg|thumb|none|A 3D version of the Mandelbrot set plot "Map 44" from the book "The Beauty of Fractals]]-->Kardioid mempunyai periode 1. Lingkaran mempunyai periode selain 1, kecuali mini mandelbrot yang ada di c = -1,75.

== Generalisasi ==
{{multiple image
 | image1 = Mandelbrot Set Animation 1280x720.gif
 | image2 = Mandelbrot set from powers 0.05 to 2.webm
 | width2 = 150
 | footer = Animasi Himpunan Multibrot untuk ''d'' dari 0 hingga 5 (kiri) dan dari 0,05 hingga 2 (kanan).
}}
[[Berkas:Quaternion Julia x=-0,75 y=-0,14.jpg|thumb|Kumpulan 4D Julia dapat diproyeksikan atau dipotong melintang menjadi 3D, dan karena itu, Mandelbrot 4D juga dimungkinkan.]]

=== Himpunan Multibrot ===
[[Himpunan Multibrot]] adalah himpunan berbatas yang ditemukan di bidang kompleks untuk anggota famili rekursi monik univariat [[polinomial]] umum

:<math> z \mapsto z^d + c.\ </math>

Untuk integer d, himpunan ini adalah lokus keterhubungan untuk himpunan Julia yang dibangun dari rumus yang sama. Lokus keterhubungan kubik penuh juga telah dipelajari; di sini orang menganggap dua parameter berulang <math> z \mapsto z^3 + 3kz + c </math>, yang dua [[titik kritikal (matematika)|titik kritikal]] adalah [[akar kuadrat kompleks]] dari parameter ''k''. Parameter berada dalam lokus keterhubungan kubik jika kedua titik kritis stabil.<ref>[[Rudy Rucker]]'s discussion of the CCM: [http://www.cs.sjsu.edu/faculty/rucker/cubic_mandel.htm CS.sjsu.edu] {{Webarchive|url=https://web.archive.org/web/20170303184004/http://www.cs.sjsu.edu/faculty/rucker/cubic_mandel.htm |date=2017-03-03 }}</ref> Untuk famili umum [[holomorphic function]], ''boundary'' dari himpunan Mandelbrot digeneralisasikan ke [[lokus percabangan]], yang merupakan objek alami untuk dipelajari bahkan ketika lokus keterhubungan tidak berguna.

<!--The [[Multibrot set]] is obtained by varying the value of the exponent ''d''. The [[Multibrot set|article]] has a video that shows the development from ''d'' = 0 to 7, at which point there are 6 i.e. (''d'' − 1) lobes around the perimeter. A similar development with negative exponents results in (1 − ''d'') clefts on the inside of a ring.

===Higher dimensions===
There is no perfect extension of the Mandelbrot set into 3D. This is because there is no 3D analogue of the complex numbers for it to iterate on. However, there is an extension of the complex numbers into 4 dimensions, called the [[quaternion]]s, that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions.<ref>http://archive.bridgesmathart.org/2010/bridges2010-247.pdf retrieved 19 August 2018</ref> These can then be either [[cross section (geometry)|cross-sectioned]] or [[Projection mapping|projected]] into a 3D structure.

===Other, non-analytic, mappings===
[[File:Mandelbar fractal from XaoS.PNG|left|thumb|Image of the Tricorn / Mandelbar fractal]]
[[File:BurningShip01.png|thumb|Image of the burning ship fractal]]

Of particular interest is the [[tricorn (mathematics)|tricorn]] fractal, the connectedness locus of the anti-holomorphic family

:<math> z \mapsto \bar{z}^2 + c.</math>

The tricorn (also sometimes called the ''Mandelbar'') was encountered by [[John Milnor|Milnor]] in his study of parameter slices of real [[Cubic function|cubic polynomials]]. It is ''not'' locally connected. This property is inherited by the connectedness locus of real cubic polynomials.

Another non-analytic generalization is the [[Burning Ship fractal]], which is obtained by iterating the following :

:<math> z \mapsto (|\Re \left(z\right)|+i|\Im \left(z\right)|)^2 + c.</math>-->

== Lihat pula ==

* [[Buddhabrot]]
* [[Fraktal collatz]]
* [[Fractint]]
* [[Perubahan Gilbreath]]
* [[Mandelbox]]
* [[Mandelbulb]]
* [[Menger Sponge]]
* [[Fraktal Newton]]
* [[Potret orbit]]
* [[Orbit trap]]
* [[Pickover stalk]]

== Referensi ==
{{Reflist}}

== Bacaan lebih lanjut ==
* [[John W. Milnor]], ''Dynamics in One Complex Variable'' (Third Edition), Annals of Mathematics Studies 160, (Princeton University Press, 2006), {{isbn|0-691-12488-4}} <br />(First appeared in 1990 as a [https://web.archive.org/web/20060424085751/http://www.math.sunysb.edu/preprints.html Stony Brook IMS Preprint], available as [http://www.arxiv.org/abs/math.DS/9201272 arXiV:math.DS/9201272] )
* Nigel Lesmoir-Gordon, ''The Colours of Infinity: The Beauty, The Power and the Sense of Fractals'', {{isbn|1-904555-05-5}} <br />(includes a DVD featuring [[Arthur C. Clarke]] and [[David Gilmour]])
* [[Heinz-Otto Peitgen]], [[Hartmut Jürgens]], [[Dietmar Saupe]], ''Chaos and Fractals: New Frontiers of Science'' (Springer, New York, 1992, 2004), {{isbn|0-387-20229-3}}

== Pranala luar ==
{{Wikibooks|Fraktal }}
{{commons}}
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* {{Curlie|Science/Math/Chaos_and_Fractals|Chaos and Fractals}}
* [http://classes.yale.edu/Fractals/MandelSet/welcome.html The Mandelbrot Set and Julia Sets by Michael Frame, Benoit Mandelbrot, and Nial Neger] {{Webarchive|url=https://web.archive.org/web/20130521110007/http://classes.yale.edu/Fractals/MandelSet/welcome.html |date=2013-05-21 }}
* [http://vimeo.com/12185093 Video: Mandelbrot fractal zoom to 6.066 e228]
* [https://www.youtube.com/watch?v=NGMRB4O922I Relatively simple explanation of the mathematical process, by [[Holly Krieger|Dr Holly Krieger]], MIT]
* [https://mandelbrot-svelte.netlify.com Mandelbrot set images online rendering]
* [https://www.rosettacode.org/wiki/Mandelbrot_set Various algorithms for calculating the Mandelbrot set] (on [[Rosetta Code]])
* [https://github.com/pkulchenko/ZeroBraneEduPack/blob/master/fractal-samples/zplane.lua Fractal calculator written in Lua by Deyan Dobromiroiv, Sofia, Bulgaria]

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