Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes
Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form <inline-formula><math xmlns="http://www...
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MDPI AG
2024-03-01
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| Online Access: | https://www.mdpi.com/2075-1680/13/3/204 |
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| author | Asifa Tassaddiq Amna Kalsoom Maliha Rashid Kainat Sehr Dalal Khalid Almutairi |
| author_facet | Asifa Tassaddiq Amna Kalsoom Maliha Rashid Kainat Sehr Dalal Khalid Almutairi |
| author_sort | Asifa Tassaddiq |
| collection | DOAJ |
| description | Iterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mi mathvariant="double-struck">C</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>p</mi><mi>n</mi></msup><mo>+</mo><mi>m</mi><mi>p</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals. |
| first_indexed | 2024-04-24T18:33:02Z |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-04-24T18:33:02Z |
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| spelling | doaj.art-1dbccc7c16ac4488b6f0654f20764da12024-03-27T13:21:06ZengMDPI AGAxioms2075-16802024-03-0113320410.3390/axioms13030204Generating Geometric Patterns Using Complex Polynomials and Iterative SchemesAsifa Tassaddiq0Amna Kalsoom1Maliha Rashid2Kainat Sehr3Dalal Khalid Almutairi4Department of Basic Sciences and Humanities, College of Computer and Information Sciences, Majmaah University, Al-Majmaah 11952, Saudi ArabiaDepartment of Mathematics and Statistics, International Islamic University, Islamabad 44000, PakistanDepartment of Mathematics and Statistics, International Islamic University, Islamabad 44000, PakistanDepartment of Mathematics and Statistics, International Islamic University, Islamabad 44000, PakistanDepartment of Mathematics, College of Science, Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi ArabiaIterative procedures have been proved as a milestone in the generation of fractals. This paper presents a novel approach for generating and visualizing fractals, specifically Mandelbrot and Julia sets, by utilizing complex polynomials of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mi mathvariant="double-struck">C</mi></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>p</mi><mi>n</mi></msup><mo>+</mo><mi>m</mi><mi>p</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. It establishes escape criteria that play a vital role in generating these sets and provides escape time results using different iterative schemes. In addition, the study includes the visualization of graphical images of Julia and Mandelbrot sets, revealing distinct patterns. Furthermore, the study also explores the impact of parameters on the deviation of dynamics, color, and appearance of fractals.https://www.mdpi.com/2075-1680/13/3/204iterationfixed pointsfractals |
| spellingShingle | Asifa Tassaddiq Amna Kalsoom Maliha Rashid Kainat Sehr Dalal Khalid Almutairi Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes Axioms iteration fixed points fractals |
| title | Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes |
| title_full | Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes |
| title_fullStr | Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes |
| title_full_unstemmed | Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes |
| title_short | Generating Geometric Patterns Using Complex Polynomials and Iterative Schemes |
| title_sort | generating geometric patterns using complex polynomials and iterative schemes |
| topic | iteration fixed points fractals |
| url | https://www.mdpi.com/2075-1680/13/3/204 |
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