A Brief Survey of Paradigmatic Fractals from a Topological Perspective
The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi>&l...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/8/597 |
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| author | Julián Patiño Ortiz Miguel Patiño Ortiz Miguel-Ángel Martínez-Cruz Alexander S. Balankin |
| author_facet | Julián Patiño Ortiz Miguel Patiño Ortiz Miguel-Ángel Martínez-Cruz Alexander S. Balankin |
| author_sort | Julián Patiño Ortiz |
| collection | DOAJ |
| description | The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow></semantics></math></inline-formula> which exceeds the topological dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi></mrow></semantics></math></inline-formula>. In this regard, we point out that the constitutive inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>></mo><mi>d</mi></mrow></semantics></math></inline-formula> can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined. |
| first_indexed | 2024-03-10T23:55:34Z |
| format | Article |
| id | doaj.art-b0cd021ee726407aa807559e2bc7e420 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T23:55:34Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-b0cd021ee726407aa807559e2bc7e4202023-11-19T01:11:15ZengMDPI AGFractal and Fractional2504-31102023-08-017859710.3390/fractalfract7080597A Brief Survey of Paradigmatic Fractals from a Topological PerspectiveJulián Patiño Ortiz0Miguel Patiño Ortiz1Miguel-Ángel Martínez-Cruz2Alexander S. Balankin3SEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, México City 07738, MexicoSEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, México City 07738, MexicoSEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, México City 07738, MexicoSEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, México City 07738, MexicoThe key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi></mrow></semantics></math></inline-formula> which exceeds the topological dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi></mrow></semantics></math></inline-formula>. In this regard, we point out that the constitutive inequality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mo>></mo><mi>d</mi></mrow></semantics></math></inline-formula> can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.https://www.mdpi.com/2504-3110/7/8/597dimension numbersconnectednessconnectivityramificationloopinessdegrees of freedom |
| spellingShingle | Julián Patiño Ortiz Miguel Patiño Ortiz Miguel-Ángel Martínez-Cruz Alexander S. Balankin A Brief Survey of Paradigmatic Fractals from a Topological Perspective Fractal and Fractional dimension numbers connectedness connectivity ramification loopiness degrees of freedom |
| title | A Brief Survey of Paradigmatic Fractals from a Topological Perspective |
| title_full | A Brief Survey of Paradigmatic Fractals from a Topological Perspective |
| title_fullStr | A Brief Survey of Paradigmatic Fractals from a Topological Perspective |
| title_full_unstemmed | A Brief Survey of Paradigmatic Fractals from a Topological Perspective |
| title_short | A Brief Survey of Paradigmatic Fractals from a Topological Perspective |
| title_sort | brief survey of paradigmatic fractals from a topological perspective |
| topic | dimension numbers connectedness connectivity ramification loopiness degrees of freedom |
| url | https://www.mdpi.com/2504-3110/7/8/597 |
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