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AbstractWe introduce a fractal operator on \(\mathcal {C}[0,1]\) which sends a function \(f \in \mathcal {C}(I)\) to fractal version of f where fractal version of f is a super fractal interpolation function corresponding to a countable data system. Furthermore, we study the continuous dependence of super fractal interpolation functions on the parameters used in the construction. We know that the invariant subspace problem and the existence of a Schauder basis gained lots of attention in the literature. Here, we also show the existence of non-trivial closed invariant subspace of the super fractal operator and the existence of fractal Schauder basis for \(\mathcal {C}(I)\). Moreover, we can see the effect of the composition of Riemann-Liouville integral operator and super fractal operator on the fractal dimension of continuous functions. We also mention some new problems for further investigation. Access this article Log in via an institution Subscribe and save Get 10 units per month Download Article/Chapter or eBook 1 Unit = 1 Article or 1 Chapter Cancel anytime Subscribe now Buy Now Price excludes VAT (USA) Tax calculation will be finalised during checkout. Instant access to the full article PDF. Similar content being viewed by others Data availabilityData sharing not applicable to this article as no data sets were generated or analyzed during the current study.ReferencesBarnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)Article MathSciNet MATH Google Scholar Barnsley, M.F.: Fractals Everywhere. Academic Press, Orlando (1988)MATH Google Scholar Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. J. Approx. Theory. 57(1), 14–34 (1989)Article MathSciNet MATH Google Scholar Barnsley, M.F.: Fractals Super. Cambridge University Press, Cambridge (2006) Google Scholar Beer, G.: Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance. Proc. Amer. Math. Soc. 95, 653–658 (1985)Article MathSciNet MATH Google Scholar Chandra, S., Abbas, S.: The calculus of bivariate fractal interpolation surfaces. Fractals 29(03), 2150066 (2021)Article MATH Google Scholar Chandra, S., Abbas, S.: Analysis of fractal dimension of mixed Riemann-Liouville integral, Numerical Algorithms. (2022)Falconer, K.J.: Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc, New York (1990)MATH Google Scholar Gowrisankar, A., Uthayakumar, R.: Fractional calculus on fractal interpolation for a sequence of data with countable iterated function system. Mediterr J. Math. 13, 3887–3906 (2016)Article MathSciNet MATH Google Scholar Jachymski, J.: Continuous dependence of attractors of iterated function systems. J. Math. Anal. Appl. 198, 221–226 (1996)Article MathSciNet MATH Google Scholar Kapoor, G.P., Prasad, S.A.: Super fractal interpolation functions. Int. J. Nonlinear Sci. 19(1), 20–29 (2015)MathSciNet MATH Google Scholar Kapoor, G.P., Prasad, S.A.: Convergence of cubic spline super fractal interpolation functions. Fractals 22(1,2), 7 (2014)MATH Google Scholar Liang, Y.S.: Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation. Nonlinear Anal. 72(11), 4304–4306 (2010)Article MathSciNet MATH

Generate a Hilbert Sequence Walk the Hilbert fractal and enumerate its coordinates.Generate a Peano Sequence Walk the Peano fractal and enumerate its coordinates.Generate a Moore Sequence Walk the Moore fractal and enumerate its coordinates.Generate a Hilbert String Encode the Hilbert fractal as a string.Generate a Peano String Encode the Peano fractal as a string.Generate a Moore String Encode the Moore fractal as a string.Generate a Cantor String Encode the Cantor set as a string.Generate a Dragon String Encode the Heighway Dragon as a string.Generate a Sierpinski String Encode the Sierpinski fractal as a string.Sierpinski Pyramid Generate a Sierpinski tetrahedron (tetrix) fractal.Cantor's Cube Generate a Cantor's cube fractal.Menger Sponge Generate a Sierpinski-Menger fractal.Jerusalem Cube Generate a Jerusalem cube fractal.Mosely Snowflake Generate a Jeaninne Mosely fractal.Mandelbrot Tree Generate a Mandelbrot tree fractal.Barnsey's Tree Generate a Barnsley's tree fractal.Barnsey's Fern Generate a Barnsley's fern fractal.Binary Fractal Tree Generate a binary tree fractal.Ternary Fractal Tree Generate a ternary tree fractal.Dragon Fractal Tree Generate a dragon tree fractal.De Rham Fractal Generate a de Rham curve.Takagi Fractal Generate a Takagi-Landsberg fractal curve.Peano Pentagon Generate a Peano pentagon fractal curve.Tridendrite Fractal Generate a tridendrite fractal curve.McWorter's Pentigree Generate a Pentigree fractal curve.McWorter's Lucky Seven Generate a lucky seven fractal curve.Eisenstein Fractions Generate an Eisenstein fractions fractal curve.Bagula Double V Generate a Bagula double five fractal curve.Julia Set Generate a Julia fractal set.Mandelbrot Set Generate a Mandelbrot fractal set.Mandelbulb Fractal Generate a Mandelbulb fractal.Mandelbox Fractal Generate a Mandelbox fractal.Buddhabrot Fractal Generate a Buddhabrot fractal.Burning Ship Fractal Generate a Burning Ship fractal.Toothpick Fractal Generate a toothpick sequence fractal.Ulam-Warburton Fractal Generate an Ulam-Warburton fractal curve.ASCII Fractal Generate an ASCII fractal.ANSI Fractal Generate an ANSI fractal.Unicode Fractal Generate a Unicode fractal.Emoji Fractal Generate an emoji fractal.Braille Fractal Generate a braille code fractal.Audio Fractal Generate a fractal in audio form.Draw a Pseudofractal

We’re sorry, something doesn't seem to be working properly. Please try refreshing the page. If that doesn't work, please contact support so we can address the problem. AbstractFractal geometry has unique advantages for a broad class of modeling problems, including natural objects and patterns. This paper presents an approach to the construction of fractal surfaces by triangulation. After introducing the notion of iterated function systems (IFSs), we prove theoretically that the attractors of this construction are continuous fractal interpolation surfaces (FISs). Two fast, parallel, and iterative algorithms are also provided. Several experiments in natural phenomena simulation verify that this method is suitable for generating complex 3D shapes with self-similar patterns. Access this article Log in via an institution Subscribe and save Get 10 units per month Download Article/Chapter or eBook 1 Unit = 1 Article or 1 Chapter Cancel anytime Subscribe now Buy Now Price excludes VAT (USA) Tax calculation will be finalised during checkout. Instant access to the full article PDF. Similar content being viewed by others Explore related subjects Discover the latest articles, news and stories from top researchers in related subjects. ReferencesBarnsley MF (1986) Fractal function and interpolation. Constr Approx 2:303–329 Google Scholar Barnsley MF (1988) Fractals everywhere. Academic Press, New York Google Scholar Demko S, Hodges L. Naylor B (1985) Construction of fractal objects with iterated function systems. Comput Graph 19:271–278 Google Scholar Falconer K (1990) Fractal geometry, mathematical foundations and applications. Wiley, New York Google Scholar Fournier A, Fussell D, Carpenter L (1982) Computer rendering of stochastic models. Commun ACM 25:371–384 Google Scholar Geronimo JS, Hardin D (1993) Fractal interpolation surfaces and a related 2D nuetiresolution analysis. J Math Anal Appl 176:561–586 Google Scholar Hart JC, Lescinsky GW, Sandin DJ, DeFanti TA, Kauffman LH (1993) Scientific and artistic investigation of multidimensional fractals on the AT&T pixel machine. Visual Comput 9:346–355 Google Scholar Lewis JP (1987) Generalized stochastic subdivision. ACM Trans graph 6:167–190 Google Scholar Mandelbrot BB (1982) The fractal geometry of nature. Freeman, New York Google Scholar Massopust PR (1990) Fractal surfaces. J Math Anal Appl 151:275–290 Google Scholar Miyata K (1990) A method of generating stone wall patterns. Comput Graph 24:387–394 Google Scholar Nailiang Z, Yiwen J, Sijie L (1991) An approach to the synthesis of realistic terrain. In: Staudhammer J, Qunsheng P (ed) Proceedings of CAD/Graphics '91, International Academic Publishers, Beijing, pp 31–35 Google Scholar Oppenheimer PE (1986) Real-time design and animation of

It's been a few days since my last post because honestly, after understanding the basics behind what generates a fractal, especially the Mandelbrot, the next inevitable step for me was to download as many different Fractal programs as I could and start experimenting :) ... It has been a virtual mushroom trip, to say the least.For now though, let me stick to Fractal eXtreme. Such a nifty little program! So much more to it than one initially thinks... You've probably played around with it a bit yourself already but for the sake of being complete, I'll start at the beginning.The first obvious thing is that you need to do is choose a Set when the program opens. It's default is the standard and much loved Mandelbrot set, but you can choose from many others.Listed below the Mandelbrot are more Mandelbrots using different powers in their formulas. As it explains in the program, the higher the exponent, the more nodes the Mandelbrot has (always one less node than the power).There's also an option called Mandelbrot Arbitrary Power, which is a lot of fun. You know that the normal Mandelbrot set has the function f(z)=z^2 + c behind it. Well, with the Arbitrary Set, you can set the exponent to any real number you want. The resulting fractals can be out of this world.Then, just when you thought the Arbitrary Power was cool, along comes: The Mandelbrot Complex Power ... That's right: z^(some complex number) + c ... Instead of jading you to the adjectives 'incredible' and 'amazing', let me show you. Examples to follow of selected Mandelbrots of which I've spoken about so far.Mandelbrot normal exponent changes :Standard Mandelbrot SetMandelbrot^3 [ f(z)=z^3+c ]Mandelbrot^8 [ f(z)=z^8 + c ]Mandelbrot^3.5Mandelbrot^2.5Mandelbrot^1.7Complex Power changes:Mandelbrot^(8,1.73i)Mandelbrot^(3.1,2.5i)Mandelbrot^(2.08,0.36i)One thing you'll notice with making changes to the exponent in these ways is that, the higher the exponent, the longer it takes for the program to render a good-looking image, especially the more you zoom in. But you don't need to zoom in very far to discover really beautiful fractals. Go ahead and try some of the different Mandelbrots, experiment with colours, etc. To change the Arbitrary and Complex powers once you've loaded the default, you need to go to Options > Plug-in Setup.And there you have it :) Hope you're having fun :) ... Fractal eXtreme has a few other very interesting options for creating new Fractals (Auto Quadratic, the "Hidden Mandelbrot", Barnsley 1, 2 and 3, Classic and Complex Newton, and Nova/NovaM), but those I'll show you in the next post.