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This chapter describes how to use intersection and closest-hit shaders to implement real-time visualizations of complex fractals using distance functions. The Mandelbulb and Julia sets are used as examples.

Observing complex 3D fractals, such as Mandelbulb and Mandexbox, and examining their interesting details is very useful in using them for visual effects in media works, 3D arts, etc. However, the observation on a 2D display is different from their actual three-dimensional appearance. In this paper, we used a 3D volumetric display to visualise them in real 3D space. Furthermore, these 3D fractals were visualised on a commercially available stereoscopic display, and the differences in stereoscopic effect were compared.

The aim of this paper is to show the origins of fractal geometry in the mathematical work of Weierstrass, Peano, Julia and Hausdorff, as well as from input from studies on Brownian motion and turbulence in physics by Richardson, Perrin and Kolmogorov. It concludes with a brief review of some of the many applications of fractal geometry in science and technology.

In this paper, we have generalized the Fα-calculus by suggesting Fourier and Laplace transformations of the function with support of the fractals set which are the subset of the real line. Using this generalization, we have found the diffraction fringes from the fractal grating Cantor sets.

Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless; 2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance; and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.

Fractal i0w(1b) (in author’s notation) of five-pointed stars is considered. The initiator of the fractal is a five-pointed star. When building the fractal, the prefractals of previous step are situated so as their centers coincide with the vertices of a five-pointed star. This star is called a “generalized star.” The size of the generalized star increases τ times at each step, where τ ≈ 1.618 is the golden mean. The orientation of the generalized star reverses at each step. A comparison of the internal parts of the fractal with the layers of the Wieringa roof (layers of Penrose tiling) shows their similarity. Computer simulation confirms that the two-dimensional Penrose tiling can be constructed from a set of four types of fractal i0w(1b).

Responsible for the entire third act battle, Weta Digital's work focused on the build and subsequent destruction of the interior of Planet Ego, a complex structural environment based on Mandelbulb fractal. Over 50% of Weta's shots required fx work including the creation of Ego's human form that surrounds his celestial framework - achieved using a clever growth system.

We implement the fractal construction process to the MV-algebra pastings. Pasted MV-algebras generate special types of difference posets (D-posets). We compute the dimension of new fractal D-posets and show that the fractal structures discussed in this paper have a non-integer fractal dimension. Furthermore we observe an interesting agreement between the fractal dimension of the classical linear fractals (Cantor sets, Koch curves) and the dimension of fractal D-posets (Cantor fractal D-posets and Koch fractal D-posets).

Fractal continuum electrodynamics is developed on the basis of a model of three-dimensional continuum Φ D 3 ⊂ E 3 with a fractal metric. The generalized forms of Maxwell equations are derived employing the local fractional vector calculus related to the Hausdorff derivative. The difference between the fractal continuum electrodynamics based on the fractal metric of continua with Euclidean topology and the electrodynamics in fractional space F α accounting the fractal topology of continuum with the Euclidean metric is outlined. Some electromagnetic phenomena in fractal media associated with their fractal time and space metrics are discussed.

In this chapter fractal functions are considered (ie, functions whose graphs are fractal sets and which are generated by certain classes of iterated function systems). The term fractal refers to the fact that the graph of such a function has, in general, a nonintegral dimension. It is shown that these fractal functions may be used for interpolation and approximation purposes, and are in this way analogous to (parameterized) splines.

Dutkay and Jorgensen developed the concept of wavelets on a σ-finite fractal measure space and demonstrated how to construct fractal wavelets on their enlarged fractal spaces using finite-dimensional orthonormal bases by detailing their construction on an enlarged Cantor fractal space. Their fractal wavelet construction was studied further by D'Andrea, Merrill, and Packer who provided details for constructing fractal wavelets on an enlarged Sierpinski gasket space. Motivated by the correspondence between finite-dimensional orthonormal bases with multi resolution analyses on these enlarged fractal spaces, in this paper we provide a construction for wavelet frames on the enlarged fractal spaces of Dutkay and Jorgensen using finite-dimensional frames.

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called Fα-integral, where α is the dimension of F. A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The Fα-integral and Fα-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and Fα-differentiability is generalized. Finally we touch upon an example of absorption along fractal paths, to illustrate the utility of the framework in model making.

This article presents a fun activity of generating a double-minded fractal image for a linear algebra class once the idea of rotation and scaling matrices are introduced. In particular the fractal flip-flops between two words, depending on the level at which the image is viewed.

A hyperhelix is a fractal curve generated by coiling a helix around a rect line, then another helix around the first one, a third around the second… an infinite number of times. A way to generate hyperhelices with any desired fractal dimension is presented, leading to the result that they have embedded an algebraic structure that allows making arithmetic with fractal dimensions and to the idea of an infinitesimal of fractal dimension.

In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.

January/February CGaA artist Paul Griffitts does all his work with a free fractal rendering program called Mandelbulb 3D (MB3D). Especially with Labyrinth, the cover image, he implements various parameters for specific effects. Griffitts says he likes the effects that can be created through various coloring options and the application of different maps-height maps, color maps, light maps.

provide an innovative method for generating 3D images of real-world objects by using computational modelling algorithms based on the imperatives of self- similarity, scale invariance, and dimensionality. Of the many different types of fractals that have come into limelight since their origin, the family of Mandelbrot Set fractals has eluded both mathematicians and computer scientists alike. And the "true" 3D realization of the Mandelbrot set has been a challenging centre piece of research with its limits extending only to that of the sky. An earlier paper co-authored by us in 2011 explained a method of realizing a "true" 3D simulation of the Mandelbrot set and the rendering of the same onto 3- dimensional space. This paper takes a step further in using this variant of the Mandelbulb as input and outlines a method of the game-enabling of the same Mandelbulb by using direction-oriented vectors that are analogous in function to that of Support Vectors in the Support Vector Graphics (SVG) domain. A real-world application of the same can translate to examples of understanding an entire coast-line set to motion in space by adding 3D-animation enabled elevation to the corresponding fractal image.

The Penrose lattice, an example of quasicrystals, is considered as a fractal set. The generation of these sets in the graph tree form is considered. The problem of percolation of informodynamic characteristics and fractal dimensions defined in terms of the information entropy is solved for the constructed graphs.

This paper proves that the Ginzburg-Landan partial differential equation admits an inertial fractal set whose fractal dimension is finite. Furthermore, We produce an exponentially approximating sequence of localizing compact fractal sets and a fractal structure of the attractor.

Analogs of the Lagrange equation for particles evolving in a space of fractal dimension are obtained. Two cases are considered: 1) when the space is formed by a set of material points (a so-called fractal continuum), and 2) when the space is a true fractal. In the latter case the fractional integrodifferential formalism is utilized, and a new principle for devising a fractal theory, viz., a generalized principle of least action, is proposed and used to obtain the corresponding Lagrange equation. The Lagrangians for a free particle and a closed system of interacting particles moving in a fractal continuum are derived.

In this paper, the principle of construction of a fractal surface is introduced, interpolation functions for a fractal interpolated surface are discussed, the theorem of the uniqueness of an iterated function system of fractal interpolated surface is proved, the theorem of fractal dimension of fractal interpolated surface is derived, and the case that practical data are used to interpolate fractal surface is studied.

In this paper, the methods of construction of a fractal surface are introduced, the principle of bivariate fractal interpolation functions is discussed. The theorem of the uniqueness of an iterated function system of bivariate fractal interpolation functions is proved. Moreover, the theorem of fractal dimension of fractal interpolated surface is derived. Based on these theorems, the fractal interpolated surfaces are created by using practical data.

This paper tackles the construction of fractal maps on the unit sphere. The functions defined are a generalization of the classical spherical harmonics. The methodology used involves an iterated function system and a linear and bounded operator of functions on the sphere. For a suitable choice of the coefficients of the system, one obtains classical maps on the sphere. The different values of the system parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of the square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal mappings.

It is demonstrated in this paper that the Euler equation which corresponds to the subdivision of a torus into a polyhedron is a rational source of plane fractals. The vertex orders and polygon edges within a plane fractal are an invariant of the fractal. The process of forming a fractal is intimately related to the topological and infinitely divisible properties of the solutions for the Euler equation. Based on this equation it is shown that only four linear plane fractals are available and the boundary of a plane fractal is a fractal curve, such as the Koch curve. Certain relations between a plane fractal and a fractal formed by the Julia set are also discussed with the aid of a new plane fractal pattern given in this work. As an application of the present analysis, brittle fragmentation of a thin plate is considered. Present results provide a new expression for estimating the average size of a fragment based on an energy balance principle and an idea about quasi-identical plane fractals. Present theoretical analysis is in agreement with experimental results and previous investigations.

The fractal is made up of the union of several copies of itself, each reproduction being transformed by a function system. Iterated function systems or (IFSs) are a method of creating fractals; the resulting structures are always self-similar. Any set of linear maps (affine transformations) and connected set of probabilities determines an Iterated function system. Each IFS has a unique “attractor” which is naturally a fractal set (object). Fractal design of different shapes has gained much recognition in recent years. In this paper two algorithms are given that create pictures of fractals using IFS. Also this paper presents the application of the theory of IFS to generate line fractal, Koch polygon, Sierpinski triangle and Pythagorean tree.

Fractals provide an innovative method for generating 3D images of real-world objects by using computational modelling algorithms based on the imperatives of self-similarity, scale invariance, and dimensionality. Images such as coastlines, terrains, cloud mountains, and most interestingly, random shapes composed of curves, sets of curves, etc. present a multi-varied spectrum of fractals usage in domains ranging from multi-coloured, multi-patterned fractal landscapes of natural geographic entities, image compression to even modelling of molecular ecosystems. Fractal geometry provides a basis for modelling the infinite detail found in nature. Fractals contain their scale down, rotate and skew replicas embedded in them. Many different types of fractals have come into limelight since their origin. This paper explains the generation of a famous category of fractals, namely the Mandelbrot Set and the Mandelbulb, which is the 3D rendering of the Mandelbrot Set that gives a real-world look and feel in the world of fractal images.

Since the advent of Mandelbrot's fractal geometry in 1975, many geological and geophysical patterns have been characterized by their fractal scaling properties. The purpose of this paper is to present and compare standard and new methods used to estimate the fractal dimension of random (natural) fractals. Examples demonstrate the relative limitations of the techniques, and the modifications necessary to more accurately estimate the scaling properties of certain types of fractals. Discussion and examples include branching diffusion-limited aggregates, fracture surfaces, and fractional Brownian surfaces.

The most relevant tools of fractal geometry are introduced, and several classes of mathematical frameworks which generate fractal configurations dynamically (in space-time as well as in parameter spaces) are examined.

In this paper, the conception of fractal geometry derived from geometric inversion is introduced. A ramified self-inverse fractal with symmetry and a self-inverse fractal dust set are constructed. The authors extend the conception of the fractal osculation and propose a new notion—fractal envelope. Finally, two examples of self-inverse fractal (soap and egg) are given.

A fractal tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. In this paper, we investigate the boundary dimension of [Formula: see text]- and [Formula: see text]-tilings. We first derive an explicit recursion formula for the boundary edges of [Formula: see text]- and [Formula: see text]-tilings. Then we present an analytical expression for their fractal boundary dimensions using matrix methods. Results indicate that, as [Formula: see text] increases, the boundaries of [Formula: see text]- and [Formula: see text]-tilings will degenerate into general Euclidean curves. The method proposed in this paper can be extended to compute the boundary dimensions of other kinds of fractal tilings.

The concept of fractal form analysis is introduced, in which fractal metrication is defined for partial objects of fractals as a cmD or b mD with the fractal dimensionality D.

The energy form on a closed fractal curve F is constructed. As F is neither self-similar nor nested, it is regarded as a ”fractal manifold”; the energy is obtained by integrating the Lagrangian on F . 2000 Math. Subj. Class.: Primary 31C25, 28A80; Secondary 58

In this paper fractal procedure of covering surface is represented. Subject of investigations is quasicrystal Penrose mosaic (MP) which has axis of symmetry of the fifth order. Elements of MP generation are pair of golden (g) rhombs divided on with conservation of g-relations. Following generations of MP are obtained by iterative procedure. Cayley trees are built for obtained fractal structures. The percolation problem of fractal dimensionality and entropic functionals is solved for obtained graphs.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Benoit B. Mandelbrot, who advanced the concept of power law scaling as a fundamental property of a broad range of natural processes and patterns in geophysics, economics, mathematics, and virtually all of science, died on 14 October 2010 in Cambridge, Mass., at the age of 85. Mandelbrot, known as the “father of fractal geometry,” was a mathematician who developed the scaling concepts of self-similarity and self-affinity and found examples in spatial, temporal, and size patterns across a broad spectrum of disciplines. He coined the term “fractal” (from the Latin noun “fractus,” meaning fragmented) for shapes and patterns that exhibit self-similarity, meaning that they are statistically scale independent. Such shapes are characterized by fractional power law exponents, between the integer (Euclidean) dimensions. He is best known through his books, including Les Objets Fractals: Forme, Hasard et Dimension; Fractals: Form, Chance and Dimension; The Fractal Geometry of Nature; andMultifractals and 1/f Noise: Wild Self-Affinity in Physics [Mandelbrot, 1975, Mandelbrot 1977, Mandelbrot 1982, Mandelbrot 1999].

A group of fractal image makers claim to have made the best three-dimensional portrayal to date of the Mandelbrot set, the most famous fractal equation

We describe and analyze a parametrization of fractal ‘‘curves’’ (i.e., fractal of topological dimension 1). The nondifferentiability of fractals and their infinite length forbid a complete description based on usual real numbers. We show that using nonstandard analysis it is possible to solve this problem: A class of nonstandard curves (whose standard part is the usual fractal) is defined so that a curvilinear coordinate along the fractal can be built, this being the first step towards the possible definition and study of a fractal space. We mention fields of physics to which such a formalism could be applied in the future.

Mandelbrot's fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension D, a number that agrees with our intuitive notion of dimension but need not be an integer. A brief mathematical charactrization of random fractals is presented with emphasis on variations of Mandelbrot's fractional Brownian motion. The important concepts of fractal dimension and exact and statisical self-similarity and self-affinity will be reviewed. The various methods and difficulties of estimating the fractal dimension and lacunarity from experimental images or point sets are summarized.

Based on the construction of bivariate fractal interpolation surfaces, we introduce closed spherical fractal interpolation surfaces. The interpolation takes place in spherical coordinates and with the transformation to Cartesian coordinates a closed surface arises. We give conditions for this construction to be valid and state some useful relations about the Hausdorff and the Box counting dimension of the closed surface.