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# 2: Ray tracing. 3: The Sierpinski Pyramid.

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Visually complex objects with infinitesimally fine features, naturally call for mathematical representations. The geometrical property of self-similarity - the whole similar to its parts - when iterated to infinity generates such features. Finite sets of affine contractions called Iterated Function Systems (IFS), with their compact attractors IFS...

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... The space of possible parametric families of tipset connected n-ary complex trees is incredibly vast and rich. Nonetheless we believe that the theoretical basis set in this work will contribute to the study of fractal dendrites [33], topological spaces admitting a unique fractal structure [34] [35], and the geometry and dimension of fractals [36] [37] [38]. We also believe that the notion of complex tree can be extended to hypercomplex spaces in three and higher dimensions [39] [40]. ...
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The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected self-similar sets Fₐ(z). In order to study topological changes of Fₐ(z) in regions R⊂ℂ where these families are defined, we introduce a new kind of set M⊆R which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set M₀⊆M related to a special type of connectivity for which we provide a theorem. Among other things, the present theory provides a unified framework to families of self-similar sets traditionally studied as separate with elements Fₐ(z) disconnected for parameters z∈R∖M.
... The space of possible parametric families of tipset connected n-ary complex trees is incredibly vast and rich. Nonetheless we believe that the theoretical basis set in this work will contribute to the study of fractal dendrites [42] [54], topological spaces admitting a unique fractal structure [7] [21], and the geometry and dimension of fractals [4] [26] [52]. We also believe that the notion of complex tree can be extended to hypercomplex spaces in three and higher dimensions [24] [27]. ...
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Advanced Mathematics Master Thesis, Universitat de Barcelona http://hdl.handle.net/2445/129385 The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets which includes Cantor sets, Koch curves, Lévy curves, Sierpiński gaskets, Rauzy fractals, and fractal dendrites. We note a fundamental dichotomy for n-ary complex trees that allows us to study topological changes in regions $\mathcal{R}$ where one-parameter families of connected self-similar sets are defined. Moreover, we show how to obtain these families from systems of equations encoded by tip-to-tip equivalence relations. As far as we know, these families and the sets $M , M_{0}$, and $\mathcal{K}$ that we introduce to study $\mathcal{R}$ are new. We provide a theorem, and a necessary condition, for certifying if a given tipset (self-similar set associated to a complex tree) is a fractal dendrite. We highlight a special class of totally connected tipsets that we call root-connected. And we provide a pair of theorems related to them. For a given one-parameter family we also define the set of root-connected trees $M_{0}$ which presents an asymptotic similarity between its boundary and their associated tipsets. By adapting the notion of post-critically finite self-similar set (p.c.f. for short), the open set condition, and the Hausdorff dimension, we arrive to an upper bound for the existence of p.c.f. trees in a given one-parameter family. We also provide a theorem that allows us to discard non-p.c.f. trees just by looking at some local properties. In relation to this theorem, we set a conjecture of an interesting observation that has been consistent in numerous computational experiments. The space of one-parameter families of tipset-connected complex trees has just begun to be explored. For the family $TS(z) := T \{z, 1/2, 1/4z\}$ we prove that there is a pair of regions contained in the set $\mathcal{K}$ with a piece-wise smooth boundary. We show that this piece-wise smooth boundary is a rather exceptional case by considering a closely related family, $T S(z) := T {z, -1/2, 1/4z}$. Finally we indicate how the general framework works for one-parameter families with non-fixed mirror-symmetric trees.
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