scholarly journals TURTLE GRAPHICS OF MORPHIC SEQUENCES

Fractals ◽  
2016 ◽  
Vol 24 (01) ◽  
pp. 1650009
Author(s):  
HANS ZANTEMA
Keyword(s):  
Fixed Points ◽  
Space Filling ◽  
Koch Curve ◽  
Drawing Direction ◽  
Sequence Elements ◽  
Self Similar ◽  
Unit Segment ◽  
Exact Relationship ◽  
Turtle Graphics ◽  
Infinite Sequences

The simplest infinite sequences that are not ultimately periodic are pure morphic sequences: fixed points of particular morphisms mapping single symbols to strings of symbols. A basic way to visualize a sequence is by a turtle curve: for every alphabet symbol fix an angle, and then consecutively for all sequence elements draw a unit segment and turn the drawing direction by the corresponding angle. This paper investigates turtle curves of pure morphic sequences. In particular, criteria are given for turtle curves being finite (consisting of finitely many segments), and for being fractal or self-similar: it contains an up-scaled copy of itself. Also space-filling turtle curves are considered, and a turtle curve that is dense in the plane. As a particular result we give an exact relationship between the Koch curve and a turtle curve for the Thue–Morse sequence, where until now for such a result only approximations were known.

scholarly journals SPACE-FILLING CURVES OF SELF-SIMILAR SETS (III): SKELETONS

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050028
Author(s):  
HUI RAO ◽  
SHU-QIN ZHANG
Keyword(s):  
Finite Type ◽  
Type Condition ◽  
Space Filling ◽  
Open Set Condition ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Neighbor Graph ◽  
Open Set ◽  
Self Similar

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

scholarly journals Space-filling curves of self-similar sets (II): edge-to-trail substitution rule

Nonlinearity ◽  
2019 ◽  
Vol 32 (5) ◽  
pp. 1772-1809 ◽  
Author(s):  
Xin-Rong Dai ◽  
Hui Rao ◽  
Shu-Qin Zhang
Keyword(s):  
Space Filling ◽  
Substitution Rule ◽  
Space Filling Curves ◽  
Self Similar

scholarly journals SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850022 ◽  
Author(s):  
D. V. STÄGER ◽  
H. J. HERRMANN
Keyword(s):  
Three Dimensional ◽  
Fractal Dimensions ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Sphere Packings ◽  
Space Filling ◽  
Contact Networks ◽  
Four Dimensions ◽  
Inversive Geometry ◽  
Self Similar

Inversive geometry can be used to generate exactly self-similar space-filling sphere packings. We present a construction in two dimensions and generalize this construction to search for packings in higher dimensions. We discover 29 new three-dimensional topologies of which 10 are bearings, and 13 new four-dimensional topologies of which five are bearings. To characterize the packing topologies, we estimate numerically their fractal dimensions and we analyze their contact networks.

Dimension of the generalized 4-corner set and its projections

2011 ◽  
Vol 32 (4) ◽  
pp. 1190-1215 ◽  
Author(s):  
BALÁZS BÁRÁNY
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Iterated Function Systems ◽  
Common Fixed Points ◽  
New Method ◽  
Dimension Theory ◽  
Box Dimension ◽  
Extended Version ◽  
Iterated Function ◽  
Self Similar

AbstractIn the last two decades, considerable attention has been paid to the dimension theory of self-affine sets. In the case of generalized 4-corner sets (see Figure 1), the iterated function systems obtained as the projections of self-affine systems have maps of common fixed points. In this paper, we extend our result [B. Bárány. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59], which introduced a new method of computation of the box and Hausdorff dimensions of self-similar families where some of the maps have common fixed points. The extended version of our method presented in this paper makes it possible to determine the box dimension of the generalized 4-corner set for Lebesgue-typical contracting parameters.

scholarly journals MICRO AND MACRO FRACTALS GENERATED BY MULTI-VALUED DYNAMICAL SYSTEMS

Fractals ◽  
2014 ◽  
Vol 22 (04) ◽  
pp. 1450012 ◽  
Author(s):  
T. BANAKH ◽  
N. NOVOSAD
Keyword(s):  
Dynamical Systems ◽  
Topological Space ◽  
Fixed Points ◽  
Metric Space ◽  
Large Scale ◽  
Fractal Structure ◽  
Complete Metric Space ◽  
Koch Curve ◽  
Sierpinski Triangle ◽  
Sierpiński Carpet

Given a multi-valued function Φ : X ⊸ X on a topological space X we study the properties of its fixed fractal[Formula: see text], which is defined as the closure of the orbit Φω(*Φ) = ⋃n∈ωΦn(*Φ) of the set *Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals [Formula: see text] and [Formula: see text] for a contracting compact-valued function Φ : X ⊸ X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.

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