On the arithmetic of fractal dimension using hyperhelices

Toledo-Suárez, Carlos D.

doi:10.1016/j.chaos.2007.01.095

Cited by 19 publications

(27 citation statements)

References 1 publication

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“…We shall call the process of replacing a given smooth curve by one that winds around it as helicalising the curve. The curve that is produced after an infinite number of helicalisations would be a fractal [1][2][3]. Notice however that this iterative helicalisation process does not result in a self-similar fractal as the n-th level is not some number of exact scaled copies of the (n − 1)-th level.…”

Section: Introductionmentioning

confidence: 99%

Helicalised fractals

Saw

Chew

2015

Chaos, Solitons & Fractals

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We formulate the helicaliser, which replaces a given smooth curve by another curve that winds around it. In our analysis, we relate this formulation to the geometrical properties of the self-similar circular fractal (the discrete version of the curved helical fractal). Iterative applications of the helicaliser to a given curve yields a set of helicalisations, with the infinitely helicalised object being a fractal. We derive the Hausdorff dimension for the infinitely helicalised straight line and circle, showing that it takes the form of the self-similar dimension for a self-similar fractal, with lower bound of 1. Upper bounds to the Hausdorff dimension as functions of $\omega$ have been determined for the linear helical fractal, curved helical fractal and circular fractal, based on the no-self-intersection constraint. For large number of windings $\omega\rightarrow\infty$, the upper bounds all have the limit of 2. This would suggest that carrying out a topological analysis on the structure of chromosomes by modelling it as a two-dimensional surface may be beneficial towards further understanding on the dynamics of DNA packaging.Comment: 25 pages, 10 figures. v3: Detailed derivation of the Hausdorff dimension included. Accepted by Chaos, Solitons & Fractal

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Section: Introductionmentioning

confidence: 99%

Helicalised fractals

Saw

Chew

2015

Chaos, Solitons & Fractals

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“…Helices and helical structures exist in the structure of proteins and biomolecules . They can be also found in the fractal geometry . Helices are also important for the tool path description, the simulation of kinematic motion, the design of highways, etc.…”

Section: Introductionmentioning

confidence: 99%

“…[5][6][7] They can be also found in the fractal geometry. 8,9 Helices are also important for the tool path description, the simulation of kinematic motion, the design of highways, etc.…”

Section: Introductionmentioning

confidence: 99%

On k‐type null Cartan slant helices in Minkowski 3‐space

Nešović

Öztürk²,

Öztürk

2018

Math Methods in App Sciences

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In this paper, we define k‐type null Cartan slant helices lying on a timelike surface in Minkowski space double-struckE13 according to their Darboux frame, where k ∈ {0,1,2}. We study these helices by using their geodesic curvature, normal curvature , and geodesic torsion. Additionally, we determine their axes and consider the special cases when the mentioned helices are geodesic curves and principal curvature lines lying on the timelike surface in double-struckE13. Furthermore, we obtain some interesting relations between 0‐, 1‐, and 2‐type null Cartan slant helices.

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“…Lancret expressed the result, which describes helices, a curve is a helix if and only if the ratio / is constant in 1802, and Saint Venant proved it in 1845. There are many highly interesting applications of helices such as helical structures in fractal geometry, DNA spirals, collagen triple helix and the helical ladder structures in architecture [2][3][4][5]. Izumiya and Takeuchi defined Slant helices.…”

Section: Introductionmentioning

confidence: 99%

New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space

Aydemir

Yerlikaya

2015

Geometry

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We obtained a new representation for timelike Bertrand curves and their Bertrand mate in 3-dimensional Minkowski space. By using this representation, we expressed new representations of spherical indicatricies of Bertrand curves and computed their curvatures and torsions. Furthermore in case the indicatricies of a Bertrand curve are slant helices, we investigated some new characteristic features of these curves.

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On the arithmetic of fractal dimension using hyperhelices

Cited by 19 publications

References 1 publication

Helicalised fractals

Helicalised fractals

On k‐type null Cartan slant helices in Minkowski 3‐space

New Representations of Spherical Indicatricies of Bertrand Curves in Minkowski 3-Space

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