Two definitions of fractional dimension

Tricot, Claude

doi:10.1017/s0305004100059119

Cited by 345 publications

(297 citation statements)

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“…To deWne the dimension of a structure, it is necessary for it to include boxes of every possible distance between two points. This requirement is always observed in nature (Gouyet 1992) and allows the fractal dimension of an object to be deWned (Tricot 1982). Fractional dimension provides a measurement of roughness of fractal curves: lines have a dimension of one, surfaces a dimension of two, and solid bodies a dimension of three.…”

Section: Introductionmentioning

confidence: 99%

Use of fractal dimensions to quantify coral shape

et al. 2007

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A morphometrical method to quantify and characterize coral corallites using Richardson Plots and Kaye's notion of fractal dimensions is presented. A Jurassic coral species (Aplosmilia spinosa) and Wve Recent coral species were compared using the Box-Counting Method. This method enables the characterization of their morphologies at calicular and septal levels by their fractal dimensions (structural and textural). Moreover, it is possible to determine diVerences between species of Montastraea and to tackle the high phenotypic plasticity of Montastraea annularis. The use of fractal dimensions versus conventional methods (e.g., measurements of linear dimensions with a calliper, landmarks, Fourier analyses) to explore a rugged boundary object is discussed. It appears that fractal methods have the potential to considerably simplify the morphometrical and statistical approaches, and be a valuable addition to methods based on Euclidian geometry.

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Section: Introductionmentioning

confidence: 99%

Use of fractal dimensions to quantify coral shape

et al. 2007

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“…See [14] for proofs of (1) and (2) and [12] for (3) and (4). All of the above inequalities in this section be strict.…”

Section: Theorem 32 (Dimension Inequalitiesmentioning

confidence: 99%

Cubes and their centers

Thornton

2017

Acta Math. Hungar.

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Abstract. We study the relationship between the sizes of sets B, S in R n where B contains the k-skeleton of an axes-parallel cube around each point in S, generalizing the results of Keleti, Nagy, and Shmerkin [6] about such sets in the plane. We find sharp estimates for the possible packing and boxcounting dimensions for B and S. These estimates follow from related cardinality bounds for sets containing the discrete skeleta of cubes around a finite set of a given size. The Katona-Kruskal Theorem from hypergraph theory plays an important role. We also find partial results for the Haussdorff dimension and settle an analogous question for the dual polytope of the cube, the orthoplex. Introduction and Statements of Results 1.1.Introduction. In [6] the authors find sharp bounds for the Hausdorff, boxcounting, and packing dimensions of sets S, B ⊆ R 2 where B contains either the vertices or boundary of an axes-parallel square around every point in S, and cardinality bounds for finite sets satisfying discrete versions of these conditions. Their results are summarized in the following table. If S has size s for the given notion of size, then a sharp lower bound for the size of B is given in terms of s: Notion of SizeVertex problem Boundary problem (0-skeleton of a 2-cube) (1-skeleton of a 2-cube)

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“…The work above relies completely on the fact that we pack by closed balls. As noted in the introduction, this follows recent practice, however in Taylor and Tricot's original definition of packing measure in [11] (see also [12]), open balls were used. If the Hausdorff function h is left-continuous, then packing measure using open balls is Borel regular, since with this definition it is easy to prove that the pre-packing measure P h 0 is the same for a set as for its closure, and therefore that each set is contained in an F σδ set of the same measure.…”

Section: The Constructionmentioning

confidence: 99%