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A smooth curve as a fractal under the third definition
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Industrial Development, IT and Land Management, Land management, GIS.ORCID iD: 0000-0001-9328-9584
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Industrial Development, IT and Land Management, Land management, GIS.ORCID iD: 0000-0002-2337-2486
2018 (English)In: Cartographica, ISSN 0317-7173, E-ISSN 1911-9925, Vol. 53, no 3, p. 203-210Article in journal (Refereed) Published
Abstract [en]

It is commonly believed in the literature that smooth curves, such as circles, are not fractal, and only non-smooth curves, such as coastlines, are fractal. However, this paper demonstrates that a smooth curve can be fractal, under the new, relaxed, third definition of fractal – a set or pattern is fractal if the scaling of far more small things than large ones recurs at least twice. The scaling can be rephrased as a hierarchy, consisting of numerous smallest, a very few largest, and some in between the smallest and the largest. The logarithmic spiral, as a smooth curve, is apparently fractal because it bears the self-similar property, or the scaling of far more small squares than large ones recurs multiple times, or the scaling of far more small bends than large ones recurs multiple times. A half-circle or half-ellipse and the UK coastline (before or after smooth processing) are fractal, if the scaling of far more small bends than large ones recurs at least twice.

Abstract [fr]

Il est généralement convenu dans les écrits que les courbes douces, comme les cercles, ne sont pas fractales, et que seules les courbes qui ne sont pas douces, comme les littoraux, sont fractales. Les auteurs montrent toutefois qu'une courbe douce peut être fractale, en vertu d'une troisième définition, nouvelle et élargie, du terme fractal — un ensemble ou un motif est fractal si l'échelle d'un nombre beaucoup plus grand de petits éléments que de grands se répète au moins deux fois. L'échelle peut être interprétée comme étant la hiérarchie, soit un grand nombre d'éléments très petits, très peu d'éléments très grands, et des éléments se situant entre les plus petits et les plus grands. La spirale équangulaire, à titre de courbe douce, est en apparence fractale du fait qu'elle affiche la propriété d'autosimilitude, ou du fait que l'échelle d'un nombre beaucoup plus grand de petits carrés que de grands se répète plusieurs fois, ou l'échelle d'un nombre beaucoup plus grand de petite courbures que de grandes se répète plusieurs fois. Un demi-cercle ou une demi-ellipse et le littoral du Royaume-Uni (avant ou après lissage) sont fractals si l'échelle d'un nombre beaucoup plus grand de petites courbures que de grandes se répète au moins deux fois.

Place, publisher, year, edition, pages
2018. Vol. 53, no 3, p. 203-210
Keywords [en]
Third definition of fractal, head/tail breaks, bends, ht-index, scaling hierarchy
Keywords [fr]
courbures, échelle, hiérarchie, indice h-t, ruptures de tête ou de queue, troisième définition de fractal
National Category
Other Engineering and Technologies
Identifiers
URN: urn:nbn:se:hig:diva-26164DOI: 10.3138/cart.53.3.2017-0032OAI: oai:DiVA.org:hig-26164DiVA, id: diva2:1183544
Available from: 2018-02-18 Created: 2018-02-18 Last updated: 2018-10-15Bibliographically approved
In thesis
1. Topological and Scaling Analysis of Geospatial Big Data
Open this publication in new window or tab >>Topological and Scaling Analysis of Geospatial Big Data
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Geographic information science and systems face challenges related to understanding the instinctive heterogeneity of geographic space, since conventional geospatial analysis is mainly founded on Euclidean geometry and Gaussian statistics. This thesis adopts a new paradigm, based on fractal geometry and Paretian statistics for geospatial analysis. The thesis relies on the third definition of fractal geometry: A set or pattern is fractal if the scaling of far more small things than large ones recurs multiple times. Therefore, the terms fractal and scaling are used interchangeably in this thesis. The new definition of fractal is well-described by Paretian statistics, which is mathematically defined as heavy-tailed distributions. The topology of geographic features is the key prerequisite that enables us to see the fractal or scaling structure of the geographic space. In this thesis, topology refers to the relationship among meaningful geographic features (such as natural streets and natural cities).

The thesis conducts topological and scaling analyses of geographic space and its involved human activities in the context of geospatial big data. The thesis utilizes the massive, volunteered, geographic information coming from LBSM platforms, which are the global OpenStreetMap database and countrywide, geo-referenced tweets and check-in locations. The thesis develops geospatial big-data processing and modeling techniques, and employs complexity science methods, including heavy-tailed distribution detection and head/tail breaks, along with some complex network analysis. Head/tail breaks and the induced ht-index are a powerful tool for geospatial big-data analytics and visualization. The derived scaling hierarchies, power-law metrics, and network measures provide quantitative insights into the heterogeneity of geographic space and help us understand how it shapes human activities at city, country, and world scales. 

Place, publisher, year, edition, pages
Gävle: Gävle University Press, 2018. p. 73
Series
Studies in the Research Profile Built Environment. Doctoral thesis ; 7
Keywords
Third definition of fractal, scaling, topology, power law, head/tail breaks, ht-index, complex network, geospatial big data, natural cities, natural streets
National Category
Computer and Information Sciences Earth and Related Environmental Sciences
Identifiers
urn:nbn:se:hig:diva-26197 (URN)978-91-88145-24-6 (ISBN)978-91-88145-25-3 (ISBN)
Public defence
2018-05-16, Lilla Jadwiga-salen, Kungsbäcksvägen 47, Gävle, 10:00 (English)
Opponent
Supervisors
Available from: 2018-04-24 Created: 2018-03-04 Last updated: 2018-04-25

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Ma, DingJiang, Bin

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