{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,28]],"date-time":"2025-11-28T17:21:03Z","timestamp":1764350463382,"version":"3.38.0"},"reference-count":23,"publisher":"University of Toronto Press Inc. (UTPress)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cartographica"],"published-print":{"date-parts":[[2018,9]]},"abstract":"<jats:p> 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 article demonstrates that a smooth curve can be fractal, under a new, relaxed, third definition of fractal \u2013 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-similarity 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. <\/jats:p>","DOI":"10.3138\/cart.53.3.2017-0032","type":"journal-article","created":{"date-parts":[[2018,10,1]],"date-time":"2018-10-01T19:03:20Z","timestamp":1538420600000},"page":"203-210","source":"Crossref","is-referenced-by-count":7,"title":["A Smooth Curve as a Fractal under the Third Definition"],"prefix":"10.3138","volume":"53","author":[{"given":"Ding","family":"Ma","sequence":"first","affiliation":[{"name":"Department of GIScience, Faculty of Engineering and Sustainable Development\u00a0\/ University of G\u00e4vle\u00a0\/ G\u00e4vle\u00a0\/ G\u00e4vleborg\u00a0\/ Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bin","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of GIScience, Faculty of Engineering and Sustainable Development\u00a0\/ University of G\u00e4vle\u00a0\/ G\u00e4vle\u00a0\/ G\u00e4vleborg\u00a0\/ Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1696","reference":[{"key":"r1","doi-asserted-by":"crossref","unstructured":"Bader, M. 2013. Space-Filling Curves: An Introduction with Applications in Scientific Computing. Berlin: Springer.","DOI":"10.1007\/978-3-642-31046-1_7"},{"key":"r2","unstructured":"Batty, M., and P. Longley. 1994. Fractal Cities: A Geometry of Form and Function. London: Academic Press."},{"key":"r3","doi-asserted-by":"crossref","unstructured":"Cattani, C., and A. Ciancio. 2016. \u201cOn the Fractal Distribution of Primes and Prime-Indexed Primes by the Binary Image Analysis.\u201d Physica A: Statistical Mechanics and Its Applications 460: 222\u201329. https:\/\/doi.org\/10.1016\/j.physa.2016.05.013.","DOI":"10.1016\/j.physa.2016.05.013"},{"key":"r4","unstructured":"Chen, Y.G. 2008. Fractal Urban Systems: Scaling, Symmetry, and Spatial Complexity [Chinese]. Beijing: Science Press."},{"key":"r5","doi-asserted-by":"crossref","unstructured":"Chen, Y.G. 2015. \u201cPower-Law Distributions Based on Exponential Distributions: Latent Scaling, Spurious Zipf's Law, and Fractal Rabbits.\u201d Fractals, 23(2): 1550009. https:\/\/doi.org\/10.1142\/s0218348x15500097.","DOI":"10.1142\/S0218348X15500097"},{"key":"r6","doi-asserted-by":"crossref","unstructured":"Chen, Y.G. 2017. \u201cFractal Analysis Based on Hierarchical Scaling in Complex Systems.\u201d In Fractal Analysis \u2013 Applications in Health Sciences and Social Sciences, ed. Fernando Brambila, 141\u201364. Rijeka: InTech.","DOI":"10.5772\/intechopen.68424"},{"key":"r7","doi-asserted-by":"crossref","unstructured":"Clauset, A., C.R. Shalizi, and M.E.J. Newman. 2009. Power-Law Distributions in Empirical Data. SIAM Review 51(4): 661\u2013703. https:\/\/doi.org\/10.1137\/070710111.","DOI":"10.1137\/070710111"},{"key":"r8","doi-asserted-by":"crossref","unstructured":"Douglas, D.H., and T.K. Peucker. 1973. \u201cAlgorithms for the Reduction of the Number of Points Required to Represent a Digitized Line or Its Caricature.\u201d Cartographica 10(2): 112\u201322. https:\/\/doi.org\/10.3138\/fm57-6770-u75u-7727.","DOI":"10.3138\/FM57-6770-U75U-7727"},{"key":"r9","doi-asserted-by":"crossref","unstructured":"Falconer, K. 2003. Fractal Geometry: Mathematical Foundations and Applications, 2nd ed. Chichester, UK: Wiley.","DOI":"10.1002\/0470013850"},{"key":"r10","doi-asserted-by":"crossref","unstructured":"Gao, P., Z. Liu, G. Liu, H. Zhao, and X. Xie. 2017. \u201cUnified Metrics for Characterizing the Fractal Nature of Geographic Features.\u201d Annals of American Association of Geographers 107(6): 1315\u201331, https:\/\/doi.org\/10.1080\/24694452.2017.1310022.","DOI":"10.1080\/24694452.2017.1310022"},{"key":"r11","doi-asserted-by":"crossref","unstructured":"Irving, G., and H. Segerman. 2013. \u201cDeveloping Fractal Curves.\u201d Journal of Mathematics and the Arts 7(3\u20134): 103\u201321. https:\/\/doi.org\/10.1080\/17513472.2013.852399.","DOI":"10.1080\/17513472.2013.852399"},{"key":"r12","doi-asserted-by":"crossref","unstructured":"Jiang, B. 2013. \u201cHead\/Tail Breaks: A New Classification Scheme for Data with a Heavy-Tailed Distribution.\u201d Professional Geographer 65(3): 482\u201394. https:\/\/doi.org\/10.1080\/00330124.2012.700499.","DOI":"10.1080\/00330124.2012.700499"},{"key":"r13","doi-asserted-by":"crossref","unstructured":"Jiang, B. 2015a. \u201cThe Fractal Nature of Maps and Mapping.\u201d International Journal of Geographical Information Science 29(1): 159\u201374. https:\/\/doi.org\/10.1080\/13658816.2014.953165.","DOI":"10.1080\/13658816.2014.953165"},{"key":"r14","doi-asserted-by":"crossref","unstructured":"Jiang, B. 2015b. \u201cHead\/Tail Breaks for Visualization of City Structure and Dynamics.\u201d Cities 43: 69\u201377. https:\/\/10.1016\/j.cities.2014.11.013. Reprinted in Capineri, C., M. Haklay, H. Huang, V. Antoniou, J. Kettunen, F. Ostermann, and R. Purves R. (eds.) 2016. European Handbook of Crowdsourced Geographic Information, 169\u201383. London: Ubiquity Press. https:\/\/doi.org\/10.1016\/j.cities.2014.11.013.","DOI":"10.1016\/j.cities.2014.11.013"},{"key":"r15","doi-asserted-by":"crossref","unstructured":"Jiang, B. 2017. \u201cLine Simplification.\u201d In International Encyclopedia of Geography: People, the Earth, Environment and Technology, ed. D. Richardson, N. Castree, M.F. Goodchild, A. Kobayashi, W. Liu, and R.A. Marston, 4059\u201365. Hoboken, NJ: Wiley. https:\/\/doi.org\/10.1002\/9781118786352.wbieg0005.","DOI":"10.1002\/9781118786352.wbieg0005"},{"key":"r16","doi-asserted-by":"crossref","unstructured":"Jiang, B., and A. Brandt. 2016. \u201cA Fractal Perspective on Scale in Geography.\u201d ISPRS International Journal of Geo-Information 5(6): 95. https:\/\/doi.org\/10.3390\/ijgi5060095.","DOI":"10.3390\/ijgi5060095"},{"key":"r17","doi-asserted-by":"crossref","unstructured":"Jiang, B., X. Liu, and T. Jia. 2013. \u201cScaling of Geographic Space as a Universal Rule for Map Generalization.\u201d Annals of the Association of American Geographers, 103(4), 844\u201355. https:\/\/doi.org\/10.1080\/00045608.2013.765773.","DOI":"10.1080\/00045608.2013.765773"},{"key":"r18","doi-asserted-by":"crossref","unstructured":"Jiang, B., and J. Yin. 2014. \u201cHt-Index for Quantifying the Fractal or Scaling Structure of Geographic Features.\u201d Annals of the Association of American Geographers 104(3): 530\u201341. https:\/\/doi.org\/10.1080\/00045608.2013.834239.","DOI":"10.1080\/00045608.2013.834239"},{"key":"r19","unstructured":"Kaye, B.H. 1989. A Random Walk through Fractal Dimensions. New York: VCH Publishers."},{"key":"r20","unstructured":"Lam, N.S.N., and L.D. Cola. 2002. Fractals in Geography. Caldwell, NJ: Blackburn."},{"key":"r21","unstructured":"Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. New York: Freeman."},{"key":"r22","unstructured":"Thompson, D.W. 1917. On Growth and Form. Cambridge: Cambridge University Press."},{"key":"r23","unstructured":"Zipf, G.K. 1949. Human Behaviour and the Principles of Least Effort. Cambridge, MA: Addison-Wesley."}],"container-title":["Cartographica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/utppublishing.com\/doi\/pdf\/10.3138\/cart.53.3.2017-0032","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T10:41:23Z","timestamp":1741862483000},"score":1,"resource":{"primary":{"URL":"https:\/\/utppublishing.com\/doi\/10.3138\/cart.53.3.2017-0032"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,9]]}},"alternative-id":["10.3138\/cart.53.3.2017-0032"],"URL":"https:\/\/doi.org\/10.3138\/cart.53.3.2017-0032","relation":{},"ISSN":["0317-7173","1911-9925"],"issn-type":[{"type":"print","value":"0317-7173"},{"type":"electronic","value":"1911-9925"}],"subject":[],"published":{"date-parts":[[2018,9]]}}}