Fractal-like networks effectively endow life with an additional fourth spatial dimension. This is the origin of quarter-power scaling that is so pervasive in biology. Organisms have evolved hierarchical branching networks that terminate in size-invariant units, such as capillaries […] Natural selection has tended to maximize both metabolic capacity, by maximizing the scaling of exchange surface areas, and internal efficiency, by minimizing the scaling of transport distances and times. These design principles are independent of detailed dynamics and explicit models and should apply to virtually all organisms.~ Abstract of “The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms” (West, Brown, and Enquist).
Living cities, like organisms, demonstrate certain fractal organization patterns:
- Hierarchical street structure; the narrowest lanes run into narrow streets, which feed into wider streets, etc… (self-similarity).
- A wide range of street widths, with the very narrowest streets accounting for the greatest total length.
These patterns are easily observed in Seoul, in Tokyo, and in the pre-industrial cores of European cities. The opposite is seen in cities surveyed and parceled in advance. These grid cities generally demonstrate a minimum of hierarchy in their street structure (usually no more than 2 tiers of street width).
Quantitatively speaking:
- By analogy with living organisms, we expect each tier of streets to account for a total length equal to 1.41 times the total length of the next wider tier (assuming the branching number, n, is equal to 2).
- Similarly, we expect each tier of streets to be 0.63 (0.79 squared) times the width of the next wider tier (or 1.59 times the width of the next narrow tier).
Further Reading
MathBench gives “simple” explanations of quarter-power scaling and branching networks.
These three papers are of interest to the fractal analysis of urban street structures:
- This paper gives a very simple method of calculating fractal dimension: http://www.urbanform.org/online_unlimited/pdf2005/200592_95-107.pdf
- This paper (by Elena Rodina et al) mocks the ideas that fractals represent chaos: “…visual chaos of Tokyo’s street patterns contains the hidden order that can be measured”: http://www.spacesyntax.net/symposia-archive/SSS4/fullpapers/45RodinaRodinDumachpaper.pdf
- This paper is based on the Tokyo paper: http://www.urbanform.org/online_unlimited/pdf2005/200592_95-107.pdf
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“One thing a minimum standard is sure to produce is uniformity. The 19th-century by-laws were such an advance that it is not surprising that no one went beyond them. The result is the uniformity and drabness of the English industrial city (Fig. 5a). Unlike the Georgian builder, who composed his terrace into a unit, focused it on a square, or varied the width of the streets, the Victorian speculative builder merely reproduced a standard unit with as few breaks as the by-laws would allow. Mumford has compared this repetition to the machine which so dominated the age”
~Emrys Jones, “Towns & Cities”(Oxford University Press, 1966)
[It’s worth noting that the English city shown in Jones’s Fig. 5a is not nearly as uniform as the section of Chicago shown in Fig. 5b!]
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