# A Map Is a Living Structure with the Recurring Notion of Far More Smalls than Larges

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## Abstract

**:**

[Maps] are no longer merely considered as aids…, but as products of scientific research which, being complete in themselves, convey their message by means of their own signs and symbols and through these furnish the basis for further geographic deduction. … [the subjectivity] must not predominate: the dictates of science will prevent any erratic flight of the imagination and impact to the map a fundamentally objective character in spite of all subjective impulses.Max Eckert (1908)

## 1. Introduction

## 2. The State of the Art of Maps and Mapping

**Figure 1.**(Color online) Geometric primitives versus geometrically meaningful entities. (Note: A street network is represented as a set of junctions or street segments (geometric primitives, which are not centers) (

**a**), whereas it is more correctly perceived as a collection of named streets (geometrically meaningful entities, which are centers) (

**b**), each of which is colored as one of the four hierarchical levels: blue for the least connected streets, red for the most connected street (only one), and yellow and turquoise for those between the most and the least connected. A curvilinear feature is usually represented as a set of line segments (geometric primitives, which are not centers) (

**c**), but it is more correctly perceived as a collection of far more small bends than large ones (geometrically meaningful entities, which are centers) (

**d**), because the notion of far more small bends than large ones occurs twice: (1) x

_{1}+ x

_{2}+ x

_{3}> x

_{4}+ x

_{5}+ x

_{6}+ x

_{7}, and (2) x

_{1}> x

_{2}+ x

_{3}).

## 3. Living Structure of Centers and Its Two Fundamental Laws

**Figure 2.**(Color online) The ten fictitious cities and their interrelationship constitute a living structure. (Note: As a structural invariant of the central place theory model (Christaller 1933) [5], the cluster of the ten cities (

**a**) is composed of the largest city (

**b**) bounded by the red square, surrounded by two middle-sized cities (

**c**) separated by the green line and bounded by the green box, and further surrounded by seven smallest cities (

**d**) separated by blue lines and bounded by the blue box, thus with three hierarchical levels, indicated by dot sizes and colors. Because of mutual relationship among the ten cities (

**e**), each city has different degree of life, as indicated by the dot sizes (

**f**)).

## 4. Mapping as the Head/Tail Breaks Process: Data Classification and Map Generalization

_{1}is followed by the two middle-sized bends x

_{2}and x

_{3}, and the four smallest bends x

_{4}, x

_{5}, x

_{6}, and x

_{7}. From the point of view of the recursively defined bends, there are three inherent hierarchical levels. Thus, the simplification of the line can be carried out in a similar manner as that of the Koch curve. However, there is one potential problem in the course of line simplification or generalization: The simplified curve may create intersections either with the curve itself or with other geographic features, the so-called “self-intersection” or “intersection with others”, which produces topologically incorrect geographic features. The solution to this problem is simple (Jiang 2017 [39] and related references therein) whenever intersections occur with a simplified curve, that part of the curve has to go back to the previous iteration, or a few trivial points have to be added to avoid the intersection, but all other parts without conflicts remain unchanged.

## 5. Further Discussions on Living Structure for Maps and Art

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Illustration of the head/tail breaks as an iterative function. (Note: The data as a whole is recursively divided into the head (for those greater than the average) and the tail (for those less than the average). The whole or data is seen as an iterated system, i.e., the head of the head of the head and so on. For the sake of simplicity, we illustrate three iterations or four classes).

**Figure 4.**Generation (

**a**) and generalization (

**b**) of the Koch curve with the first four iterations (Note: Beginning with a segment of scale 1 (n = 0), it is divided into thirds, and the middle third is replaced by two equilaterals of a triangle, leading to four segments of scale 1/3 (n = 1). This division and replacement process continues for scales 1/9, and 1/27, leading respectively to 16 segments, and 64 segments (n = 2, and 3). This is the generation of the Koch curve, as shown in panel (

**a**). On the other hand, the Koch curve (Level 0) can be generalized in a step-by-step fashion, as shown in Table 3, resulting in the outcome in panel (

**b**)).

**Figure 5.**(Color online) A living structure with four hierarchical levels of natural streets. (Note: The natural streets that are represented—on the surface—by geometrical details of locations, sizes, and directions (

**a**) are transformed into the topology of the streets or living structure, in the deep sense, with far more less-connected streets than well-connected ones (

**b**).

**Figure 6.**Composition II by Piet Mondrian (

**a**) and its evolution from the empty square. (Note: It meets the minimum condition of being a living structure, and it is simple enough to illustrate how it is differentiated in a step-by-step fashion in panels (

**b**–

**e**), thus with a gradually increasing degree of life or beauty; there are far more smalls (4) than larges (1) from (

**b**,

**c**), and again far more smalls (6) than larges (4) from (

**c**,

**d**), so the ht-index is 3, which meets the condition of being a living structure. Thus both (

**b**,

**c**) are non-living structure, for their ht-index is less than 3. In addition, there is a violation of far more smalls (7) than larges (6) from (

**d**,

**e**), for 6 and 7 are more or less similar. If we consider the evolution in the opposite direction (from (

**e**,

**b**) then it can be viewed as a generalization process, very much like that of map generalization).

**Table 1.**Two fundamental laws of living structure. (Note: These two laws—scaling law and Tobler’s law—complement each other and recur at different levels of scale of living structure).

Scaling Law | Tobler’s Law |
---|---|

There are far more small centers than large ones | There are more or less similar centers |

across all scales, and | available at each scale, and |

the ratio of smalls to larges is disproportional (80/20). | the ratio of smalls to larges is closer to proportional (50/50). |

Globally, there is no characteristic scale, so exhibiting | Locally, there is a characteristic scale, so exhibiting |

Pareto distribution, or a heavy-tailed distribution, | Gauss-like distribution, |

due to spatial heterogeneity or interdependence, indicating | due to spatial homogeneity or dependence, indicating |

complex and non-equilibrium phenomena. | simple and equilibrium phenomena. |

Number | Mean | # Head | # Tail | % Head | % Tail |
---|---|---|---|---|---|

39 | 0.11 | 9 | 30 | 23% | 77% |

9 | 0.31 | 3 | 6 | 33% | 67% |

3 | 0.61 | 1 | 2 | 33% | 67% |

**Table 3.**The four iterations of the Koch curve as shown in Figure 4a.

Iteration | Scale | # Segment |
---|---|---|

0 | 1 | 1 |

1 | 1/3 | 4 |

2 | 1/9 | 16 |

3 | 1/27 | 64 |

# Segment | Mean | # Head | # Tail | % Head | % Tail |
---|---|---|---|---|---|

85 | 0.08 | 21 | 64 | 25% | 75% |

21 | 0.20 | 5 | 16 | 24% | 76% |

5 | 0.47 | 1 | 4 | 20% | 80% |

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**MDPI and ACS Style**

Jiang, B.; Slocum, T.
A Map Is a Living Structure with the Recurring Notion of Far More Smalls than Larges. *ISPRS Int. J. Geo-Inf.* **2020**, *9*, 388.
https://doi.org/10.3390/ijgi9060388

**AMA Style**

Jiang B, Slocum T.
A Map Is a Living Structure with the Recurring Notion of Far More Smalls than Larges. *ISPRS International Journal of Geo-Information*. 2020; 9(6):388.
https://doi.org/10.3390/ijgi9060388

**Chicago/Turabian Style**

Jiang, Bin, and Terry Slocum.
2020. "A Map Is a Living Structure with the Recurring Notion of Far More Smalls than Larges" *ISPRS International Journal of Geo-Information* 9, no. 6: 388.
https://doi.org/10.3390/ijgi9060388