Allometric scaling originates in biology, where it refers to scaling relations between the size of a body part and the size of the whole body when an organism grows. In cities, various allometric relations have also been discovered, such as those between the complexity of traffic networks and urban quantities. Metro networks are typical traffic networks in cities. However, whether allometric relations with metro networks exist is still uncertain. In this study, “fractal dimension” was employed as the complexity measure of metro networks, and potential allometric relations between fractal dimensions and urban indicators in 26 main cities in China were explored. It was found that fractal dimensions of metro networks had positive allometric relations with gross domestic product (GDP), population, particulate matter with a diameter less than 2.5 microns (PM_2.5), the road congestion index and the average price of second-hand housing (with Spearman’s R of 0.789, 0.806, 0.273, 0.625 and 0.335, respectively) but inverse allometric relations with sulfur dioxide (SO₂) and residential satisfaction (with Spearman’s R of −0.270 and −0.419, respectively). Such discoveries imply that allometric relations do exist with metro networks, which is helpful in deepening our understanding of how metro systems interact with urban quantities in the self-organized evolution of cities.

Personalized mappings become popular among the public with the support of data diversity and device diversity. To develop personalized maps, constructing map symbols through automated ways is beneficial. The formal representation of map symbols (i.e., expressing map symbols by mathematical operators) is fundamental to the automated construction of map symbols. A previous study to evaluate the feasibility of structures of Chinese characters for representing map symbols shows that 77.5% of map symbols can be represented by them, although there are imperfections in some cases. It means that: (1) the other 22.5% of symbols should be formally represented by other mathematical solutions, and (2) those imperfect cases should be made perfect through some modification or refinements. In this study, we solve the representation problems of these two types of map symbols (i.e., the map symbol did not or imperfectly fit the structures of Chinese characters) by employing additional basic operators and proposing some metric and color modifications. To validate these proposed solutions, experiments have been carried out by using eight sets of symbols that are publicly available (e.g., Google Icons). The results indicated that almost all the map symbols can be formally represented with additional operators and metric and color modifications. The percentages of map symbols that did not fit structures of Chinese characters solved by these operators and modifications are 2.4% and 20.1%, respectively. The percentages of map symbols that imperfectly fit them solved by these operators and modifications are 8.7% and 8%, respectively. This work could not only enrich cartographic theory but also prompt the mathematization of map symbol construction.