The total population of the United Kingdom is expected to increase from 63 million in 2011 to 73 million people in 2037. This is an increase by 16% in 26 years, which puts the urban infrastructure, particularly the transportation infrastructure, under great pressure. Urban infrastructure, especially the transport infrastructure, to a large degree determines the overall shapes, productivity, and energy efficiency of cities. There has been considerable work on modelling the relations between urban growth patterns and transportation networks. By contrast, few studies have focused on the relations between the geometry of the transportation infrastructure and energy efficiency, in particular between innermost part (the core) and outer parts (the periphery) of cities. In many cities, the inner parts (constituting the core) of street networks are much older than, and commonly recognised as being geometrically different from, the newer and outer parts (constituting the periphery). To understand better the differences between the core and periphery parts of street networks, as well as their general structure and growth, in particular in relation to energy and efficiency, we analysed 41 street networks of small to large British cities with a total of more than 750 thousand streets. We also developed methods for quantifying the geometric (structural) differences between the core and periphery parts of the cities, and use the results to assess the efficiencies of the street-network in relation to city populations. The Gibbs/Shannon formula (a measure of spreading or dispersion) is used to calculate the street orientation-entropies and length-entropies of the resulting probability distributions. The street orientations vary from roughly orthogonal (grid-like) street sets with low entropy to close-to uniformly distributed street sets with high entropy. The length entropy shows a strong positive linear correlation with average street length and spacing and a negative correlation with street density. In comparison with the parts of the network periphery, the network cores have generally shorter streets, constituting low-entropy networks. This is in agreement with many of the cores being several hundred years old, or more, and thus constructed when much less energy was available than now. Therefore, many of the cores are the constructs of comparatively low-energy societies. There are two principal mechanisms by which a street network can accommodate increasing number of streets so as to be able to transport efficiently a growing population. One mechanism is through adding streets within the existing network. The other mechanism is through adding streets at the margins, that is, in the periphery parts, of the network. The first mechanism is referred to as expansion; the second is referred to as densification. We selected two British cities for a detailed analysis of the growth of their networks, namely Sheffield in England and Dundee in Scotland. Sheffield has a population of 555,500 (in 2010) and a total street number of 23,501. Tracing the evolution of the street network of Sheffield from 1736 to 2010, the results show that there was significant densification of the network in Sheffield, particularly during the period 1890-1950. Nevertheless, expansion has been the dominating mechanism of street-network growth during the entire period from 1736 to 2010. Dundee has a population 144,000 (in 2010) and a total street number of 9,616. Tracing the evolution of the street network of Dundee from 1600 to 2010, the results are significantly different from those for Sheffield. More specifically, whereas expansion dominated in the periods 1600–1776 and 1846–1912, expansion and densification are essentially similar in the other two periods considered, namely in 1776-1821 and 1912-2007. Results from cities in other countries show more extreme cases, where during certain periods densification entirely dominates. When as a city grows its network necessarily increases in total length but also spreads out (covers a larger area) and produces more entropy. This might indicate that when a city expands, its energy efficiency decreases. Clearly, the cost of the street network increases as the city expands. This follows because there is a certain constructional energy per kilometre for various types of roads and streets. Thus, for streets of a given type, to make them longer requires gradually more constructional energy. In order to assess the efficiencies of the street networks in the 41 cities, we compared the city populations with the street number and cumulative street length for each city. The number of streets ranges from 2226 (Winchester) to 240,611 (London), while the population ranges from 28,156 (Dover) to 7,825,200 (London). When plotting the population versus the number of streets, it is clear that cities with large populations need proportionally fewer streets and smaller cumulative total street length to transport that population than smaller cities. What this means is that the larger cities have fewer streets and less total street length per capita than the smaller cities. The results suggest, at least for British cities, that as the city size increases its transportation capacity becomes used more efficiently, so that the network becomes more energy efficient.