mathematical fact that the area of a two-dimensional circle is Pi r^2, while the volume of a three-dimensional sphere is 4/3Pi r^3, the volume of a four-dimensional hypersphere is 1/2 Pi^2 r^4, and so on.
Below I show pictures of various networks. In each case the first picture is drawn to emphasize obvious regularities in the network. But the second picture is drawn in a more systematic way—by picking a specific starting node, and then laying out other nodes so that those at
Captions on this page:
Examples of various networks, shown first to emphasize their regularities, and second to illustrate the number of nodes reached by going successively more steps from a given node. For networks that in a limiting sense correspond to ordinary d-dimensional space, this number grows like r^(d-1). All the larger networks shown are approximately uniform, in the sense that similar results are obtained starting from any node. Network (e) effectively has limiting dimension Log[2,3]≅1.58.