Fig. 3

State-of-the-art construction paradigm for configurations of streets: the frames (\({{\bf s}}^{*}_{\star }\)) show this paradigm steps for constructing street \(r_{c}\) on street map (\({{\bf m}}\)) from the notional example in Fig. 1. Our illustration assumes that streets \(r_{a}\) and \(r_{b}\) were constructed previously and that the remaining streets will be constructed afterward. The superscript and subscript of each frame label indicate the street under construction and the step order, respectively. Each street-segment in colour already belongs to a street: when the colour is vivid and the line is solid, the street was committed; when the colour is pallid or the line is vividly dashed, the street is under construction. Each street-segment in grey is a candidate for belonging to a new street. We attribute to each street a particular colour. The construction goes like this. Initial stage (\({{s}}^{c}_{0}\)): no street-segment is yet assigned to street \(r_{c}\). Seeding step (\({\mathrm{s}}^{c}_{1}\)): pick at random one candidate street-segment—the seed street-segment is in pallid orange and marked with an orange bold square. Orientation step (\({\mathrm{s}}^{c}_{2}\)): orient at random the seed street-segment and move toward the head junction—the square mark is now a pentagonal “home plate” indicating the orientation, the path moved along from the mark to the head junction \(j_{1}\) is now vividly dashed, the excluding pie aligned with the incoming tangent at \(j_{1}\) immediately identifies along which outgoing street-segments the street might continue (see Fig. 2). Appending loop steps (\({\mathrm{s}}^{c}_{3}\))–(\({\mathrm{s}}^{c}_{5}\)): arbitrarily continue at the head junction along any valid outgoing street-segment (see Fig. 2) while applicable—at \(j_{1}\) the street might continue toward either \(i_{1}\) or \(j_{3}\) (see Fig. 2\({\mathrm{a}}_{2}\)), the latter choice was arbitrarily taken; at \(j_{3}\) the street might continue toward either \(i_{3}\) or \(j_{5}\), as the former choice was no more possible only the latter could be taken; at \(j_{5}\) the street can only continue toward \(i_{7}\); at \(i_{7}\) the street can no more continue so that the recursion ended. Inverting step (\({\mathrm{s}}^{c}_{6}\)): move toward the tail junction and formally invert orientation—the forward recursion lets now place to a backward recursion, the pentagonal mark has flipped its orientation and has rounded its tail to mark the epoch. Prepending loop steps (\({\mathrm{s}}^{c}_{7}\))–(\({\mathrm{s}}^{c}_{8}\)): arbitrarily continue at the tail (formal head) junction along any valid outgoing street-segment (see Fig. 2) while applicable—at \(j_{2}\) the street might continue toward either \(j_{4}\) or \(j_{6}\) (see Fig. 2\({\mathrm{b}}_{2}\)), the former choice was arbitrarily taken; at \(j_{4}\) the street can only continue toward \(i_{6}\); at \(i_{6}\) the street can no more continue so that the backward recursion ended. Commit step (\({\mathrm{s}}^{c}_{9}\)): commit the new achieved street and loop forward to build the next street until applicable—the now achieved street \(r_{c}\) is in solid line, its mark is an unbold circle, and it has a label; retrospectively, this step leads to Initial stage (\({\mathrm{s}}^{d}_{0}\)) for the next street \(r_{d}\) while the above Initial stage (\({\mathrm{s}}^{c}_{0}\)) appears to result from Commit step (\({\mathrm{s}}^{b}_{6}\)) for the previously committed street \(r_{b}\); the construction of streets loops until no more street-segment is unassigned. The arbitrary choices in Appending and Prepending loop steps are actually join principles (see “A street is an exclusive joined sequence of street-segments” section). Supplementary Animation A1 (Additional file 1) shows a complete construction of the configuration of streets on street map (\({\mathrm{m}}\))