Technical Terms:
chirality, component, cutnode, cycle, directed graph, indegree, mutually reachable, outdegree, strongly connected, subgraph, tree, unilaterally connected.
Link to Chapter 3 Complement.
The most extensive, and by all accounts the best known, pneumatic message network was "Reseau Pneumatique" or "Le Petit Bleu," the message network that served Paris for 120 years (1865–1984). Little notes, written on special blue stationery (hence "petit bleu"), swished through metal tubing suspended in the sewers of Paris en route from one pneumatic substation to another. Even Walt Disney found a certain humorous charm in sending Fred MacMurray on a frenzied tour through this subterranean Paris maze (Disney, 1962). From its beginnings as a modest cycle focused on the Bourse on the Right Bank and on 103 Rue de Grenelles (General Office) on the Left Bank, the "Pneu" grew, by 1907, to include two hundred ten miles of tubing joining 120 pneumatic stations (Figure 3.1) (Postmaster General, 1909).
Unlike the Rohrpost in Berlin, which began as a graph theoretic tree and grew as a tree, the "Pneu" in Paris began as a single cycle, grew to include not only this single cycle, but also subtrees, and directed subgraphs. Redundancy in network connection grew with the system. By 1907, there were two central offices, at the Bourse and at the head office of post, telephone, and telegraph. There were a number of "head-of-line" substations offering a full range of services; there were local offices (often incident with neighborhood telegraph offices) serving quartier needs; and, there were press offices linking directly into governmental facilities (Figure 3.1 illustrates a representation of this hierarchy) (U.S. Congress, 1909). Within these geographic layers determined by difference in function, techniques involving graphs and directed graphs serve as structural models (Harary, Norman, and Cartwright, 1965).
Figure 3.1. Paris, pneumatic network (Reseau Pneumatique), 1907 is represented in this figure as a graph. North is at the top. Largest circles represent Central Offices (the most northerly is on the Right Bank of the Seine and the most southerly is on the Left Bank of the Seine); next largest circles represent Head of Line Offices; next largest represent Local Offices; and, smallest circles represent Press Offices. Solid lines represent pneumatic linkage. Source: Arlinghaus, 1985, modified from U.S. Congress, 1909, p. 141.
Many of the head-of-line pneumatic substations were hubs for localized pneumatic transmission, directed to telegraph offices in various quartiers. The Rohrpost in Berlin was centered on one office; the Parisian network reflects its geographic circumstance and is based on two centers, one on the Left Bank and one on the Right Bank of the Seine River (nodes L and R in Figure 3.2 below). These two centers have a skeletal structure similar to the Rohrpost, but with some redundancy built in for river crossings (as offered by nodes b1, g1, and i1 in Figure 3.2 below), that reach across Paris.
Figure 3.2. This figure shows a Mapplet on Two Centers. Java™ Applet showing the structural model of two-way tubing, or connection pattern, on which the Paris pneumatic network was based. In this network, R and L are the centers on the Right and Left Banks of the Seine River. Redundant cross-river connection is provided through nodes b1, g1, and i1. Drag subgraphs within the Mapplet to alter arrangement to appear more similar to that of the map from which it was derived. Sun Microsystems, Java™. Source: Graph.java. Used with permission from Sun Microsystems, Inc. Copyright 1998-2001 Sun Microsystems, Inc. All rights reserved.
The two-way connection pneumatic tubes serve as a skeletal, underlying graph for the entire network. There are, in addition, eight separate, smaller digraphs (Figure 3.3). Often, the head-of-line office is a cutnode of the digraph--that node, which, if removed, would force the graph to split apart into separate components. Generally, the circulation pattern within the digraph components is cyclic; the arrows within the directed subgraphs illustrate this fact. In some cases, the flow is clockwise, and in others, it is counterclockwise; the orientation of the flow is suggested by the sequences of arrows on the edges of the directed subgraphs in Figure 3.3.
The digraph labeled D1 in Figure 3.3 is unilaterally connected (for any two nodes at least one is reachable from the other) but is not strongly connected (not every two nodes are mutually reachable). (See Theorem 3c.2.) The node d1 in subgraph D1 has outdegree 1 and indegree 0. Once a message has left the pneumatic station at nodes d1, it is trapped within the digraph and can never return to its origin (without the use of the rest of the system). Thus, D1 is not as self-sufficient relative to the entire system as are the remaining seven digraphs, all of which are strongly connected—every two nodes are mutually reachable without recourse to the rest of the pneumatic system. In systems formed by overlaying digraphs on graphical structures, it is evidently wise to require that the digraphs be strongly connected to optimize the self-sufficiency of the entire system.
Figure 3.3. Directed subgraphs (digraphs) within the Pneumatique.
Another fundamental geometric issue in forming overlays is to determine whether it is necessary to flip one layer through the third dimension in order to achieve accurate alignment of the layers. In a traditional cartographic view, this problem is simply that of aligning map layers using tick marks to register the position of one overlay in relation to another. When one layer is the mirror image of another, it is necessary to perform a flip through the third dimension in order to form an accurate overlay of otherwise identical objects. As an example, consider a lower case letter ‘p’. The lower case letter ‘d’ may be made to overlay the lower case letter ‘p’ using a simple rotation within the plane. The letter lower case ‘q’ has the correct shape to use as an overlay for ‘p’; but, to ensure correct registration, mere rotation within the plane is insufficient; a flip (reflection) outside the plane is required. The former model, all within the plane, is termed achiral; the latter, requiring a flip through the third dimension, is chiral. Chiral, from the Greek, means pertaining to the hand. To superimpose the left hand onto the right hand a flip is needed; Maurits Escher captured this idea in a familiar line drawing in which one hand traces the other. Whenever a flip is required, the motion is chiral; otherwise, it is achiral. Harary and Robinson (1975), Harary, Robinson, and Balaban (1976), and Harary and Mezey (1988) used the concept of chirality to create structural models in chemistry. Agranoff (1978) employed it to clarify biochemical nomenclature tied to relations among hydroxyl groups (equatorial and axial).
Thus, one might designate pairs of directed cycles within the Pneumatique context as "chiral" or "achiral"; arrows label the direction of the flow as clockwise or counterclockwise. Two cycles are said to be chiral if they cannot be directly superimposed on each other in the plane. A flip outside the plane is required for accurate superimposition of orientation. The cycles C1 and C2 in Figure 3.3 that share a common node v have opposite orientation and are chiral. When two cycles have the same orientation and can be superimposed in the plane, they are said to be achiral. The cycles C1' and C2' in Figure 3.3 have the same orientation and are achiral. Figure 3.4 shows animated renditions of this situation, labeled as in Figure 3.3.
Figure 3.4. This figure shows detail of pneumatic intersection. In the chiral case, the tubing crosses at v and an attendant has access to both digraphs simultaneously; however, collisions can occur. In the achiral case, the tubing need not cross at v'. Simultaneous access is not possible nor are collisions of pneumatic carriers at v'.
The physical implications of variations in relative orientation of cycles within a directed graph become significant when one considers sending messages in a continuous stream through (rather than to) substations, such as v and v', that are cutnodes within these digraphs. When two cycles are chiral, the tubing is forced to cross in the plane; an attendant or other monitor, placed at the cutnode v, would have direct access, using one hand, to each cycle C1 and C2 and also to their union. However, pneumatic carrier collisions could occur at v. In the other situation, in which C1' and C2' have the same orientation, the tubing need not cross in the plane at the cutnode v'. Thus, the possibility of carrier collisions can be eliminated, but direct simultaneous access to both cycles, to clear clogged tubes and other difficulties, is also eliminated. Using structural models to understand implications of the direction of flow in a system can influence decisions for improvements.
A combination of structural models (including trees, cycles, and digraphs), employed at different geographical scales described simply, as a single structural model, the entire strategy (including built-in redundancy) for transmitting and receiving pneumatic messages. More important, perhaps, it offered a means for understanding how the efficiency of the system might be improved—in terms of chiral and achiral pairs of directed cycles—and where optimum locations for such improvements might be located.