Euler Walks and Graph Theory

This piece also appears in slight variation on Urban Backpacking.

The new South Park Bridge is set to open this year in Seattle.  This is cause for celebrate for many reasons.  One you may not be aware of is that it, and to the delight to mathematicians everywhere - in the spirit of  Leonhard Euler - it returns Seattle to a traversable graph. 

One will now be walk across each of Seattle’s pedestrian accessible bridges, that separate the three major landmasses, North, Central, and West Seattle, in one cycle. The significance of this might need some explaining:

Euler was concerned about the walkability of Kaliningrad in the early 1700’s. He wanted to see if he could walk across each of the cities seven bridges once, and only once, on a single walk.  This amazing feat should be the goal of many an afternoon.  Yet, Euler found it impossible. In order to reach each bridge, he was required to retrace a path across at least one other bridge. He sought to discover what conditions would be necessary as so this would not be the case. 

In discovering this conundrum, he also stumbled upon the first theorem in the field of graph theory and thus invented combinatorics. What he found was that it depended on the number of connections (edges) each node has.  In Kaliningrad’s case there were 4 sections of the city, each with 3 or 5 bridges.  To be able to cross each bridge only once, to have a traversable graph, you need each node to be connected by an even number of bridges.

With the opening of the South Park Bridge, West Seattle/South Park now has two bridges connecting it to Central Seattle, which in turn has 6 bridges connecting it to North Seattle. Each of the three nodes has an even number of connections.

This arrangement means that you can walk each of these 8 bridges (Ballard Locks, Ballard Bridge, Fremont Bridge, Aurora Bridge, University Bridge, Montlake Bridge, Spokane Street Bridge, and the new South Park Bridge) once on a nice hike that will take you across each of them. Multiple paths will work.

The consideration of Seattle of as a traversable graph is more than fun, it shows application of reason to complexity, utilizing the tools of  contemplation on the space that matters most.

The infusion of  mathematical theory into urban thinking may appear at first to be a contradiction, an oversimplification of phenomena that suffer simplification. However, it is in practice that the simplification is far more severe, and damaging. This analysis approach allows for aspects of the urban condition to be more considerately assessed. The works of Salingaros, Alexander, K. Lynch, among others have shown that there is much unrealized value simply in applying existing lines of ecological thinking to human dominated space.

Particularly to this point is Salangaros (2005), who explicitly separates the notions of hierarchical organization and simplification. But states that the potential confusing of the two has "catastrophic consequences". Moreover, Alexander (1966) explicitly calls out hierarchical organization itself as the source of of the problems of urban structure. The takeaway here is perhaps that for specific purposes an organization can help distill single pieces of information, in this case pathways, but it should never be used in extrapolation or generalization.

Christopher Alexander “A city is not a tree.” Design, London: Council of Industrial Design, 1966.

Nikos A. Salingaros "Principles if Urban Structure." Design/Science/Planning, Techne Press, Amsterdam, The Netherlands