the eulerian walk

 

The Eulerian Walk and Other Networks

Network structure was formally introduced in 1736 by the German mathematician Leonard Euler with his publication of a solution to the most perplexing conundrum of the day, sent to him from the citizens of Königsberg, a city of four separate districts connected by seven bridges:  Was it possible to take a walk so that one crosses every bridge exactly once?  In his lifetime, Euler (1707-1783) published more than 800 books and papers and is credited with inventing the science of topology and graph theory.  He first simplified the problem by reducing the four districts and seven bridges to a graph of four nodes and seven connecting edges:

He then proved that such a walk in Königsberg is impossible.  Euler’s method was a proof by contradiction, i.e., suppose such a walk is possible:  One could choose any of the districts, say “A,” in which the walk does not begin or end, so a bridge must be taken to arrive there.  Then at some other time, we leave by another bridge; we arrive again, leave again, and arrive again.  Since there are five bridges in this district, we need to use all of them, but now we are stuck at point A.  So we must either begin or end our walk here.  However, there are three other districts in town, each connected to the rest of the town by three bridges, again an odd number of bridges, meaning that we will be unable to leave any of the other points as well.  Therefore, we cannot start or end the walk in each of the four districts, which proves that no walk can cross every bridge exactly once.  An Eulerian walk is a walk on a connected graph that goes through every edge (across every bridge) exactly once.

Instead of making an exhaustive list of all possible routes, Euler formulated a general statement about any network and he specified that if a connected graph has more than two nodes with an odd degree (i.e., an odd number of “bridges”), then it has no Eulerian walk.  Furthermore, he stated that if a connected graph has exactly two nodes of odd degree, then it is an Eulerian walk; if a connected graph has no nodes with odd degree, then it is also an Eulerian walk – in this case, it is also called a closed Eulerian walk, meaning that one can start and end up at the same place.

We can also stipulate that every node be used exactly once.  This was a problem formulated in 1856 by the mathematician William R. Hamilton.  A Hamiltonian Cycle is a walk that contains all nodes of a graph.  In other words, you must pass through all points only once, not necessarily using all the bridges, like a delivery courier who wants to visit every city just once in his route.  Unlike the Eulerian walk, there are no formal statements that can prove or disprove the Hamiltonian cycle, and it can only be accomplished by trial and error.

A network, or graph, then, is a series of nodes or vertices that are connected to other nodes by edges – the network increases in size and complexity when new edges are created, linking new nodes to the original foundation.  Formerly separate networks can also be joined to form an aggregate mass once connecting edges are made.  However, in making concept maps of individuals’ mental impressions, the “bridges” can be well-worn paths and need not be restricted to a single use.  When “walked upon” more than once, the action establishes a connective power in binding one island of thought to another – either new (discovery) or old (memory) perceptions – forming a continuous chain of discrete synaptic states, traveling from node to node.  Likewise, individual nodes can be experienced by taking any appropriate bridge to reach them, as a Hamiltonian graph; it is not necessary to use all of the available ones.  If there is a connected node with no further pathways, then it remains a “dead end” until a new connection branches away in the individual’s mind, leading to either further inquiry or resolution.

Analytic poems write this graph with a specific format to create a network of impressions born uniquely in each person who experiences it by reassembling pre-existing representations and observing the collection from more than one viewpoint, thus gaining insight from original juxtapositions.  Specifically, by building a series of networks, one dimensional level at a time, with each system defined by common characteristics, we can form a coherence among previously disparate elements, i.e., those subsets now distinguished by their relative intersections.  Overlapping relationships and sources resulting from selected affinities (text, image, and concept) will become a map of interconnected pathways leading to unknown territories.  Thus, the poem is simultaneously fixed and free – limitless possibilities from a given framework.

                                                                                                                                         

To summarize, the analytic poem is a lyrical composition that utilizes an algorithm for creating a network of concept maps.  These are usable by anyone, since every individual owns their relevant histories and impressions, based upon how they identify phenomena.  As more individuals contribute to a core poem, the network grows and gains strength – in so doing, the poem aspires to first define, and then supersede, ordinary boundaries of perception by using an inspirational approach to examine Nature.  The method uses a structured taxonomy of ascending cognitive levels that are built one at a time by adopting a geometric understanding of dimensions, and the map proceeds incrementally until a coherence is attained – the method of dissecting perceptions into understandable levels builds complexity transdimensionally with additional nodes.  Since network vertices represent units of both cognition and spacetime, we can willfully add or subtract dimensions as we become aware of new ideas.  Therefore, the process of creating the analytic poem is itself a type of geometry that likewise grows from a set of axioms; in this case, while each dimension contains postulates and affinities related to Nature, the whole network of all dimensions stands as the final statement.

Given such possibilities, observers have the ability to utilize a network as a transformative tool by exchanging projected representations for authentic impressions.  Since for each source-entity there is an exemplar (a geometric pre-image) imprinted onto the phenomenon (its image) that we apprehend, information is transferred from an idealized state to the functioning world of matter and energy.  The geometry of transformations neatly parallels the structuralist approach in semiotics that defines a “sign” as the composite of both the signified (signifié), i.e., the mental perception of the meaning (pre-image), and the signifier (signifiant), i.e., its physical manifestation (image).

However, between the two states, the concept and the representation, the content is reduced due to an imperfect transmission vehicle – the signified essence, which is indicated by the signifier, can be variable with each observer – dynamically, energy is always lost from friction, resistance, and entropy –– and dimensionally, perceptions outside any given construct are limited by the requirements of a meta-state (any two-dimensional being is incapable of three-dimensional perceptions).  An analogy, as Plato wrote in The Republic, is that our common three-dimensional reality could be envisioned as the shadows cast from an unknown exterior location.  Hypothetically, this higher dimension acts as a collective intellect holding the archetypes that imbue world-impressions with meaning: if attainable, we could identify the true values of phenomena based upon source-entities from this wellspring.  And geometry potentially allows for complete transformations in this idealized state, without a loss of information; as a structured system, its purpose can describe all phenomena, beginning with intrinsic axioms, to interpret Nature.  Edmund Husserl best described this notion in 1936:

Now the problem would be to discover, through recourse to what is essential to history [Historie], the historical original meaning which necessarily was able to give and did give to the whole becoming of geometry its persisting truth-meaning.  It is of particular importance now to bring into focus and establish the following insight:  Only if the apodictically general content invariant throughout all conceivable variation, of the spatiotemporal sphere of shapes is taken into account in the idealization can an ideal construction arise which can be understood for all future time and by all coming generations of men and thus be capable of being handed down and reproduced with the identical intersubjective meaning.  This condition is valid far beyond geometry for all spiritual structures which are to be unconditionally and generally capable of being handed down.”

 

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ENDNOTES

image:  Steve Deihl, “Celestial Navigation,” 22” x 15”, mixed media on Rives BFK paper, 2008.
image:  Schematic representation of the seven bridges of Königsberg, showing the four districts, A, B, C, and D. The graph is famous for its simplicity, and as Euler proved, each of the four districts has an odd number of bridges, making it impossible to cross each one only once.

image:  hamiltonian cycle: 220px-Hamiltonian_path.svg.png en.wikipedia.org

Plato:  In Plato’s Republic (ca. 380 B.C.), he describes a group of prisoners bound inside a cave who witness only the representations of forms (shadows) projected against the back wall. The true objects pass unseen behind the prisoners who mistakenly ascribe reality to the shadows, knowing of no other.

Husserl further states: “It is a general conviction that geometry, with all its truths, is valid with unconditioned generality for all men, all times, all peoples, and not merely for all historically factual ones but for all conceivable ones. The presuppositions of principle for this conviction have never been explored because they have never been seriously made a problem. But it has also become clear to us that every establishment of a historical fact which lays claim to unconditioned objectivity likewise presupposes this invariant or absolute a priori. Only [through the disclosure of this a priori] can there be an a priori science extending beyond all historical facticities, all historical surrounding worlds, peoples, times, civilizations; only in this way can a science as aeterna veritas appear. Only on this fundament is based the secured capacity of inquiring back from the temporarily depleted self-evidence of a science to the primal self-evidences.”

Edmund Husserl, The Origin of Geometry, 1936. Manuscript originally published by Eugen Fink in the Revue internationale de philosophie, Vol. I, No. 2, 1939, under the title, “Der Ursprung der Geometrie als intentional-historisches Problem.” Jacques Derrida, Edmund Husserl’s Origin of Geometry: An Introduction. Presses Universitaires de France, 1962. Translated, John P. Leavey, Jr., University of Nebraska Press, Lincoln, Nebraska, 1989.