Spatial Thinking in Planning Practice: An Introduction to GIS

23 topology, nor will any amount of rubber-sheeting or other data transformations change relations from one form to another. Containment is the property that de!nes one entity as being within another. For example, if an isolated node (representing a household) is located inside a face (representing a congressional district) in the database, you can count on it remaining inside that face no matter how you transform the data. Topology is vitally important to the Census Bureau, whose constitutional mandate is to accurately associate population counts and characteristics with political districts and other geographic areas. Connectedness refers to the property of two or more entities being connected. In Figure 2.1, Topologically, node N14 is not connected to any other nodes. Nodes N9 and N21 are connected because they are joined by edges E10, E1, and E10. In other words, nodes can be considered connected if and only if they are reachable through a set of nodes that are also connected; if a node is a destination, we must have a path to reach it. Connectedness is not immediately as intuitive as it may seem. A famous problem related to topology is the Königsberg bridge puzzle (Figure 3.3). Figure 3.3. "e seven bridges of Königsberg bridge puzzle. Source: Euler, L. “Solutio problematis ad geome- triam situs pertinentis.” Comment. Acad. Sci. U. Petrop. 8, 128-140, 1736. Reprinted in Opera Omnia Series Prima, Vol. 7. pp. 1-10, 1766. "e challenge of the puzzle is to !nd a route that crosses all seven bridges, while respecting the following criteria: 1. Each bridge must be crossed; 2. A bridge is a directional edge and can only be crossed once (no backtracking); 3. Bridges must be fully crossed in one attempt (you cannot turn around halfway, and then do the same on the other side to consider it “crossed”). 4. Optional: You must start and end at the same location. (It has been said that this was a traditional require- ment of the problem, though it turns out that it doesn’t actually matter – try it with and without this re- quirement to see if you can discover why.) "e right answer is, there is no such route. Euler proved, in 1736, that there was no solution to this problem. Chapter 3: Topology and Creating Data