Form versus Function : Walking the LineAmrish Deshmukh, Niko Stahl, Tarun ChitraNovember 18, 20081AbstractThe Arts quad is a space that is subject to several competing interests. Its design is expectedto oer the opportunity for an individual to move conveniently from one end to another, to keepgeneral maintenance costs feasible and to allow for activities ranging from lectures to snowball ghts.This paper proposes a mathematical model, which provides a way of intelligently designing thepath network of the quad. The centerpiece of our model is a cost function, which evaluates thefeasibility of a given path conguration.To explore the set of feasible path conguration we wrote an algorithm that randomly generatessamples of this set. We then improved on this search by constructing an optimization algorithminspired by Markov chain Monte Carlo methods. We believe this improved search has found a localminimum path conguration, as it appears stable under perturbation.2ContentsIIIIIIIVVVIProblem StatementThe Arts Quadrangle as a GraphThe Length of a PathUnocial Path Induced by Human BehaviorThe Cost FunctionAssumptions:446789VIIVIIIIXAlgorithms for Finding Optimal Path CongurationsResults:Recommended Solution91011XXIFuture WorkBibliography31516Part IProblem StatementThe task is to redesign the Arts quad walkways using a mathematical model that will help usdetermine a preferred design. Beyond the general fact that minimizing the total length of the pathsand maximizing the areas of contiguous lawns are preferable, we are asked to consider the followingcriteria: Path maintenance costs Landscaping costs Pedestrian trac and behavior The creation of unocial paths and its impact on the lawn The general appeal of the quadTo implement these criteria in our model, we are provided with the following principles: The path maintenance cost is proportionate to the total path length. The landscaping cost depends on the number of contiguous lawns, the creation of unocialpaths (as a result of pedestrians leaving the paved paths to arrive at their destination morequickly) and the geometry of contiguous lawn. If the path between two points is 15% longer than the straight line connecting the points, apedestrian will leave the path and cut across the quad. An average pedestrian might leave the path if it implies saving more than 10% of the totallength the path.Part IIThe Arts Quadrangle as a GraphGraph theory has been an important tool in exploring problems which range from determining theneural network of nematode C. elegans to nding the cause of failure in electrical power grids1.By framing our walkway design problem in the language of graphs, we can readily extract the keyrelationships between structure and function.We describe the Arts quadrangle (hereby referred to as the Arts quad or simply quad)as a graph of 10 nodes, which represent the most common points of entry and exit to the quad(see gure 1 below).1See 14Figure 1: Cornell Arts QuadLet the set of nodes be A = x|x 1, 2, . . . , 10. Now we can dene a path to be anordered pair (a, b) and the set of all paths as the relationR = (a, b)|a, b A, a = b (1)since every pair of distinct nodes will dene a line segment, or one-way path, in the plane. Then theset of all possible congurations of paths is given by the power set P (R). This set has 290 elements(the cardinality of a power set of a set with 90 elements)This presents an overwhelming set of possibilities, but fortunately there are three constraints,which we imposed to make our set less unwieldy. We will only model:1. Non-directed graph : Currently the space (1, 3) is distinct from the path (3, 1). We nd this tobe unreasonable as pedestrian paths are very rarely “one-way”2. Connected graph : Aesthetically and functionally it makes little sense to allow a building to besurrounded completely by grass. Furthermore, we picked the ten nodes because we consideredthem to be essential circulation points of the quad. Therefore, having one of them disconnectedfrom the network would be unreasonable3. Graphs including the perimeter : This is again chosen in line with our opinions on aestheticsand utility. While pedestrians are likely to accept longer distances than a straight line toremain on the ocial path, it seems unlikely that a person going from A to B will abide witha path that strictly increases the distance to B before allowing the pedestrian to actuallyapproach B (see gure 2 below).5Node 1 2 3 4 5 6 7 8 9 10Location (0, 0) (0, 3) (0, 9) (0, 15) (0, 18) (5.6, 18) (7, 18) (7, 14) (7, 6) (7, 0)SW MorrillHallMcGrawHallWhiteHallTjadenHallSibleyHallNE LincolnHallGS Hall SEiFigure 2: In the rst graph, we see that travelling from (0, 1) (1, 0) never increases the distancefrom (1, 0), whereas in the second graph, going from (0, 0) (1, 0) incurs this costOnce we have attained potentially optimal congurations that satisfy these three constraints, wewill remove the co