A mathematical excursion in the isochronic hills

Philip Scarf, Centre for OR and Applied Statistics, University of Salford, Salford M5 4WT, UK Introduction Looking out over the South Pennines in mid-winter, through the driving rain, we can see flat topped hills, drab and uninviting. The heather, dark at this time of year, gives them a foreboding aspect. When walking and faced with a choice to go over or around, their flat tops mean that the over-the-top route is usually quicker. Of course, we may be out for a leisurely day in the hills and may not be concerned with speed. Then again, the rain may be stinging our face and a warm fireside and a pint of ale may be beckoning. Muddy, coal-black paths may define our route. Perhaps the choice is not so simple. There may be a multitude of paths, relics of local industry and agriculture with greater human input than at present. The population density on the moorland fringe was significantly higher in 1900 than at the end of the millennium. As elsewhere, the human population has given way for sheep. These hill flocks play a role in maintaining the upland path network. They also assist the mountain runner and navigator—grazed upland makes good running. This runner, when competing in events such as the Original Mountain Marathon (2006), will be concerned with fastest routes and not with route aesthetics. Then, a good over-or-around decision will save time. Choosing the best line between checkpoints, one that minimizes climb and distance travelled and has good running, is part of the art of mountain navigation. The Original Mountain Marathon is an “adventure race” for teams of two. It takes place over two days in late October. Competitors in the elite class cover over 80km in two stages in mountainous terrain, navigating from point to point, and camping overnight at a remote location. All pairs have to be self-supporting, carrying all their food and gear for the two stages. In the Lake District, the fells have more of a conical quality. Here it may be quicker to go around than over. The Howgill Fells, sandwiched between the Lakes and the Pennines, are geographically and topographically intermediate—they are more rounded in character. Well grazed and offering a multitude of routes over, around or in between, difficult route choices abound. With a pair of checkpoints carefully selected by an event planner, perhaps the execution times for all routes between them are approximately equal. That is, all routes may be isochronic (of equal duration). A mountain runner and navigator may then ask the question: can isochronic routes be characterized? We may imagine a hill such that all routes over it are equal it time terms. What does such a hill look like? We might call such a hill, if it existed, isochronic. A cone is a simple hill. Does there exist an isochronic cone? If a runner knew what an isochronic hill looked like when represented topographically (with contours), he or she could potentially make faster decisions regarding over or around when competing in mountain navigation races. The runner might be trained to spot a hill that was flatter than isochronic (go over) or steeper than isochronic (go around). The recreational walker may also want to find the quickest way to the pub! In this paper, we attempt to find descriptions of simple hills (cones, pyramids, and domes) that are isochronic. First we have to consider a rule that relates climb to distance. (Note, in this article, climb will refer to the vertical component of distance. Distance will mean the horizontal component of distance.) Consider the problem of travelling on foot from one side of a hill to the other in the shortest time. The obvious solution is to run faster. Therefore, consider this problem for an athlete who runs at a fixed speed. Then the solution will be to go by the shortest route—over the top. But the effect of the climb is to slow the runner on the ascent. If the over-the-top route involves significant climb, it may not be quicker than going round. Therefore, to pose the problem more usefully, we will assume that the athlete travels horizontally, on level ground, at a constant speed, and ascends (travels vertically) at a rate that implies 1 unit of distance vertically is equivalent in time terms to α units of distance horizontally. This equivalence between distance and climb was proposed by Scarf (1998), and is based on Naismith’s rule (Naismith, 1892) which states that “men in fair condition should allow for easy expeditions, namely, an hour for every three miles on the map with an additional hour for every 2000ft of ascent”. Thus, 2000 feet of climb is equivalent to 3 miles (=15,840 feet) of distance and so Naismith’s rule implies α = 7.92. We call α = 7.92 Naismith’s number. If a route comprises of a horizontal distance component of x units and a vertical distance component of y units, then Scarf calls x y + α the equivalent distance of the route. Naismith did not provide any empirical evidence to substantiate his rule. However, the record times for fell races provide support for Naismith’s number (Scarf, 2007). Norman (2004) finds evidence for a smaller value of α (in road running and treadmill experiments). Others (e.g. Langmuir, 1984; Rees, 2004) have proposed refinements to the rule particularly for steeper ground. α may vary between runners. It is important to note that in the analysis of Scarf (2007), and in Naismith’s original proposal, there is a presumption that the rule applies to routes that start and finish at the same elevation—what goes up must come down—and therefore empirical values of α should be based on the times for journeys or events that start and finish at the same elevation. Thus the effect of ascent is confounded with the effect of descent. While we can therefore only calculate the equivalent distance of a route that starts and finishes at the same elevation, we can compare competing routes between two points at different elevations because Naismith’s rule implies that the difference in time only depends on the difference in climb and the difference in distance between the routes. Alternatively, one can make an additional assumption that descent has no effect on speed. Whatever the exact underlying nature of the relationship in time terms between climb and distance, we will assume that the equivalence rule holds for our idealized athlete. For our idealized athlete, the shortest route in time terms from one side of a hill to the other will be the route with least equivalent distance. There will be an infinite number of routes and so this approach is only helpful if one can narrow down the choices to a small number. The two simplest routes are (i) over and (ii) around. There may be reasonable routes that lie between these.