András J. Molnár
Co-Founder, Pilgrimage Academy, Viatorum Trail Experts
Research Fellow, HUN-REN SZTAKI
Budapest, Hungary
Borbála Benkhard
Assistant Professor
Department of Landscape Protection and Environmental Geography
University of Debrecen, Hungary
A matematika számos ága kapcsolódik az túraösvényekhez. Amikor arról beszélünk, hogy ‘Trailology’ (magyarul „Ösvénytan”) néven a túraösvények (turistautak, zarándokutak, zöldutak; kerékpáros vagy vízi útvonalak és hasonlók) tanulmányozását szeretnénk középpontba helyezni multidiszciplináris megközelítésben, rögtön eszünkbe juthat – egyebek mellett – a geometria és a statisztika, de továbbgondolva akár a gráfelmélet és a kombinatorika is, sőt, még mélyebbre menve a logika és a formális rendszerek tudománya is kapcsolódik a témához. Írásunkban áttekintünk néhány érdekes, matematikai köntösben felmerülő témát és kérdést, mellyel a turistautakkal kapcsolatos gyakorlati munkáink (kutatás, fejlesztés) során találkoztunk; és utalunk arra, hogyan tud az „Ösvénytan” mertíteni a matematika kimeríthetetlen eszköztárából és szemléletmódjából.
Looking at what mathematics may bring to trailology seems to be trivial at first sight: one can measure the length of a trail, its altitude range, its grade, and make computations such as summarizing trail lengths in an area, or calculating quantities or ratios related to certain phenomena along them. But even simple questions may become tricky when it comes to calculations and computations: How can we compute the difficulty of a route given the difficulty of its parts? What is trail difficulty at all? Well, it depends – exertion difficulty, related to the length and ascent, adds up but overall technical difficulty should be taken as the most difficult part. Or: How to summarize the total length of a set of routes when they have shared sections? What do we mean by ‘total length’ in such a case? Should we count the shared sections twice or only once? Well, it depends – it is not the same talking about signage or trail tread work. And if we think about a trail being a geographical entity, the whole world of geometry comes in, with questions such as what exactly we mean by length; is it the trail’s projected length on a map, its measured length in a geographical coordinate system, or on the terrain with a mechanical device? In either case, what is the resolution of the recorded data, and the measurement unit? Or, just going into simple statistics, how many people visit a trail section? Shall we count twice if some hikers go forth-and-back…? And so on. … We, as trail researchers and practitioners have been facing such questions during our trail studies in visitor monitoring and trail maintenance planning.
This is where the rigorous formalism of mathematics can help a lot, requiring precise definitions and reasoning. Math forces us to be profound and to clarify things. At the same time, it gives us an outstanding freedom by allowing us to make our own definitions. What is trail difficulty? Well, depends on your definition. What is the trail length – depends on how you define and measure it. Mathematics, when applied to Trailology, can give us an approach to look at phenomena related to trails in a more subtle way, avoiding pitfalls such as comparing apples and oranges. Moreover, mathematics gives us a wide variety of tools and methods for use, just like any of the sciences. An interesting topic for example is computing the walking time along trails – is there a way to model it by a formula? Empirical studies using regression analysis tend to confirm the usefulness of experience-based calculations of hiking communities and authors who have indicated the ‘length’ of a hike (and with that somewhat of its difficulty) by an indicative walking time on signposts and in trail guidebooks: if you walk slower or faster than indicated, you can count on that personal ratio of walking time when planning your next hikes in a trail system of consistent time indication – despite the tendency we have observed among hikers to think that indicating time instead of distance may look arbitrary and subjective at first sight.
But mathematics is not just about numbers. It entails logic, formal languages, geometry, combinatorics, and much more. It relates to semiotics, modeling, data management and other fields. Talking about, for instance, trail marking (blazing) or signage systems, they can also be looked at as a simple language with symbols and rules – having their syntax, semantics and pragmatics, just like any other formal system or language. And we can analyze and compare them based on their properties and expressive power: some marking systems can easily express certain phenomena, while it is more challenging or even impossible with others. As we look at the repertoire of colors, patterns, shapes, arrows, textual and pictographic elements used, we can recognize what aspects are prioritized by one system over others, and so we can put these systems into ‘families’ based on their similarities. And when it comes to an improvement of one’s marking or signage system, examples from other systems can be adapted and so, by extending the system, similarly to how mathematics works, by introducing new concepts such as negative, fractional or imaginary numbers while keeping and generalizing rules and relations of the existing system. When, for instance, the Hungarian trail marking system was to be revised in 2013, it was not hard to extend it in a consistent way, due to its robustness and flexibility.
Looking at trails as spatial entities, besides geometry, an area of particular use is graph theory. It looks at objects with their connections, their relative arrangement, layout or topology by abstracting away their exact particular geographical locations or shapes. Since Euler’s classical ‘Seven Bridges of Königsberg’ problem – where he wanted to find a roundtrip through the bridges crossing each one only once, and eventually found out it was not possible with the given configuration – concepts and methods of graph theory have been developed through the ages. Transportation or utility systems, but biology and more recently, social sciences have been widely using these concepts and tools, among others. They are effective in looking at shortest paths, round trips with certain properties, flows and capacity analysis or planning, or the significance of certain sections or junctions (nodes). Graph theory has found its natural way into transportation systems, which are somehow similar to trail systems. However, looking at the use and dynamics of them, it turns out that trails need different concepts – for instance, hikers tend to like round trips instead of seeking the shortest path from one place to another. And in some cases, 3-dimensional solutions can mitigate conflicts arising from shared use (e.g. bridges and tunnels where footpaths meet cycling routes) rather than 2-dimensional trail networks. One may analyze hikers’ movement patterns along a trail network, and categorize their trip shapes – such as forth-and-back, circular, or petal, etc. Some of our visitor surveys have, for instance, shown discrepancies between the actual patterns and hikers’ preferences or ideals, which poses interesting questions about how they perceive the area and its trail systems and why they do not (/want? / can?) actually hike the way they wish.
Another question comes, for instance, when talking about a closure of a trail section or an establishment of a new trail section and its potential impacts of possible trips along a network. Knowing hikers’ preferences, possible starting and destination points, trip ranges, etc., how do the possible trips change by removing or adding a section into the graph? Which trailhead in a network offers the most variety of certain types of hikes? We could use combinatorics to enumerate all possibilities, and recognize the huge number of options, but the actual number will usually not be too informative. Instead, we have developed a new type of indicator – called the trip variety index – which is more understandable, taking into account the possible repetitions of trail sections among the variety of trips. Such concepts, specifically developed or adapted for trails, seem to be more indicative for assessments in trail systems than some of the conventional indices used for transportation networks.
Identifying and classifying trip types and movement patterns is only a part of a more holistic visitor monitoring approach used for areas with extensive trail systems. The main interest to get to know hikers visiting an area is driven by a need for effective visitor management measures, including decisions on how the system should be improved and what probable effects of a particular improvement will have on the larger system. There has been a discussion, for instance, on introducing additional trail connections and/or public transport (shuttle) services in the Pilis Biosphere Reserve (located in Transdanubia and Central Hungary), to improve sustainability and better distribution of hikers in the area. Regular and systematic visitor monitoring has shown how the recently added trail sections are used and how the usage of other related trails have been changed, and how adding a public bus service will allow a new variety of possible trips.
And this is where the numbers come in again: visitor numbers. Our specific visitor method includes not only counting hikers at specific spots, but also surveying their trip routes. If this is done systematically, so that potential visitors entering the area can be ‘captured’ (by a so-called ‘closure method’ in selecting the survey points) this makes it possible to aggregate visitor numbers potentially along the full trail network graph, beyond the observation points. By adding questionnaires and linking their data to the routes the questioned hikers take, one may get valuable information on the types of visitors along certain trail sections and the trips they tend to take, informing visitor management measures, such as information, signage, etc. along those trail sections. Aggregation of numeric values and looking at trends in time can, however, be tricky. Beyond the already mentioned vague phenomena of how to summarize trail lengths and visitors, one must be careful when comparing trends based on ratios such as expressed in percentages. The well-known Simpson’s paradox is an example of a seemingly increasing trend given some percentages and particular variables which diminishes or turns to the opposite when one aggregates over a distinctive variable and looks at the totals, again, in percentages. Statistics should be done in the right way, and so done with trails as well.
And we have not yet even mentioned the emerging techniques of artificial intelligence, the surveys and data available through online channels, the power of logics and rule-based signage planning, etc. If we look at the variety and robustness of the multitude of fields of mathematics brings into the study of trails, there seem to be almost endless possibilities for Trailology to utilize and develop further.
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