The Coastline Paradox is the counter-intuitive observation that it is in fact impossible for a land mass to have a well-defined perimeter. The length of a coast is limited only by the instrument used to measure it. A longer and, hence, less accurate unit of measurement means a greater degree of cartographic distortion. For example: if one were to measure the coast of Great Britain with a 100 kilometer ruler, it would be approximately 2,800 kilometers. With a 50 kilometer ruler, which allows for a greater degree of accuracy, the coast would measure 3,400 kilometers. This is a near twenty five percent increase in beach front property just by changing our ruler.
Similarly, if we constrain ourselves to the point to point measurements of GPS systems we short change ourselves in the total distance ridden on any given day. This never really occurred to me until we spent several days riding the back roads of the Northern Chilean Altiplano. As we continued slogging our way over the trocha and calaminas (dirt roads and washboards) I took some time to brush up on my Non-Euclidian Geometry in order to figure out just how far we were actually traveling.
We’d been rocking up and down bumps in the road all morning. I noticed they were quite uniform, typically about a foot in width and two inches in height measured from crest to trough. The following drawing (not to scale) represents a four foot section of road. As you can see, it has two crests and two troughs.
With these assumptions, the total length traveled can be determined with some relatively simple geometry in which we use the chord (C) of a circle to determine the length of the arc (A).
In order to find length A, we need to assume that the arc is part of a complete circle and find the radius (r) as well as the angle (a) between the two end points of C.
We can determine the diameter (D) and radius by inserting values B and C from the above example into the following formula:
With our radius, we can now reverse engineer the chord equation of a circle to find angle a.
From here, the length of the arc, A, is simply equal to the ratio of angle a to 360 degrees (the sum of all angles in a circle) and then multiplied by the circumference of the circle (which is of course 2πr).
This means that in 1 foot of road with our assumptions of a 2 inch drop from crest to trough and a width of 1 foot there will actually be 1.018 feet of travel. This is a nearly 2% increase in distance. So in a 100 kilometer day we would travel closer to 102 kilometers. This may seem insignificant, but when you’re dead tired and trudging 4km/hr uphill and into the wind it is an extra thirty minutes at the end of your day that you’d probably rather spend with your feet up.
In addition to more linear traveling, you will climb an extra 8% of whatever distance you ride. So over 100 kilometers, the washboards alone will account for 800 meters of climbing. Throughout the Andes, we have been averaging about 2,000 meters of climbing for every 100 kilometers we ride, so this effectively makes the grade feel twice as steep as what the map will show.
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