Wikipedia:Reference desk/Archives/Mathematics/2014 August 8

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August 8

Geography: How do they compute the mathematical area of oddly-shaped land masses?

This is a follow up to my questions above (Wikipedia:Reference desk/Mathematics#Perimeter of the USA). In fact, this was going to be my original question, before I got sidetracked with the concept of perimeter. Officials often report the "area" of a geographic entity. For example, the area of the United States is 3,678,190 square miles (according to this: List of countries and dependencies by area). How do they calculate this? Obviously, land masses are not perfect squares or rectangles; they are very oddly shaped. Furthermore, on the borders of the country or state, there are usually many "jagged" edges, lots of "nooks and crannies". How do the officials measure all of that? Also, when they report the area of a land mass, do they "minus out" the bodies of water that are located within that area of land mass? Thanks. Joseph A. Spadaro (talk) 01:18, 8 August 2014 (UTC)

I think before computers they used something called a planimeter, a mechanical gadget that you use one end to trace the map and the other end tells you the area. With computers it's not a big problem to compute the area of a polygon with thousands or millions of sides. The map of the country would be stored in a computer as such a polygon (or a set of polygons if you count islands and whatnot). Unlike a coastline, the area converges to a fixed value as you take finer and finer measurements, so if you want more accuracy then use more sides in your polygons. The formulas for areas are more complicated for a shape on the surface of a sphere, but still no problem for computers, you just need a slightly smarter programmer. --RDBury (talk) 09:12, 8 August 2014 (UTC)

Thanks. But what is the difference between calculating a coastline (and, hence, perimeter) versus calculating an area? Don't you need the coastline measurement to, in fact, compute the area? In other words, isn't the coastline one of the "sides" of the polygon? Joseph A. Spadaro (talk) 13:19, 8 August 2014 (UTC)

See How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension for the problem involved with measuring the coastline. Due to its fractal nature the length of the coastline grows as you take smaller steps.--Salix alba (talk): 14:24, 8 August 2014 (UTC)

There's a standard surveying technique for calculating the area of an irregular polygon, such as a plot of land, by making only one measurement of length: it's triangulation. You go somewhere near the middle and pick two fairly widely separated points from each of which all the vertices may be seen; this is your baseline. You measure the straight-line distance between the ends of the baseline, and then (using a theodolite or similar), measure the angles from each end of the baseline to all of the vertices. You then calculate the area of a triangle, several times. --Redrose64 (talk) 15:45, 8 August 2014 (UTC)

Here's a fact that might surprise you, and I think will address your confusion. It is possible for an object of have an infinite perimeter, but still have a finite area! See e.g. Koch snowflake. So, even if we can debate about the 'true' length of the coastline of Britain, the difference between a two finite notions of perimeter may only result in a 0.001 m^2 difference in area. SemanticMantis (talk) 17:28, 8 August 2014 (UTC)

And you don't even need fractals; on the hyperbolic plane, a triangle with infinitely long straight sides (that are asymptotic to each other) has finite area. This theorem, they say, provoked Dodgson to dismiss hyperbolic geometry as absurd. —Tamfang (talk) 22:22, 9 August 2014 (UTC)

And you don't even need non-Euclidean geometry; in the Cartesian plane, the region where $0<x^{n}y<1<x;n>1$ has infinite perimeter and finite area. —Tamfang (talk) 18:34, 10 August 2014 (UTC)

For an example, say we have two figures:

+-+ +-+
| | | |     +---+
| +-+ |     |   |
|     |     |   |
+-+ +-+     |   |
  | |       |   |
  | |       +---+
  +-+ 

The left one is the more detailed version, and the right is the more approximate. Both have areas of 12, but the left has a perimeter of 18, while the right has a perimeter area of 14. The more detail is added, the larger the perimeter tends to get, while the area remains about the same. StuRat (talk) 23:37, 9 August 2014 (UTC)

I think you meant to say "while the right has a perimeter of 14." El duderino ^(abides) 20:12, 10 August 2014 (UTC)

Correct (and corrected). Thanks, StuRat (talk) 16:31, 11 August 2014 (UTC)

Let me just emphasize, the area of a landmass will be well-defined and finite even if the coastline has a fractal nature (as previously linked). As you heard before when you zoom in on a coastline or use a finer-grained ruler, the length of the coastline changes dramatically from what you measured before, while zooming in on a coastline to measure area will result in your measurements continuously converging on a single number, with the change in your calculation getting smaller on each zoom.

You can do this experiment at home: tear a jagged edge on a piece of paper. Measure the the area and perimeter using intervals of only inches, only cm, only mm, and with a magnifying glass, get even closer. Area will get closer and closer to some fixed number, while perimeter will almost certainly grow larger as you zoom in, especially with a magnifying glass if you can zoom in on the fuzzy bits of the tear.

One more thing: a "3D" fractal (like the Menger sponge) has a finite volume and infinite surface area, and it scales up from there. SamuelRiv (talk) 00:34, 10 August 2014 (UTC)

Thanks, all. Joseph A. Spadaro (talk) 19:02, 14 August 2014 (UTC)