<p>It's actually more complex than you would think because landscapes are approximately <a href="https://en.wikipedia.org/wiki/Fractal_landscape" rel="nofollow noreferrer">fractal surfaces</a>. An analogue is the <a href="https://en.wikipedia.org/wiki/Coastline_paradox" rel="nofollow noreferrer">coastline paradox</a>. Basically, the problem is to measure the length of a coastline. Simple, right? But, as it turns out, it's totally dependent on the resolution used. The smaller your measuring stick, the more little inlets you measure, and the perimeter grows to infinity in the limit of small measuring sticks as you measure around every little sand grain and rock. Lines like these are, statistically, fractals, in the sense that, looking at smaller and smaller scales you will see about the same amount of looping back on itself. These are common in nature; a better example (because it is a fractal to much smaller scales) is a cloud.</p>
<p>Similarly, there is no preferred scale when looking at a landscape, because large mountains look approximately like hills (hills in the same region-the statistics are not the same around the world). The way to measure the roughness of these surfaces is by their <a href="https://en.wikipedia.org/wiki/Hausdorff_dimension" rel="nofollow noreferrer">fractal or Hausdorff dimension.</a></p>
<p>Of course, however, all of these things are not perfect fractals and are really only fractals for a certain size range. Big mountains look like little mountains, but rocks under your shoe look different. And in my explanation above, it's questionable to compare the sand grains to the coast as you would see it on a map. And, finally, I don't have expertise regarding what actual geographers use as their measure of roughness, but they certainly have to contend with fractal statistics.</p>

CC BY-SA 4.0
2021-08-12T20:44:57.140Z
2021-08-12T20:44:57.140Z
Thomas Dewitt
<p>I'm very interested in Russian history, and this led me the other day to look at a beautiful painting of the famous Vladimir Highway. The painting is famous in its own right, and is said to express deep truths about the Russian soul. It made me ask myself: <strong>Is Russia flatter than the UK?</strong> (I'm English.)</p>
<p>The question seems fairly straightforward, but I can't see any straightforward way of answering it. On the one hand, Russia's highest peak, Mount Elbrus, is some 5642 m above sea level, and thus over four times higher than Ben Nevis, the UK's highest point. On the other hand, Russia's famous lack of defensible terrain has led to Moscow being besieged and/or razed numerous times, whereas the moorland which constitutes most of the border between England and Scotland hindered the English from ever subjugating their northern neighbours militarily.</p>
<p><strong>Average elevation</strong> seems like a sensible statistic to look at - until one remembers countries like Botswana, which is located on one large plateau, and which thus has a high average elevation, and yet (I'm told) is rather flat.</p>
<p>I suppose one would want something like an <strong>interquartile range</strong> of elevations. But how would one pick, say, 100 elevations in the country in question, from which to pick the 25th and the 75th highest?</p>
<p><strong>Is there already one single, generally accepted statistic for measuring how flat or hilly a given country, county, parish, etc is?</strong> If so, how do Russia and the UK compare to other countries?</p>
<p><a href="https://i.sstatic.net/x1ESe.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/x1ESe.jpg" alt="vladimir_highway" /></a></p>

CC BY-SA 4.0
2021-08-13T00:28:22.297Z
If specifically interested in flatness, suggestions that come to mind might be (percentage of nation where all terrain within 10mi of point is no more than x meters of variation) or (percentage of nation where slope averaged over some area is smaller than x meters).  Interesting question.  I don't see the fractal problem here as we're not really interested in measuring distance or even interested in slopes over small areas, but instead area averages... certainly the averaging scale is important, but should be connected to human perceptions reasonably enough.
JeopardyTempest
CC BY-SA 4.0
2021-08-13T02:46:42.357Z
[The roughness of natural terrain: A planetary and remote sensing perspective](https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JE001429)
Keith McClary
CC BY-SA 4.0
2021-08-13T07:08:09.063Z
Kansas is flatter than a pancake: https://www.improbable.com/airchives/paperair/volume9/v9i3/kansas.php
Jean-Marie Prival
CC BY-SA 4.0
2021-08-13T08:58:15.680Z
@Jean-MariePrival: I hope the author of that "paper" was nominated for an [Ig Noble prize](https://en.wikipedia.org/wiki/Ig_Nobel_Prize) ;-)
Fred
CC BY-SA 4.0
2021-08-15T00:50:47.740Z
@JeopardyTempest An alternative is a wavelet analysis, with cut offs at frequencies about which humans don't care.
David Hammen
<geography>
Is there any single, generally accepted statistic describing the flatness of a given territory?
Tom Hosker
# Is there any single, generally accepted statistic describing the flatness of a given territory?

(Asked by: username_0 on 2021-08-12)

I'm very interested in Russian history, and this led me the other day to look at a beautiful painting of the famous Vladimir Highway. The painting is famous in its own right, and is said to express deep truths about the Russian soul. It made me ask myself: **Is Russia flatter than the UK?** (I'm English.)

The question seems fairly straightforward, but I can't see any straightforward way of answering it. On the one hand, Russia's highest peak, Mount Elbrus, is some 5642 m above sea level, and thus over four times higher than Ben Nevis, the UK's highest point. On the other hand, Russia's famous lack of defensible terrain has led to Moscow being besieged and/or razed numerous times, whereas the moorland which constitutes most of the border between England and Scotland hindered the English from ever subjugating their northern neighbours militarily.

**Average elevation** seems like a sensible statistic to look at - until one remembers countries like Botswana, which is located on one large plateau, and which thus has a high average elevation, and yet (I'm told) is rather flat.

I suppose one would want something like an **interquartile range** of elevations. But how would one pick, say, 100 elevations in the country in question, from which to pick the 25th and the 75th highest?

**Is there already one single, generally accepted statistic for measuring how flat or hilly a given country, county, parish, etc is?** If so, how do Russia and the UK compare to other countries?

[![vladimir_highway](https://i.sstatic.net/x1ESe.jpg)](https://i.sstatic.net/x1ESe.jpg)

**Tags:** geography

> **Comments:**
> **username_1**: If specifically interested in flatness, suggestions that come to mind might be (percentage of nation where all terrain within 10mi of point is no more than x meters of variation) or (percentage of nation where slope averaged over some area is smaller than x meters).  Interesting question.  I don't see the fractal problem here as we're not really interested in measuring distance or even interested in slopes over small areas, but instead area averages... certainly the averaging scale is important, but should be connected to human perceptions reasonably enough.  
> **username_2**: [The roughness of natural terrain: A planetary and remote sensing perspective](https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JE001429)  
> **username_3**: Kansas is flatter than a pancake: https://www.improbable.com/airchives/paperair/volume9/v9i3/kansas.php  
> **username_4**: @username_3: I hope the author of that "paper" was nominated for an [Ig Noble prize](https://en.wikipedia.org/wiki/Ig_Nobel_Prize) ;-)  
> **username_5**: @username_1 An alternative is a wavelet analysis, with cut offs at frequencies about which humans don't care.  

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## 1 answers

### Answer by username_6 on 2021-08-12. Score: 2

It's actually more complex than you would think because landscapes are approximately [fractal surfaces](https://en.wikipedia.org/wiki/Fractal_landscape). An analogue is the [coastline paradox](https://en.wikipedia.org/wiki/Coastline_paradox). Basically, the problem is to measure the length of a coastline. Simple, right? But, as it turns out, it's totally dependent on the resolution used. The smaller your measuring stick, the more little inlets you measure, and the perimeter grows to infinity in the limit of small measuring sticks as you measure around every little sand grain and rock. Lines like these are, statistically, fractals, in the sense that, looking at smaller and smaller scales you will see about the same amount of looping back on itself. These are common in nature; a better example (because it is a fractal to much smaller scales) is a cloud.

Similarly, there is no preferred scale when looking at a landscape, because large mountains look approximately like hills (hills in the same region-the statistics are not the same around the world). The way to measure the roughness of these surfaces is by their [fractal or Hausdorff dimension.](https://en.wikipedia.org/wiki/Hausdorff_dimension)

Of course, however, all of these things are not perfect fractals and are really only fractals for a certain size range. Big mountains look like little mountains, but rocks under your shoe look different. And in my explanation above, it's questionable to compare the sand grains to the coast as you would see it on a map. And, finally, I don't have expertise regarding what actual geographers use as their measure of roughness, but they certainly have to contend with fractal statistics.


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