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Why does the path of the Great Wall of China bend back and forth so much? Naïvely, it would have taken less material and guards to defend if it had taken a straighter path, so there must have been some reason. But what?

Great Wall of China, #1

Great Wall of China, #2

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    Ruddy mountains refused to line up straight to have that wall built on top of them. Ruddy guards refused to man a wall in the valley to have not-so-nice things shot at them from above. ;-) – DevSolar Aug 11 '16 at 15:32
  • Not just the Chinese wall. Much of Hadrian's Wall en.wikipedia.org/wiki/Hadrian%27s_Wall is built at the top of hills and scarps. – jamesqf Aug 21 at 3:52
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The Chinese designed the wall to be an effective barrier; that was the goal. And to answer your question, we need to ask: what land barrier stopped foreign troops the best?

The answer in China, and everywhere else, is Mountains. The Great wall is nothing more than a fortification of existing natural barriers. Look closely at the pictures you've posted, and those posted by Ken Graham. You'll see that they built the great wall directly on the mountain ridge. So this is the easy reason the route is bendy:

Because it follows a mountain ridge

I don't know about you, but I wouldn't enjoy building a wall along a mountain ridge. It was a ton of work just to survey the land, and clear trees in both directions from the construction site, and then carry the materials to the site. The expenditure of materials might have been the least costly component.

All this being said, the the real barrier was the mountains. On these mountains, the wall served 3 major functions:

  1. To fortify the mountain passes, yet allow trade to continue

    At the passes, the wall was actually the primary barrier. The wall in passes are very high, with multiple wings.

  2. To provide a location for watchtowers at regular intervals.

    These watchtowers were not just for watching the terrain north for gathering troops, but also for communications between towers.

  3. Logistics: To provide a direct path between mountain pass fortifications for the transfer of troops, equipment, food, and other materials.

When I've hiked the wall, I found it very difficult, but far easier than traveling along a natural mountain ridge. I think the men who ran along the wall wearing armor and carrying weapons were strong men indeed.

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    Note, also, that in the absence of ports the wall is horse-proof even if unmanned. (That is, short of actually tearing it down.) – Loren Pechtel Aug 11 '16 at 23:28
  • So, despite taking more resources, it's cheaper to build, cheaper to maintain, and more useful once it's built? Explains everything. Thank you. – BenRW Aug 12 '16 at 11:30
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    The very short section I've walked along seemed to be placed to give a good view out from China. It would have been difficult to mass an army anywhere close to it, on the outside, without being noticed. – Patricia Shanahan Aug 12 '16 at 22:18
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    Not only horse-proof but cart-proof. This means you cannot invade with a baggage train. – user207421 Aug 13 '16 at 0:16
  • "was could might have been"? – Lightness Races in Orbit Aug 13 '16 at 22:52
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The Great Wall of China:

It is the longest man-made construction in the world. In the old times, it was of great military importance of preventing the enemies' intrusion and was regarded as the 'Guardian Angel' of the central plain in the past. - Great Wall of China FAQs

Why did they build the Great Wall of China?

The Great Wall of China is the longest man-made structure in the world, stretching 21,196.18km (13,170.6956 miles) long. It was built to keep out raiding parties of nomadic tribes, such as the Mongol, Turic and Xiongnu, from modern-day Mongolia and Manchuria. The first Emperor of China, Qin Shi Huang, is often considered the father of the Great Wall, but even before he united the nation in 221BC, individual states built walls to keep out invaders as early as the 7th Century. Qin connected, lengthened and fortified the walls to protect the northern border between 221-206BC. Subsequent dynasties, most notably the Ming, maintained and rebuilt it.

Always maintained as a military defence – at its peak the Ming Wall was guarded by more than one million men – the Wall evolved other uses. Aside from being a transportation corridor, it was used to regulate trade, such as collecting duties on goods transported along the Silk Road. It was also used to restrict both immigration and emigration.

But why does the Great Wall of China follow such a bendy route?

The wall was generally built at the boundary line of the day. In normal conditions, taking the commanding position was the most important factor to win a battle. In this case, lofty mountain top was always the preferred location to build the wall. - Great Wall of China FAQs

The Great Wall of China

The Great Wall of China

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High ground is also easiest to defend. As noted before, it was built in top of the ridges. This offers a defensive advantage, as well as good visibility.

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    This is too brief to be a good answer. It contains no sources or references and adds nothing that isn't covered in other answers and comments. – KillingTime Aug 14 '16 at 8:21
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    No reason to be overly verbose. Occam's razor. – user1780242 Aug 14 '16 at 21:35
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Mountain roads are "best" built along ridge lines that minimize (to the extent possible), the surface area of the mountains containing the roads to minimize the necessary excavation. This, in turn, is dictated by the process of building the road through the parts of the mountain, where the slopes are the "minimum."

At its heart, the Great Wall is just a long "road" with walls on both sides. You can see from the pictures that the mountains on either side tend to have steeper slopes than the ridges through which the wall is actually built. These ridges are the points of minimal slope changes. (Think of these slopes as "hypotenuses" over the x axis with the ideal slope being zero.)

A path that was "straight line" (as the crow flies) would usually violate the above rule, and therefore be inefficient in a real environment, even though it would look "optimal" on a map. So a wall with appropriate "bends" will "take less material and guards to defend [than] if it had taken a straighter path."

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    This would be a reasonable guess if you haven't looked at the photographic evidence that others have posted: the wall almost always follows the mountain ridge. – Max Aug 11 '16 at 18:41
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    @Max: Ridges, by definition, are points on the mountain where the slope changes are closest to zero, hence a minimum. Added a line and a link to make this point. – Tom Au Aug 11 '16 at 18:57
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    It's quite clear in the pictures that the ridges are not contour lines (i.e., they are not at constant height). – Max Aug 11 '16 at 19:44
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    @TomAu I think you're confused about what a contour line is. Like Max says, the Great Wall of China is clearly not at a constant elevation. In terms of your several variables: given two variables, x,y (lat,long) the same z (elevation) is always found if the wall were built on a contour. But here the wall just stays as high as possible. Also, If it were built on a contour the wall would encircle a mountain. – Brad Aug 11 '16 at 21:07
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    It sounds like you're talking about a geodesic, not a contour line. This is also plausible, and is going to get you the shortest amount of material, but it will often hug the sides of mountains, and would overall look much closer to an as-the-crow-flies straight line than the pictured curve. If you intend to talk about contour lines given some 3D objective function, what is the function, and how does it relate to the surface? – Mario Carneiro Aug 12 '16 at 6:23

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