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The fortifications of Copenhagen, Fredericia and Acre are star-shaped. (Defense of Copenhagen is shown in the picture below.) Why not have defenses or fortifications in the shape of a pure square or circle?

enter image description here

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Star forts or bastion forts are designed to enable enfilade (or flanking) fire: shooting on the line of attackers from the side, significantly increasing firing efficiency of the defender.

Flanking fire allows guns placed in the side wall of the bastion (protected from direct fire from attackers) to safely and effectively shoot at an entire line of attackers (enfilade them) as they are closing to the neighboring bastion. Flanking fire is very effective, because if you miss one attacker, you hit the next one. Check the enfilade link for images.

Bastion with flanking fire

Here the red arrows show direction of the flanking fire from the guns placed in the protected side walls ("flanks", #3) of the bastions (#1 and #2). More details about the placement of guns in bastion's flank side is in this image of bastion with caponier. Guns in "face" walls (#4) would be much more exposed to enemy's fire.

The star fort has no "dead zones", where an attacker can hide from flanking fire, like a round turret allows.

The star design for bastions was first introduced in Italy in the 15th century, and later perfected by Vauban - in the Citadell of Lille and others. Wikipedia has more nice images and a history of the approach.

Other answers mentioned an inability to fire from above, but that is not the main reason for the star design: its goal is to enable flanking fire. And yes, star shaped bastions do not allow to fire on attackers from above, but that is not necessary (and would be dangerous for defenders), because the walls are subject to more effective flanking fire from the side walls (protected from direct fire of attackers) of neighboring bastions.

Of course Vauban also designed a way how to attack such star fortress, by digging 3 lines of parallel trenches connected by zigzag trenches to avoid enfilading fire down the trench line, with 3rd parallel coming close to the attacked fort's glacis (outer edge of the fortifications).

Vauban's design to attack start fort

Images are from Wikimedia/Wikipedia.

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    This should be the accepted answer. The mention of "dead zones" is indeed incorrect (and sounds like a rationalization for the inevitability of star forts and related designs). In fact, square and round forts served their purposes just fine for centuries, but the star fort was an innovation that made fortifications even more effective by providing for flanking fire (once the tools for such fire became widespread, at least in the West). Additionally bastions made it possible to use heavy fixed weapons (which could be pivoted to cover longer stretches of wall). – orome Feb 27 '17 at 14:14
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    Except you still reference the “accepted answer,” which may not remain the accepted answer, or might be edited, or any number of other things. Referencing other answers is a poor idea in general, and when you must it is best done by link (see share under the answer). Finally, your reference doesn't actually make sense—that answer talks about having no angle but from directly above against enemies against the wall of a circular fort, as a drawback of such forts. There is no claim that stars suffer from the same. The claim is really just another way of saying they lack angles for flanking. – KRyan Feb 27 '17 at 17:05
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    The point of other answers were that round and square forts didn't provide a good flanking angle and often times invaders at your walls would have to be defended from above. No one else seemed to suggest that there was a dead zone in a star fort. They were saying what you were saying, the goal was to enable flanking fire so you didn't have to try and fight them off from above. – JMac Feb 27 '17 at 19:39
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    Crossfire is important. In a 5-sided star-shaped fort, any approach is exposed to at least two walls worth of fire. – ttw Feb 28 '17 at 17:48
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    Aerial photograph of a beautifully preserved Vauban-style fort at Ciudadela Jaca Vista in Spain. Note the surrounding trench (NOT a moat - as this would then flood the fort), which is covered by multiple cross-firing enfilades from musket slits as an integral part of the fort's defences, and protects the base of the curtain wall from artillery fire. – Pieter Geerkens Aug 8 '18 at 9:15
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The big disadvantage of a square or circle is that once an attacker had reached the wall, they are more or less immune to danger from the defenders. The only place they can be attacked from is directly above, having stuff dropped on them. This means that being at the base of the wall is a place of relative safety for the attacker.

Conversely, for star forts, or the Copenhagen example in the question, it is easy to see that being close to the wall provides no real benefit to the attacker: there are always plenty of firing positions for defenders in cover to shoot the attackers from multiple angles. This makes the job of breaching the walls, or of moving around the base of the wall to a weak point, much harder: attackers are exposed everywhere

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    Technical term for this in "enfilade", as I explained in my answer. – Peter M. Feb 27 '17 at 14:22
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    Another reason not really so far is that the skewed angles of a star-like shape allows to deflect incoming cannon ball. If you take a square or circle, the area of impact is quite plane which maximize the impact of attacker's cannon. – gdupont Feb 27 '17 at 18:23
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    Episode 9 of the original "Connections" series from 1978 discusses this at length with visual demonstrations at real fortifications. – JimmyJames Feb 27 '17 at 22:24
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Imagine that you are defending a fort with old-fashioned round bastions in the Middle Ages. Your walls are high, but once the enemy comes close and starts scaling the walls, very few of your defenders can engage them.

old fort

After the invention of cannons it's even worse. Your walls have to be thicker and much shorter in height, otherwise they would be easily demolished by cannon fire. Therefore, the enemy can more easily scale them with ladders. Also, you have very few places to put your own cannons, and they would also be open to enemy fire.

Now look at the star fort instead.

star fort

You have a lot more wall surface you can shoot from, and more importantly, while the enemy is scaling the bastion 2 with ladders, your own men can fire at them from bastions 1 and 3.

Wherever the enemy decides to attack from, they will be under fire from multiple directions at the same time.

Also, the cannons indicated on the image can fire from a very safe spot, protected from enemy artillery. The enemy can no longer destroy your cannons via firing their own cannons from the safety of their camp: in order to even be able to take a shot at your cannons, they have to move very close to the walls. Moving cannons is very slow, and they are very vulnerable in the open.

  • That's interesting in that I wonder why round bastions were prevalent at all/ever. Limited construction technology? – Kzqai Mar 5 '17 at 21:19
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    @Kzqai, prior to the invention of flintlock muskets, most fighting was at hand-to-hand range. In such a situation, tall walls are better than thick walls, and it's easier to make a tall round tower than a tall pointy tower. – Mark Oct 4 '17 at 1:50
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These outcroppings are called Bastions and they became common when muskets and cannons became common in European warfare.

The problem with a square or circle forts is that when the enemy has reached the wall, they are in your dead zone. You can't fire at them with your musket, unless you lean over the wall, which would make you a very easy target for the opponent's musketeers. Using your cannon against them is completely impossible. All you are left to do is throw stuff at them, just like people did in the early middle-ages.

But in the same situations on a bastion fort, your comrades on the adjacent bastion can help you. They have a perfect line of fire to the enemies which try to climb up to you. And you can pick off those enemies which climb up to them.

And anyone stupid enough to assault the wall between two bastions will find themselves being fired upon from two sides at once.

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