A mathematical proof of why Turrets are better than Archers

[xRubiks]

So let's compare stats:

Turret cost: 2100 (tower + upgrades)
Archer cost: 226 (tower + upgrades)

Turret DPS: 27.5 (.5 from fire damage)
Archer DPS 6

Turret space: 25
Archer space: 9

You need 10 Archers to equal the cost of a Turret.
10 Archers is 60 DPS, which is about 2.18 times as much DPS (we'll say 2.2)
10 Archers takes up 90 blocks, which is 3.6 times as much space. This space could be used for something else.
75 blocks of Turrets and 9 blocks of archers (84, maximum amount) is 88.5 DPS. This costs 6526 coins.
Repeating - getting 6526 coins in Archers is 29 Archers (rounded).
29 Archers is 174 DPS, which is about 1.9 times the DPS as the Turret-Archer combo.
29 Archers takes up 261 space, which is about 3.1 times the 84 blocks of the TAcombo.

Remembering land defense, Turrets are amazing.
That's why.

 

[betty's oldies]

Not even close. You're forgetting the fire rate and also that each arrow does NOT deal 9 damage. Tower damage actually varies. You also compared a level 3 turret to a level 1 archer in terms of DPS. You also assume that the spaces are perfect such that no area is lost.

How arrow-based towers (turret/archer) work: arrows have to have a clear path to travel from the tower to the mob. Any building in the way will block the shot. Additionally, more arrows from a tower do not allow it to shoot all of them at 1 target. A level 4 archer shoots 4 arrows means it locks onto up to 4 targets, 1 target per arrow. That means that each upgrade allows the archer/turret to lock onto an extra target (+1 arrow per upgrade), increases the range of the tower, and increases the fire rate.

Below we will assume both towers are fully upgraded and that there's plenty of space (they give you more than enough space for a decent defense). A higher arrow/cost ratio indicates higher cost-effectiveness.

Turret:
-Fires 3 arrows per 0.5 seconds. That's up to 3 targets per attack interval.
-2100 coins as you noted above. In terms of arrow/cost, this is 0.00143 arrows per coin.
-Fire DoT. This is redundant if you have mage coverage and all but silverfish are immune to fire! Against fire-resistant mobs, fire DoT does nothing, effectively making it an extremely expensive archer!
-5x5 space
*Comment: if this tower were to have a major overhaul such as the ability to deal AoE air damage, I'd take it. Until then, no.

Archer:
-Fires 4 arrows per second. That's up to 4 targets per attack interval.
-226 coins as you noted above. In terms of arrow/cost, this is 0.0177 arrows per coin.
-You need 1.5 archers to equal the DPS of 1 turret. In other words, 2 turrets = 50 blocks while 3 archers = 27. Archers still win.
-3x3 space.
-No fire DoT. You'll get that naturally when you place mages.
*Comment: your bread-and-butter and probably one of the best anti-air towers in the game due to its high cost-effectiveness.

Spacing
In theory, you're right that 10 archers would use more space than a turret at the same value. However, as stated under "archer," when it comes to DPS then the archers use less space than the turret.

In practice, you're always going to have to lose some free space to fit towers in certain spots. 3x3s are more common than 5x5s. If you have a 6x6 area (some maps have this) and you fit a 5x5 in it, you can't build anything else, resulting in 11 blocks of space lost in that confined 6x6 area! On the other hand, you can fit 4 archers in this space without losing any space. Archers towers have another advantage because you can fit this tower in tight spots where a turret wouldn't fit.

What about against ground units? Archers are the bread-and-butter against air and are ineffective against them when TNT is already the bread-and-butter against ground. One doesn't build archers and turrets against ground because a TNT or mage tower can target more than 4 mobs at once, thus giving them better damage for cost!

TL;DR: Turrets are trash and rip you off. Get archers instead. In most cases, the basic towers are your bread-and-butter and are better than the unlockable ones.