Quick note before getting into the pictures and details, the pictures are done with the Enemy side of the field being towards the Bottom Left. Of course, my defining of Incorrect and Correct are just my own opinion.
Quite often I'll see a Heimerdinger placing his turrets in a pattern like one of the two shown here.
The first picture, all 3 turrets are placed across the lane in a line. The turrets in this setup have poor overlapping, and the only real danger area is that small oval-like area towards the center as that is where all 3 turrets can fire on the same target(s). These turrets are also close enough to each other that most standard AOE abilities will commonly hit 2 turrets. Also, this limits your turret's coverage to that line and you are provided no safety against enemies who chase as you retreat behind this line.
In the second picture, all 3 turrets are spread out way too much and have little to no overlap. while turrets placed like this will be almost impossible for most AOEs to hit multiple turrets, there is very little concentrated damage output.
These are the two turret placements that I commonly use.
In the first layout, the turrets are spaced out enough to avoid AOE damage on multiple turrets. However, they are positioned with a much larger overlapping area. In addition, only two turrets are placed forward, these are pretty much your offensive turrets (which is commonly all you need for minion waves). The 3rd turret is placed at the center towards the back, forming a triangular shape. This back turret is essentially a defensive turret and often won't be in range of incoming minion waves (though it will occasionally be able to fire on minions).
You should quickly notice that this covers a larger portion of the map, while also extending the area towards the back (using the defensive turret) that can fire on attackers should you need to retreat. In addition there is a decent sized section in the center of all three turrets where they can all fire on the same target.
The second layout is very similar to the first. The difference being that the turrets are much closer together as they are not threatened by AOE damage from your opponent. Immediately you'll see that the overlapping area in the center is much larger. This provides a bigger threat to champions such as Diana as they will be in range of all 3 turrets for a longer period of time than they would be if you used a more spread out pattern.
Of course, you don't want to stack each turret on top of each other as that reduces the total area that is covered by the turrets. These remain spaced out a little for that reason.
Moving / Refreshing the Turrets
With the two above formations, moving and refreshing the turrets is simply remembering in what order you placed the turrets. Typically you'll want to have your oldest turret as your defense turret (the one in the back) and then the two side turrets can be any order.
Moving my turrets forward, I'll simply move the back turret forward to the front, and then the two side turrets will be moved forward.
Of course, turrets that are already placed on the field will not gain extra damage as your AP increases or as you rank up your Q. So when I find the difference in damage to be significant enough, I will go through each turret and replace them with a new one that has increased damage over the old one.
Who knows, maybe the majority of you Heimerdinger enthusiasts already understand much of this...
Or perhaps not, and its for those people I provide this for. Enjoy.
Now with Color!