Multiplier Theory: Stat Synergy explained with Polytopes

(Would like to start out apologizing for the somewhat awkward terminology used, I wanted to be clear, and the day-to-day terms for these things are very ambiguous in English)

The Concept
Imagine a quadrate. The length of one side is your AD, the length of another is your ASPD. The area of the quadrate is your DPS. The best ratio between circumference and area for a quadrate is found when all sides are equally long, i.e. when it's a square. So when you have a lot of AD, more ASPD gives the quadrate more area than AD does. Now, make sure you understand this metaphor, as I'll be using it a lot.

The two lengths of the sides of the quadrate is what I call a "multiplier" or an "axis" in mechanics lingo. And they're very important for understanding scaling and stat interacting. When there are two multipliers involved, it's a quadrate. Now, if I add Crit to it, what happens? It turns into a hexahedron, going from a 2-dimensional shape to a 3-dimensional shape. Now, the DPS is the volume. Once again, the longer the other sides, the better the shorter sides get.

The problem with this metaphor is that it gets fairly abstract once we add flat percentage damage increase like those from masteries or various abilities. Now it's the 4-dimensional equivalent to the 3D hexahedron and the 2D square. So it gets harder to visualize. I'll be sticking to 3D in the examples of this post though.

Concrete Examples
Lets look at the hexahedron with AD, ASPD and Crit as the length of its sides. How does stat extensions interact with this? Like Malphite's Shield scaling off a percentage of your health. I've been met with a lot of skepticism when I say it scales just as well off armor and magic resistance. It actually scales off EHP - effective health points. The defensive equivalent of DPS.

What about Infinity Edge? That multiplies Crit damage much like Malphite's shield multiplies HP. Well, the thing is, Malphite's shield always applies, it directly multiplies EHP. Crit damage is only applied to the Crit part of your DPS. So if you had 100% Crit, then yes, the Crit damage part of IE could be considered a full multiplier. Otherwise, it's worth 50% of your Crit Chance.

What about an item like Deathcap, that multiplies the main damage stat of AP which many considers the direct equivalent of AD. But it's not, because AD carries auto attack damage comes entirely off one polytope, where all the stats interact with eachother. AP only interacts with the scaling of abilities, not the base. So while a 10% increase in AD is a total auto attack DPS increase of 10%, an increase of 30% in AP is not 30% more ability damage dealt.

And that is the mechanical explanation of why the AD carry has the highest damage scaling. Almost all of the stats interact with each other in one polytope with many effective dimensions.

This is also why on hit effects, such as the one on BoRK, are less good than they seem. Especially when it first came out, BoRK's on hit damage was often compared directly to the AD of Bloodthirster. But that's not the case. BoRK does not scale with Crit or abilities, but AD does. It's the same with other on-hit effects and abilities.

There's a reason the timeless Zerker Greaves, IE, PD, LW build is so damn good on ADCs. It's two ASPD items, two AD items, two Crit items, an ARP item and a Crit Damage item.

Multiplier Theory: Stats stacking with themselves

Non-synergistic self-stacking

I stumbled upon two quotes from the Wiki.

"Liandry's Torment is probably best for ability power champions that can deal percent health magic damage like"

"Blade of the Ruined King is a great item choice for practically any autoattacking champion that does percent health damage like"

This is incorrect. Not that it cannot be true for Liandri's or BoRK to be good on one such champion, but that is never on merit of those two items dealing percent health damage as well as that champion having percent health damage. These two stats do not have increasing returns, they do not synergize with themselves. If anything it opens you more up to counterbuilding by focusing on resistances and offensive/utility items. 6 percent plus 4 percent is 10 percent. It's not like CDR or Penetration, which have increasing returns. It's like AD, ASPD, crit, AP. The first 10 points does exactly the same as the last 10 points.

Remember back in season 1 when everyone was really bad, building tons of HP on Cho'gath because "he already has a lot of HP"? I do. Maybe my friends were bad. Maybe I was bad and in the bronze of normals. But it annoyed me to no end. But I see some of the same a lot today.

It's not that having a lot of something makes more of that stat worth less - it just makes other stats that interact with it worth more.

Synergistic Self-stacking

There are however stats that synergize with themselves. That have increasing returns. One is Penetration. Flat penetration and % penetration works well together, flat penetration and flat reduction works good together as well. Penetration builds are particularly good on champions with great base damages and on champions that generally go for low-MR/Armor targets.

Y=Damage; X=ARP

Another is CDR. Up to the cap of 40% naturally.

y=percent more casts; x=CDR

Then there's two other stats with some technicalities.
One is movement speed, which has 3 different stats that scale multiplicatively with each other. Flat movement speed, percentage movement speed, and the other percentage movement speed from certain abilities. Not all movement speed increases from abilities fall in the "other" group though.
However, movement speed is affected by two softcaps - 20% reduction after 420 ms and 50% reduction after 490. So movement speed increases act a bit odd, gaining efficiency until 420ms, then losing a bit before regaining even more efficiency, then losing a lot at 490 before gaining more and more and more efficiency. Reference the vids of Hecarims and Rammuses that go halfway across the map in a few seconds.

Y=Post caps, X=pre caps

The other technicality is healing, through any means. Net change in health is damage taken subtracted by healing taken. Once healing taken rises above damage taken, you're immortal. The reason why I state this as a technical but not practical self synergy is because of burst damage and, in case of lifesteal and spellcasts, crowd control. It is, however, still a phenomenon that can be abused in some situation. Particularly low CC and low damage scenarios, like one-on-ones and small skirmishes. (And why Aatrox is busted in 3v3. Aatrox Spreadsheet)