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**Xypherous**:

Thornmail offers pure armor and a very smooth build-up into that armor. Being able to go Cloth Armor into Chain Vest is a very powerful build-path that wastes no time in getting to that 100 armor. The major advantage of Thornmail has generally been the ability to get all the armor and to get it without needing to spend gold on any other statistic along the way.

@Xypherous, my maths might be a little off, as I'm calculating that the value given at any level as being base + 17 levels of adjustment, as I'm not entirely sure if adjustment is included in level 1, if there is an adjustment for level 1, or if as I've calced level 1 is as printed and adjustments are on level ups only. I hope you take the time to read this.

Thornmail costs 1000g for the recipe, 700g for a chain vest (45 armor) and 300g for a cloth armor (18 armor) for a total of 63 armor pre-recipe. This means that 1000g is paying for 36 armor and the passive. Using chain vest as a base, at 15.5 gold per armor point, that means it costs 558 gold for the armor, leaving the passive costing 442 gold.

Even using the values for cloth armor (16.66... gold/armor) that gives the armor a cost of 600 gold and the passive 400 gold. since it first calculates damage before your armor affects it, a 500 damage strike against 195.69 (rounded to 196) armor (Shen level 18 base with armor seals) results in about 33% damage taken, or 168.91 (round to 169) damage from the attack, reflecting 30% of 500 = 150.

Say it's a Vayne hitting for 500, once per second, and she's level 18 too- No crits, though, or that'll add more issues to the calculation (in her favour) than I'd care to input right now. She has a flat 30 MR, + MR/level glyphs for +24.3 MR, total 54.3 MR. This results in Vayne taking 97.21 (round to 97) damage from the proc. Assuming she has a stacked Thirster (and she should have one by now) she has 20% lifesteal, 20% of 168.91 = 33.782 (round to 34). 97 - 34 = 63 damage given from the Thornmail for 169 damage taken. About a 37% efficiency.

Now, for comparison, Frozen Heart. Total cost is 775g more than a Thornmail, which isn't that expensive all things considered, since it has a median build with Glacial Shroud that is 475g cheaper than thornmail. This provides 45 armor and 425 mana, so less defensive at just a Shroud, but also provides a much-needed 15% CDR, which can severely help scale ability usage and increase your TPS and DPS. Now, the mana is wasted on Shen, true, but a manatank wouldn't have the same issue.

Add another cloth armor and it goes from 45 armor to 63 armor, the same as Thornmail pre-recipe, and the cost is at 175g cheaper than Thornmail. Pure survivability lost to more mana to use for casting spells- which are often utility or defensive in nature, or aggressive such as in the case of Maokai. 475 mana and 15% cdr I think is worth 36 armor and 30% reflect, at that point, for a mana champ. 300 g later, they are over the cost of Thornmail by 125g, and have increased armor from 63 to 81. Only 18 less than that of Thornmail. the remaining 650g of the recipe accounts for 5% CDR, 18 armor (a 300g value), 25 mana, and most importantly, a 20% reduction in attack speed aura against the entire enemy team.

Against the above Vayne example, with a 1.0 speed attack, she is reduced to 0.8 attacks/second. over 10 seconds she makes 2 attacks less than against Thornmail. this reduction in damage taken means that she instead deals 1,351.28 damage instead of 1,689.1 damage, a reduction in damage of 337.82. This does lose out on the retaliation procs, which would return a total of 630 damage after MR.

Assuming the one wearing the Frozen Heart is Shen, and it is his only item, his Vorpal Blade goes from a CD of 4 seconds to 3.2 seconds. This means he gets three Vorpal Blades within the same timeframe as he would have gotten two with Thornmail- one at 3.2, one at 6.4, one at 9.6 seconds, compared to one at 4 and one at 8 seconds, respectively. Vorpal blade does 220 damage +0.6 AP and causes melee swings to restore 22 + 1.5% of his max health which is (assuming no other items or masteries or HP runes, so worst case) 28.095, total 50.095 (round 50) health per second from Vorpal's effect- This is the same regardless of thornmail or FH.

Since Shen has no AP, he deals 220 damage modified by Vayne's MR (54.3) for a total of 142.579 (round 143) damage per vorpal. Since he gets a bonus vorpal, he gets a bonus of 142.579 damage with vorpal over Thorn, reducing the damage bonus from 630 to 487. Feint's CD goes from 5 to 4 seconds, so isn't really affected in a 10 second window; Neither is Shadow Dash from 8 to 6.4 seconds (though this would give a third cast within a 20 second window).

With this all said, calculating the damage Vayne dealt reduced by FH into the damage bonus that Shen does with Vorpal against the value of the reflect of Thornmail, you get a bonus of 149.18 damage in favour of Thornmail in a 1v1 Shen vs. Vayne over 10 seconds; 292.18 if you ignore the extra Vorpal to showcase a straight autoattack fight. The moment you add a second champion though, or a champion not targetting Shen, FH's worth immediately jumps up because it is not selfish- it is shutting down the enemy regardless of whether or not he's the one being hit.

Reducing 500 damage hits incoming against an Ezreal from 1/second to 0.8/second means that he's saving Ez (84.19 armor, armor seals no additional armor) from taking 271.45 (round 271) damage hits twice in a 10 second timeframe- A total of 542.9 (round 543) damage reduced in his favour. So long as Shen isn't fighting solo, FH actually -helps him more than Thornmail does-. And the mana is a totally wasted stat for him currently, too. Not only that, but the longer the fight goes, the more FH's CDR helps Shen even more than the damage reflect, both in doing his own damage and in using his defensive abilities.

His ult goes from 160 to 128 seconds, cutting off 32 seconds, more than half a minute, off its timer. Over a 40 minute game, if you have FH at 20 minutes you can ult an entire -five- more times than if you'd taken Thornmail. Three more uses of Feint per minute, giving 750 more shield a minute, or a whopping -15,000- extra shield over the course of the remaining 20 minutes (if used on CD, not likely to be quite as effective, this is just best case, I do acknowledge) Shadow Dash 7.5 times every minute normally, 9.375 times every minute with FH. That's 37.5 times more total that you can shadow dash- much better for placement, and a 1.6 second CD reduction means that it's much more relevant during teamfights as well.

In short, Thornmail is a noob trap of the worst sort. Note that a solo Shen taking FH at the end of those 10 seconds is still alive at 521.72 health, allowing him to continue living for 4 seconds more due to the reduced attack rate. A Shen with Thornmail would be at 183.9 HP, alive for only a single second longer. 14 seconds alive compared to 11 (before Vorpal HP gain allowances extend it further), able to dish out more damage and abilities, and able to defend against more damage? I'll take the FH on Shen, or in fact anyone, be they mana champ or otherwise, before I take a Thornmail.

TL;DR: Thornmail sucks unless all you do is 1v1, here's the maths, Frozen Heart is better on even Shen than Thornmail simply by stats, even with wasted mana.