This guide was written by RealWolfM and gives facts and figures about Ryze and Tear in Dominion.

“Doing some mathematics because I am a strange person, I was investigating tear because of a guide which was giving false information advising people its not worth it.

Looking at the easy numbers:
Mana crystal costs 400 gold for 200 mana
Tear costs 995 gold for 350 mana + 4*casts

At the 37th cast of your skills, which on ryze should be fairly easy to do, your tear is now giving you equal mana/gold compared with a mana crystal.

At a MAXED tear, which is 250 casts, you are getting 2.71 times the efficiency, which is very good value for money.

And at a reasonable 100 casts towards end game, you are getting 1.51 times the efficiency, which is still very very good value.

Moving on to more interesting matters:

Ryze spells have ratios as follows
Q:
0.08mana 0.2 AP
W
0.05 Mana 0.6 AP
E
0.35 AP

Assuming a simple combo of QWEQRQ which is easily manageable, and also (usually wrongly) assuming E only bounces once, we get the following values.

For a straight mana crystal, you get +58 damage from a single combo
With an unfilled tear, you get +101.5 damage

Looking at the damage increase, simply buying a tear increases your damage from that mana crystal by 1.75 times
At the turning point of 37 casts, you deal 2.5 times more damage with your combo than you did with just a mana crystal.
At the acceptable 100 casts, its 3.75 times
250 casts is 6.75 times the damage output.

In terms of damage per cost, you get significant damage output increase, and its also very cheap.

Now on to ARCHANGELS STAFF, the item which everyone gets lategame (hopefully)
In terms of its stats:
+50 mana
+45 AP
+3% mana as AP
1860 cost

For varying values of mana, we look at its bonus damage, and its efficiency vs a mana crystal for QWEQRQ

Assuming 0 base mana for Ryze, essentially a blank slate look at the item we get that
For 0 stacks/0 casts, 3.52 times the damage output of a mana crystal.
It should be mentioned that it is 0.5 times the efficiency of a mana crystal at that point

For the break even, 37 stacks, it deals 4.38 times the damage output, with an efficiency of 0.62 times a crystal
100 stacks, deals 5.84 times the damage, 0.82 efficiency
250 stacks, 9.33 times the damage, 1.31 efficiency

Break even in terms of efficiency is about 150 stacks for 0 mana.

Now 0 base mana is a bit of a joke really, so lets try some more acceptable numbers in the mix.

If we were to say something simple like 1000 mana with no archangels, then:
For 0 casts, 9.14 times the damage output of a crystal, 1.28 times the efficiency
Break even 37 casts, 10 times the damage, for 1.4x efficiency
100 stacks, 11.46x the damage, 1.61x the efficiency
250 stacks, 14.95x the damage, 2.09x the efficiency

Now Archangels is a lategame item. Most Ryze players (me included) would not get it until you have a frozen heart + odyns veil. So those give 850 mana together, combined with Ryze starting mana of 250 + (+55 / per level), that 1000 mana part isn’t even accurate.

Lets say its about level 12, you’ve been dominating hard so you are rich and you have a frozen heart + odyns veil, that is 850+250+12*55, 1760 mana not including tear.

Comparing the damage without an archangels or tear, with the archangels, you get
Base damage output based on mana + 0 AP of 510

By having an archangels at this point you get
0 stacks, 1.49 times the damage output
37 stacks, 1.59 times the damage output
100 stacks, 1.89 times the damage output
250 stacks, 2.15 times the damage output.

That means that with just 100 stacks, you are almost doubling the damage you deal with one combo. Pretty decent right?

Now lets go to the final values, level 18, frozen heart + odyns veil, no other mana items
Base damage of 606
0 stacks gives 1.43 times the damage
37 stacks gives 1.52 times hte damage
100 stacks, 1.66 times
250 stacks, 1.99 times

CONCLUSIONS

A number of conclusions can be drawn from these numbers. It says that lategame, upgrading to an archangels staff is always valuable to increase your damage significantly (1.4 times was the minimum with his core items, even with a tear uncharged). It also says that in terms of damage, 37 casts to break even for its gold cost makes tear an incredibly valuable item. Even if you start with a catalyst this is very very efficient.

And finally it says that there is never a reason not to get a tear, and upgrading it is always valuable if what you seek is DAMAGE (remember it gives very little else, thus other items can still be prioritised instead of this).

Also to note this is all calculated with a QWEQRQ combo, which is not the most efficient damage output, and it assumed E bounces ONCE and once only, doesn’t hit twice. Just for a quick comparison a QWQEQRQWQEQ combo (lvl 18 + items) which is easily possible gives:
Base damage of 1212
Tear:
0 stacks, 1415 dmg
37 stacks, 1500 dmg
100 stacks, 1647 dmg
250 stacks, 1995 dmg

AA:
0 stacks, 1738 dmg
37 stacks, 1838 dmg
100 stacks, 2007 dmg
250 stacks, 2411 dmg