Beyond Mechanics and Calculations - by Guffin#9843

This document aims to explain the math behind Beyond, the staple of "juicing" your maps in PoE, and to show its actual power plus that of its scaling options.

I recommend to always check the graphs linked in the

blue boxes

as these do a nice job of showcasing things.

I'm well aware many people won't bother to read this document as soon as they see maths (it's only highschool level though), but check out at least the

red boxes

as the core "findings" of each segment are presented there.

If you spotted some mistakes, just message me on Discord and I'll fix them as soon as possible.

Contents:

 


Beyond Mechanics

If we want to make some conclusive calculations with regards to Beyond, it is imperative to understand how exactly it works:

When an area has a certain percent chance to "attract monsters from beyond", it enables the beyond mechanic, i.e. the spawning of Beyond demons.
These are the currently (Patch 3.17) available sources of Beyond:

Source Beyond Chance Notes
Map Roll 13% Scaled by "Effect of Modifiers on non-unique Maps", rounded down
Zana Mod 15% -
Atlas Notable Passive "Scent of Blood" 3% -
Sextant Roll 6% if awakened, 8% if elevated -
Effect of Modifiers on non-unique Maps 0% to 4% 15 Nodes with 2% increase each, for 30% total

Here are some example combinations of beyond sources, all with full Effect passives because a juice-focused mapping Setup should include these anyways:

Combination Beyond Chance Modifiers
Maximum (Elevated Sextant) 42% ⌊13 * 1.3⌋ + 15 + 3 + 8
"Triple Beyond" 40% ⌊13 * 1.3⌋ + 15 + 3 + 6
"Double Beyond" 34% ⌊13 * 1.3⌋ + 15 + 3
"Basic Beyond" 19% ⌊13 * 1.3⌋ + 3

When your map has any of the sources of Beyond chance active:

Killing a monster has an "X" percent chance (= beyond chance) to create a small red orb.

4 red orbs in a half-screen-range of each other (at 1920x1080) make one beyond pack spawn.
1 beyond pack has 4 monsters (3 magic, 1 rare), which is not scaled by pack size (See here).

4 beyond packs in one screen mean one beyond boss spawns. This also doesn't get scaled by pack size.


An Equation for Beyond

We are going to simplify reality a bit for our calculations, and assume that all the mobs killed are always within beyond merging radius of each other.
In reality this will not always be the case, as in open maps (especially with lacking density and/or low beyond chance) there will surely be a couple of red orbs somewhere that go to waste.
This just means that the absolute numbers that are discused in this document / shown in the graphs can be slightly lower in reality.

If you juice your maps hard enough, and/or run narrower layouts, it is entirely possible to get maps where not a single orb was wasted though.
I personally experience that quite often in my maps.

So, according to the Beyond Mechanics:
the amount of monsters "Y" spawned by beyond when killing an arbitrary number of monsters "N" in an area with beyond chance "X" can be expressed as:

Y = ((N * X) / 4) * 4 + ((N * X) / 4) / 4

(N * X) / 4

This is the average number of Beyond packs, as you need to proc your Beyond chance 4 times to get a pack to spawn.

* 4

The number of beyond packs times 4 results in the total number of magic and rare beyond mobs, as each portal spawns 4 mobs.

+ ((N * X) / 4) / 4

This is the average number of Beyond bosses, as for every four packs, one of them spawns.

Please note that with a Beyond chance of 50% you wouldn't put 50 as X but 0.5 instead, obviously.

The amount of monsters added is visualised here:

https://www.geogebra.org/graphing/vqfavwgy
Also check the table of values.

The amount of monsters gained is already nice to look at, but I want to try to put it into perspective in the next few segments.


More Multiplier of Beyond on the toal Amount of Monsters

Raw numbers don't mean much for many people, so it is worthwhile to think about beyond as a more multiplier for the amount of monsters you kill.
Let Z be that more multiplier, then:

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Z = Y / N
Z = (((N * X) / 4) * 4 + ((N * X) / 4) / 4) / N
Z = (17 * X) / 16

This more multiplier is visualised here:

https://www.geogebra.org/graphing/syaqyaa2
Also check the table of values.

As you can see, even with only the basic 18% Beyond chance from the map roll + atlas passives, you get a hefty 19% more multiplier.
That means that by just running the most basic Beyond setup, you basically an "extra" map of monsters every 5 maps.

And that's not counting the rarity of the added Monster, which we will look into next.


Effect of Beyond on the Amount of Loot

Remember that monster rarity affects their drops:

Magic Monsters have 600% increased Quantity, 200% increased Rarity.
Rare Monsters have 1400% increased Quantity, 1000% increased Rarity.
Unique Monsters have 2850% increased Quantity, 1000% increased Rarity.
=
Magic Monsters drop 7 times the loot of a Normal Monster at 3 times the Rarity.
Rare Monsters drop 15 times the loot of a Normal Monster at 11 times the Rarity.
Unique Monsters drop 29.5 times the loot of a Normal Monster at 11 times the Rarity.

When looking at the worth of added monsters, ideally we would look at how many Items they would drop (before map and player quant scaling).
But while the mechanics thread quotes a generic drop rate of 16% for normal monsters, it also says that this number can differ depending on the monster type. Since these differences aren't known AFAIK, there's no way to make a conclusive statement about the amount of items dropped.
The only thing we can "safely" say is how much more loot a magic/rare/unique monster drops compared to a normal monster, not how many items that are in total.
So just think of the numbers you see here as a "loot multiplicator" of sorts: the higher the number, the better. That's all there is to it.

Said multiplicator can be found as such:

A Beyond pack consists of 3 magic and 1 rare monster.
Beyond bosses only appear solo.
That is a combined Quantity Multiplier "QMulti" and Rarity Multiplier "RMulti" of:

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QMulti = ((N * X) / 4) * (3*7) + ((N * X) / 4) * (1*15) + (((N * X) / 4) / 4) * 29.5
QMulti = 10.8438 * N * X
RMulti = ((N * X) / 4) * (3*3) + ((N * X) / 4) * (1*11) + (((N * X) / 4) / 4) * 11
RMulti = (91 * N * X) / 16

Now let's compare these values to the values of the base monsters in a hypothetical example map.
As average mob rarity distribution in a no-mod map isn't known (if there even is any), and it also wildly changes depending on map mods/scarabs/sextants/random procs, we have to make some assumptions here:
A generous one for that distribution would be 6:3:1 for Normal:Magic:Rare.
So the base monsters in the map you killed to spawn the beyond mobs would have a combined quantity and rarity multiplier of:

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QMultiWholeMap = N * (6/10) * 1 + N * (3/10) * 7 + N * (1/10) * 15
QMultiWholeMap = (21 * N) / 5
RMultiWholeMap = N * (6/10) * 1 + N * (3/10) * 3 + N * (1/10) * 11
RMultiWholeMap = (13 * N) / 5

This graph showcases the comparison in loot multiplier:

https://www.geogebra.org/graphing/a9ehtrxz
You can play with the slider(s) and also check the table of values.

As you can see, the Beyond monsters give more loot than ALL the base monsters in the (hypothetical) map from 38% Beyond chance onwards.
This is obviously disregarding how certain map rolls, sextants, prophecies, scarabs etc. may shift the, hypothetical, rarity distribution of the non-beyond monsters in a map away from the example 6:3:1 spread like I said.
It is just meant to show how absolutely ridiculous the power of beyond mobs is for loot due to their superior rarity distribution.

The more multipliers beyond has on the total quantity / rarity of Items dropped can be expressed as:

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MoreMultiQ = ((10.8438 * N * X) + ((21 * N) / 5)) / ((21 * N) / 5)
MoreMultiQ = (2.58186 * X) + 1
MoreMultiR = (((91 * N * X) / 16) + ((13 * N) / 5)) / ((13 * N) / 5)
MoreMultiR = ((35 * X) / 16) + 1

This graph showcases these more multipliers:

https://www.geogebra.org/graphing/hnraaqdz
Also check the table of values.


Diminishing Returns

As is the case with any multiplier that is additive with itself, stacking more and more Beyond chance has diminishing returns though. Going from 10% to 20% Beyond chance is a higher relative increase in mobs gained than going from 30% to 40%.

The value you gain on the total amount of monsters in a map per additional percent investment into Beyond chance can be expressed as such:

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IncreaseFromNextPercent = Z(X+1) / Z(X)
IncreaseFromNextPercent = ((17 * (X+1)) / 16) / ((17 * X) / 16)
IncreaseFromNextPercent = (X + 1) / X

This graph showcases these diminishing returns:

https://www.geogebra.org/graphing/jnaermpq

As you can see, going for higher and higher beyond chances gives less and less relative improvement very quickly.

The value you gain on the total added IIQ/IIR from monsters added by beyond per additional percent investment into Beyond chance can be expressed as such:

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QuantIncreaseFromNextPercent = (10.8438 * N * (X + 1)) / (10.8438 * N * X)
QuantIncreaseFromNextPercent = (X + 1) / X
RarityIncreaseFromNextPercent = ((91 * N * (X + 1)) / 16) / ((91 * N * X) / 16)
RarityIncreaseFromNextPercent = (X + 1) / X

This graph showcases these diminishing returns:

https://www.geogebra.org/graphing/arcnj2sx

Obviously this behaves just the same as in the case of the total amount of monsters added: the IIQ/IIR from added monsters gets hit by diminishing returns heavily at higher beyond levels.

All that said though; "the more beyond the better" is always true. You are, obviously, not actually getting less monsters total by stacking more and more beyond, the relative impact of the next percentage points just becomes smaller and smaller, meaning that spending dozens of Exalts to squeeze out an additional percent of Beyond chance might not be worth it.


Conclusions about Basic Beyond

Beyond is absolutely insane juice.

It ranges from being roughly a 19% more multiplier for mob count in its most basic state (map roll + awakening completion bonus) to a 42% more multiplier at higher investments.
Added on top of that, the monsters being magic rarity and upwards makes it an absolutely unprecedented boost for loot;
As beyond mobs are of the highest rarity, they alone might even drop more loot than all other monsters taken together at higher beyond levels.

It is a fact that beyond is a ridiculous scaling mechanism for maps (and probably deserves to be gone or nerfed, don't hit me).

 


Atlas Passives

The introduction of the new Atlas passive tree revamped the scaling options we have for many mapping mechanics, and Beyond is one of them.
While generally things remained the same, with all of the previous Glennach-specific modifiers available to us, Beyond lost a lot of scaling from effect of non-unique modifiers and watchstones.

For proper map juicing, you will generally want to take all of the Beyond passives in any case. Which makes checking their individual impact relatively pointless, as there is no reason to compare them to each other (as opposed to what you had to consider with Watchstones previously).
That said, it is still fun looking into.


The "Torn Veil" Atlas Passive Node

Torn Veil:

Powerful Beyond Demons require 1 fewer Portal to summon in Areas.
Beyond Portals in Areas have 30% increased Merging Radius.

This decreases the amount of successful Beyond portals required to spawn a Boss from 4 to 3.
If you doubt this, and think that is also lowers the amount of orbs required to spawn a portal (which wouldn't make sense as the wording clearly implies it's about portals and not orbs), here is proof: https://www.youtube.com/watch?v=irlU-FweGN0
Other than that, it is currently the only method in the game to increase the merging radius of portals. As initially explained in the segment about the equation for beyond, we disregard that aspect anyways though. But it certainly is powerful in reality, depending on the map layout.

Following this, our math changes as such for the total amount of monsters spawned:

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If the amount of monsters spawned by basic Beyond is
Y = ((N * X) / 4) * 4 + ((N * X) / 4) / 4
The the amount of monsters spawned with Torn Veil is
T = ((N * X) / 4) * 4 + ((N * X) / 4) / 3

A comparison between added monsters with and without Torn Veil is visualised here:

https://www.geogebra.org/graphing/wyp55enq
You can play with the slider(s) and also check the table of values.

While this isn't impressive at first glance, it can have pretty nice effects as we will see in the following Segments.


Amount of Monsters gained

In raw numbers of monsters added, the difference between Torn Veil and basic Beyond doesn't seem large
Let's see whether that remains the same when looking at the more multiplier:
Let V be that more multiplier, then:

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V = T / Y
V = (((N * X) / 4) * 4 + ((N * X) / 4) / 3) / (((N * X) / 4) * 4 + ((N * X) / 4) / 4)
V = 52 / 51

Meaning, Torn Veil Beyond has 1.96% more monsters than basic Beyond.

If we want to find out how much of a more multiplier Torn Veil is for the amount of monsters in a map, we check how many mobs Torn Veil adds itself and divide that by the amount of normal monsters + basic Beyond monsters:

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V = (T - Y) / (N + Y)
V = ((((N * X) / 4) * 4 + ((N * X) / 4) / 3) - (((N * X) / 4) * 4 + ((N * X) / 4) / 4)) / (N + ((N * X) / 4) * 4 + ((N * X) / 4) / 4)
V = X / (51 * X + 48)

This multiplier is visualised here:

https://www.geogebra.org/graphing/mgp7ceph
You can play with the slider(s) and also check the table of values.

As shown, at 40% Beyond chance (roughly what is called "triple beyond"), Torn Veil is a 0.58% more multiplier for mobcount compared to a beyond map without it.

An increase of roughly 0.6% may seem small, but as all those monsters are Unique the impact can be quite large as explained in the next segments.


Effect on the Amount of Loot

Refer back to the Segment of Effect of Beyond on the Amount of Loot

Given that all the monsters Torn Veil adds are unique, it makes sense to look at how much quantity/rarity gets added compared to basic Beyond.

We need to get the combined Quantity "QV" and Rarity "RV" with Torn Veil:

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QV = ((N * X) / 4) * (3 * 7) + ((N * X) / 4) * (1 * 15) + (((N * X) / 4) / 3) * 29.5
QV = 11.4583 * N * X
RV = ((N * X) / 4) * (3 * 3) + ((N * X) / 4) * (1 * 11) + (((N * X) / 4) / 3) * 11
RV = (71 * N * X) / 12

The total IIQ/IIR bonuses are visualised here:

https://www.geogebra.org/graphing/vf2ezn4w
You can play with the slider(s) and also check the table of values.

As usual, a relative more multiplier on the IIQ/IIR bonuses would be more comfortable to think about:

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QV / Q = (11.4583 * N * X) / (10.8438 * N * X)
QV / Q = 1.0566
RV / R = ((71 * N * X) / 12) / ((91 * N * X) / 16)
RV / R = 284 / 273

Meaning that added beyond monsters in a Torn Veil Map have:

  • 5.66% higher Quantity multiplier
  • 4.03% higer Rarity multiplier
    than the beyond monsters in a normal map.

Divine Farming and the Amount of Unique Bosses

The Sextant mod:

"Unique Monsters drop Corrupted Items"

This is a cornerstone of divine farming (or at leat one of the possible ones).
Pretty much all unique monsters in your map are going to come from Beyond anyways, but how many does Torn Veil exactly add?

Comparing the amounts is easy, as:

UniquesBase = ((N * X) / 4) / 4
UniquesTV = ((N * X) / 4) / 3

The amount of added Uniques is visualised here:

https://www.geogebra.org/graphing/zr3ptrkq
You can play with the slider(s) and also check the table of values.

As usual, looking at a more multiplier is going to be more useful:

UniqueMulti = (((N * X) / 4) / 3) / (((N * X) / 4) / 4)
UniqueMulti = 4/3

I don't think I have to plot a graph for this: it's a 33.33% more multiplier.

33.33% more Unique monsters in your Map from a single Atlas Passive are gigantic for Divine farming.

Conclusions about Torn Veil

Torn Veil only adds very few mobs to the map in total (roughly 2% more mobs than added from basic Beyond, or roughly a 0.6% total more multiplier for the map), but the few mobs it does add offer a lot of IIQ and IIR.

But importantly, it is simply a 33.3% more multiplier for the amount of unique mobs. This makes it the best scaling in the game (besides Beyond itself) for any farming method centered around unique monsters, such as divine farming.

 


"Beyond Demon Chance" small Passives

Beyond Demon Chance:
Beyond Portals in Areas have a 5% Chance to spawn an additional Beyond Demon.

There are two such passives to pick, for a total chance of 10%. They're right next to the valuable Torn Veil Notable, so you'll pick that anyways.

Beyond "Portals" are what spawns the four magic and rare monsters. This node doesn't affect Boss spawns.

If we look back at the equation, picking both nodes would result in the following:

Y = ((N * X) / 4) * (4 + 0.1) + ((N * X) / 4) / 3

Amount of Monsters gained

The more multiplier these two passives have on the total amount of Monsters in a map can be found by dividing the amount of base Monsters + Monsters added from the buffed Beyond by the amount of base Monsters + Monsters added from basic Beyond:

MoreMulti = ((((N * X) / 4) * (4 + 0.1) + ((N * X) / 4) / 3) + N) / ((((N * X) / 4) * 4 + ((N * X) / 4) / 3) + N)
MoreMulti = (13.3 * X + 12) / (13 * X + 12)

This Multiplier can be seen here:

https://www.geogebra.org/graphing/tgnp3h8v
Check the table of values.

As you can see, it comes out to roughly a 0.7% more Multiplier at "triple beyond". The relative Impact it has diminishes somewhat the higher you go with the Beyond chance, but the differences are minimal.

If we want to see how much of a more multiplier these nodes are for beyond itself, and not the total Monster count, then we divide the amount of monsters with the nodes by the amount of monsters added from basic beyond:

MoreMulti = ((((N * X) / 4) * (4 + 0.1) + ((N * X) / 4) / 3)) / (((N * X) / 4) * 4 + ((N * X) / 4) / 3)
MoreMulti ~ 1.023

So it is roughly a 2.3% more Multiplier for the amount of Monsters added from Beyond itself.

This means that effectively, these 2 passives leading up to Torn Veil add more monsters to your map than Torn Veil itself.
Whether or not they're more profitable will be discussed in the next segment.


Effect on the Amount of Loot

Refer back to the Segment of Effect of Beyond on the Amount of Loot

For loot considerations, it is important to know that these passives can only cause an additional magic demon to spawn and not any rares.
That is, according to my testing.

Adapting the previous formula for these passives makes the comparison in Loot gained to basic Beyond pretty easy:

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QMultiPassives = ((N * X) / 4) * (3.1*7) + ((N * X) / 4) * (1*15) + (((N * X) / 4) / 4) * 29.5
QMultiPassives = 11.0188 * N * X
RMultiPassives = ((N * X) / 4) * (3.1*3) + ((N * X) / 4) * (1*11) + (((N * X) / 4) / 4) * 11
RMultiPassives = 5.7625 * N * X

QMultiPassives / QMulti = (11.0188 * N * X) / (10.8438 * N * X)
QMultiPassives / QMulti ~ 1.016
RMultiPassives / RMulti = (5.7625 * N * X) / ((91 * N * X) / 16)
RMultiPassives / RMulti ~ 1.013

Meaning that these passives are a 1.6% more multiplier on the quantity of Items gained form Beyond, and a 1.3% more multiplier for the rarity.
You'll notice at this point, that these values are only about a quarter of the gains from Torn Veil, despite Torn Veil adding way less monsters than these small passives. This is due to the superior rarity of the Torn Veil monsters.


Conclusions about "Beyond Demon Chance" small Passives

These Passives work out to be roughly a 0.7% more Multiplier in terms of Mob Count for the whole map, and a very minor increase in Rarity.

This might not sound like a lot, but 0.7% more monsters is worth more than 1% increased Pack Size in many cases (especially so due to the superior rarity distribution of Beyond mobs, which I will add something about after I have stopped being delirious from a lack of sleep), due to diminishing returns on packsize as it stacks additively with itself.
This is an amount of added monsters that is very rare on the new Atlas Tree for small nodes and even some notables.

 


"Beyond Demon Item Quantity" small Passives

Beyond Demon Item Quantity:
8% Quantity of Items dropped by Beyond Demons in Areas.

There are three such passives to pick, for a total increase of 24%.

These should stack additively with the existing quantity multipliers on Beyond monsters, as that's pretty much always the case.
As a proper multiplier, these would be truly absurd.
We will know with certainty once poedb has datamined the exact information, and I will update this segment accordingly if need be.


Effect on the Amount of Loot

Refer back to the Segment of Effect of Beyond on the Amount of Loot

Adding 24% quantity changes the quantity multipliers of added Beyond monsters as such:

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QMultiPassives = ((N * X) / 4) * (3*7.24) + ((N * X) / 4) * (1*15.24) + (((N * X) / 4) / 4) * 29.74
QMultiPassives = 11.0988 * N * X

QMultiPassives / QMulti = (11.0988 * N * X) / (10.8438 * N * X)
QMultiPassives / QMulti ~ 1.023

QMultiPassives - QMulti = (11.0988 * N * X) - (10.8438 * N * X)
QMultiPassives - QMulti = 0.255 * N * X

Meaning that these passives cause Beyond monsters to drop 2.3% more loot than before, and
that they contribute a 0.255 * N * X worth of quantity multiplier to the map themselves.

If we want to draw a comparison in relation to our hypothetical example map, we can just adapt the respective formula from the linked segment:

MoreMultiQPassives = ((11.0988 * N * X) + ((21 * N) / 5)) / ((10.8438 * N * X) + ((21 * N) / 5))
MoreMultiQPassives = (1.02352 * X + 0.387318) / (X + 0.387318)

This graph showcases the more Multipliers that Passive-Boosted Beyond offer for the quantity of Items dropped in a map:

https://www.geogebra.org/graphing/xyttmesy

At "triple Beyond" values, these passives offer roughly 1.2% more loot than not taking them.


Conclusions about "Beyond Demon Item Quantity" small Passives

These Passives work out to be roughly a bit more than a 1% more Multiplier in terms of Items dropped for the whole map.

This might not sound like a lot, but 1% more loot is worth more than 1% increased Quantity in all cases, due to diminishing returns on IIQ as it stacks additively with itself.


todo: add a comparison between loot multipliers of TV and the demonic small passives compared to the hypothetical map example. Kinda forgot that

Edit
Pub: 07 Dec 2021 03:07 UTC
Edit: 18 Mar 2022 00:01 UTC
Views: 2720