How good are shields?

[Tiktacy]

How good are shields exactly? I have a rough idea of what effect they have but i'm not exactly sure on the specifics.

What is the exact effect of SH, how good is it in general? How good is it when used early on in the dungeon? I guess my question is: Is there a guide for how to use shields most effectively with the intention of playing optimally?

[Shard1697]

Shields are good. Whether you think shields are worth having over 2handers(I'm kind of split, I think 2H is often the better choice but I do also really like a good 1H build), if you don't have spells to cast, it's a good idea to wear a shield early on without training(the SH benefits are worth the slight EV and attack speed reduction from not having shield skill), and then later once you find a good 2H weapon drop the shield. Or once you find a good 1H weapon, maybe train up shields.

SH is usually pretty similar to EV. Some attacks can't be blocked, some can't be dodged(so that's kind of a wash), and SH is a bit worse when multiple enemies are attacking you in the same turn(but you should be doing your best to avoid that anyhow, and it's still good).

[Sandman25]

SH is roughly equal to EV vs a single enemy (that was the reasoning when SH was halved in game screen), it is decreased after every successful block until your next turn.For basic idea you can check http://crawl.chaosforge.org/Shields. For accurate decisions you should use fsim, there is no other non-crazy way for humans.

[goodcoolguy]

Shields are not good. Here is a list of situations where it is okay to use shields in spite of the fact that shields are generally bad:

1. You are a small species that can't use any good two-handed weapons anyway, for example a halfling or kobold.
2. You don't melee at all, e.g. if you use conjurations for all damage, then a buckler is good.
3. You are a large species going unarmed. Trolls are the main use case here.
4. You're a formicid and there's no downside to shields other than having to train shields skill.
5. You're an octopode and you really need that defense.

Outside of these situations, it takes exceptional circumstances to make shields worthwhile, like outstanding shield and 1-handed weapon drops early enough in the game for adapting not to waste a lot of xp. All of this said, using shields outside of these situations won't make you unable to win, it will just make your character unnecessarily weak.

[Sar]

Shields are good for Nagas IMO if you're lucky to get one, as a Naga your AC/EV kind of sucks for a lot of the game so getting a bunch of badEV relatively cheaply isn't the worst idea ever.

Spoiler: show

The worst idea ever is going Chei.

[Sandman25]

Difference between 2h and 1h weapon is pretty small with low Str.
For example, HO with Str 10 and Fighting 20 vs Stone Giant
+9 eveningstar of freezing

  Code:

       16 |     14.4 |     58 |      93% |  13.4 |    70  |  1.43 |     19.2
        17 |     14.9 |     63 |      94% |  14.0 |    70  |  1.43 |     20.0
        18 |     15.2 |     62 |      93% |  14.2 |    70  |  1.43 |     20.3
        19 |     15.2 |     60 |      93% |  14.2 |    70  |  1.43 |     20.3
        20 |     15.1 |     66 |      93% |  14.1 |    70  |  1.43 |     20.2
        21 |     15.9 |     69 |      93% |  14.8 |    70  |  1.43 |     21.1
        22 |     16.1 |     68 |      93% |  15.1 |    70  |  1.43 |     21.5
        23 |     15.7 |     68 |      93% |  14.7 |    70  |  1.43 |     21.1
        24 |     16.4 |     72 |      93% |  15.4 |    70  |  1.43 |     22.1
        25 |     16.9 |     71 |      93% |  15.9 |    70  |  1.43 |     22.7
        26 |     17.1 |     72 |      93% |  16.0 |    70  |  1.43 |     22.9
        27 |     17.6 |     75 |      94% |  16.6 |    70  |  1.43 |     23.8


+9 great mace of freezing

  Code:

        16 |     16.1 |     65 |      93% |  15.1 |    90  |  1.11 |     16.7
        17 |     16.7 |     68 |      93% |  15.5 |    85  |  1.18 |     18.3
        18 |     16.6 |     62 |      93% |  15.6 |    80  |  1.25 |     19.5
        19 |     17.7 |     69 |      93% |  16.5 |    75  |  1.33 |     22.0
        20 |     17.8 |     74 |      93% |  16.6 |    70  |  1.43 |     23.8
        21 |     17.8 |     71 |      93% |  16.6 |    70  |  1.43 |     23.7
        22 |     18.3 |     69 |      93% |  17.1 |    70  |  1.43 |     24.4
        23 |     18.3 |     70 |      93% |  17.0 |    70  |  1.43 |     24.3
        24 |     18.7 |     75 |      93% |  17.5 |    70  |  1.43 |     25.0
        25 |     19.2 |     78 |      94% |  18.1 |    70  |  1.43 |     25.8
        26 |     19.0 |     77 |      94% |  17.8 |    70  |  1.43 |     25.5
        27 |     18.9 |     82 |      94% |  17.7 |    70  |  1.43 |     25.3


+9 demon whip of freezing

  Code:

        12 |      9.6 |     45 |      93% |   9.0 |    50  |  2.00 |     18.0
        13 |      9.8 |     44 |      93% |   9.2 |    50  |  2.00 |     18.3
        14 |      9.7 |     45 |      94% |   9.1 |    50  |  2.00 |     18.2
        15 |      9.9 |     44 |      92% |   9.2 |    50  |  2.00 |     18.5
        16 |     10.2 |     45 |      93% |   9.6 |    50  |  2.00 |     19.2
        17 |     10.4 |     53 |      94% |   9.8 |    50  |  2.00 |     19.7
        18 |     10.5 |     46 |      93% |   9.8 |    50  |  2.00 |     19.6
        19 |     10.5 |     52 |      92% |   9.8 |    50  |  2.00 |     19.6
        20 |     11.2 |     51 |      94% |  10.5 |    50  |  2.00 |     21.1
        21 |     11.2 |     48 |      94% |  10.6 |    50  |  2.00 |     21.1
        22 |     11.3 |     50 |      93% |  10.7 |    50  |  2.00 |     21.3
        23 |     11.1 |     53 |      94% |  10.5 |    50  |  2.00 |     21.0
        24 |     11.4 |     56 |      93% |  10.7 |    50  |  2.00 |     21.4
        25 |     11.9 |     53 |      93% |  11.1 |    50  |  2.00 |     22.3
        26 |     11.9 |     54 |      94% |  11.2 |    50  |  2.00 |     22.5
        27 |     12.0 |     56 |      93% |  11.3 |    50  |  2.00 |     22.5


Edit. With some slaying that demon whip becomes superior to great mace even at high skill and with high str.

[sanka]

I think that the main reason shields are not really recommended that generally you do not have access to a good one handed weapon early on, and you should not play on the assumption that you will find one quickly. So it is much more reasonable to drop the shield and train for the twohander which you most likely have found, since they are common.

If you cast spells than it's also not the best place to put experience in the shield skill for a medium shield for most races - I feel it is too much experience to remove the penalties. The non-spellcasting penalties are not that big, so I most likely would use a shiled if
1. I have found a good one-handed weapon early
2. I do not cast spells
(If only the first is true, than I would use a buckler instead).

There are some exceptions, there are races where shields are better and I may train it higher.

[ydeve]

If your real killdudes is spells, you should use a buckler instead of a 2h weapon. A one-hander is plenty for popcorn.

[Cheibrodos]

I think that the main reason shields are not really recommended that generally you do not have access to a good one handed weapon early on, and you should not play on the assumption that you will find one quickly. So it is much more reasonable to drop the shield and train for the twohander which you most likely have found, since they are common.

This is usually the case for me. I'd rather have a great sword in the hand than a demon blade and shield in the bush. But, if I get really lucky and find a really early endgame one-hander, I can leave my weapon skill low and save exp for other important things.

[crate]

You don't need fsim to see that dwhip is basically equal to great mace if you get to 20 skill. 17/7 is basically equal to 12/5.

But as said above, you don't actually get to choose whether to use a dwhip in lots of games, and morningstars or scimitars and such are bad compared to great swords, dire flails, halberd/glaive, etc.

[radinms]

wow. I heard Sandman25 had retired dcss and tavern.
I am surprised

[Hands]

I think that the main reason shields are not really recommended that generally you do not have access to a good one handed weapon early on, and you should not play on the assumption that you will find one quickly. So it is much more reasonable to drop the shield and train for the twohander which you most likely have found, since they are common.

This is usually the case for me. I'd rather have a great sword in the hand than a demon blade and shield in the bush. But, if I get really lucky and find a really early endgame one-hander, I can leave my weapon skill low and save exp for other important things.

Without extra slaying I find demon weapons slightly underwhelming in the late game compared to solid 2-handers like great swords, but if I find an early demon blade (and training long blades is reasonable) then I'm going to drop my enchant weapons on it without hesitation and doing this will probably make it the best available weapon for the course of a standard game.

[goodcoolguy]

Sandman25 --

A more logical comparison would show numbers for 1-handed weapons with shield penalties or with weapon skill deductions equal to the amount of xp required to remove those penalties. Of course, the latter deductions would be small with a buckler, but not with a shield and certainly not with a large shield. As others have pointed out, availability of quality 1-handed weapons is rarely good until late in a 3 rune game, so late that you've already more or less won. The same could also be said for a +9 weapon, frankly, which of course makes the gap proportionally smaller in your simulations.

Your numbers do not seem to tell us much about the merits of shields in the part of the game one is most likely to die. The main takeaway seems to be that bucklers are a strong option when an outstanding 1-handed weapon is available.

[4Hooves2Appendages]

I use shields fairly often. It happens that I'm training for the 2h weapon I'm using and find a good 1h weapon while also having a decent shield. I usually stop the 2h training and get some shields then. Yes, it does end up doing less damage, but a normal shield trained to no penalty does give appreciable defences.

In my opinion someone who fights carefully, mostly 1v1 and knows when to use consumables doesn't need the extra damage from 2h. Arguably that person doesn't need the extra defences from a shield either. Hmm.

Ogres benefit from shields early on too, especially until they get a dire flail or better.

[Sandman25]

goodcoolguy --

Early game was tested with fsim many times, that's why I didn't bother to do it again: 1h with shield is better than 2h or 1h without shield on D1. Large shield is very bad and should not be used by normal size species unless you are berserker. I am not sure what shield penalties you are talking about, no one uses shield with penalties late game while early game the penalty should be ignored because it is less than bonus for blocking attacks. When you are fighting something scary like Ogre, Hill Giant or Juggernaut ability to block attack is much more valuable than extra AC or damage you can get from training Armour/Weapon.

Here are my tests for XL 1 HuFi (I changed stats to Str 20, Dex 15, Fighting 20, Armour 10, scale mail).
Vs Stone giant, +9 scimitar and +0 shield, long blades 14, shields 15:

  Code:

           AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDam
Attacking:     10.0 |     39 |      93% |   9.4 |    70  |  1.43 |     13.4
Defending:     17.8 |     41 |      50% |   9.0 |   100  |  1.00 |      9.0

+9 great sword, long blades 18:

  Code:

           AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDam
Attacking:     14.7 |     56 |      93% |  13.7 |    70  |  1.43 |     19.6
Defending:     18.0 |     41 |      78% |  14.1 |   100  |  1.00 |     14.1

As you can see the shield blocks about 35% of attacks (28/78).

[goodcoolguy]

These are totally unrealistic parameters. Nevertheless, they capture the basic picture with shields through most of a 3 rune game: If you look at ratio of damage in to damage out, it's usually about even for shield vs. 2-handed weapon.

The problem is that low player damage adds turns to encounters in which the worst thing that can happen is generally not just taking more turns of melee from a single opponent. Typical nontrivial encounters involve multiple monsters in layouts that don't lend themselves to creating the 1v1 fights of so-called optimal play, monsters that summon or recall other monsters to you, monsters with attacks that are not affected by SH (!!) or that set your SH to 0 (!!!), and so on. Even by the numbers, trading offense for reduced incoming melee damage is not what you want to do.

[Sandman25]

It depends. As some good player mentioned recently, they almost never die to melee monsters so shield is more useful in that situation. Early Centaur is a typical example. My experience is that with great defense offense is not important, you can win as Berserker with spear without berserking ever, taking as many turns as you need or just tele away.

You are welcome to provide tests for your typical scenarios.

[goodcoolguy]

Hm, indeed, since you can always do more resetting and luring to compensate for poor overall strategy, it really doesn't matter whether shields are good or not.

[Sandman25]

Not always, you are more likely to die without shield

[4Hooves2Appendages]

Hm, indeed, since you can always do more resetting and luring to compensate for poor overall strategy, it really doesn't matter whether shields are good or not.

Actually, luring judiciously and resetting fights when you have to are good strategy.

[goodcoolguy]

Actually, that is called tactics and it isn't what he's talking about.

Shields are bad and everyone knows it. Why am I even talking about this lol

[Sar]

I know a lot of good players who value shields. It's not exactly a black-and-white issue.

[Sandman25]

I think the better player you are the more you value shields. Why? Because new players are famous for rushing towards crowds why good players use luring and pack splitting. Also the harder the combo is (less AC/EV), the more valuable SH becomes. Again new players don't play Op or Mu often.
Yes, shields are bad when you don't use tactics, why are we even talking about this lol

Edit. Also anecdote: in my current game I found +3 dire flail of draining (base damage 13) early as MuCK so I enchanted it to +8. Soon I found +3 great mace (base damage 17) of draining. Stupidly I started training M&F above 14... Why stupidly? Because at M&F 17 I finally forced myself to do some fsim tests and found that the +8 dire flail does more damage even at M&F 20 than the +3 great mace. So rushing was a mistake.
I am writing it because if you enchanted 1h, it can do more damage (in my previous game +11 scimitar was dealing much more damage than +5 battleaxe). And if you don't enchant your early weapon, you can die (unless you are playing some OP combo of course).

[crate]

Dire flail is a demon blade (13/6 at skill 14+), so it's almost just as good as great mace (17/7 at skill 20+). Not a mystery that great mace is worse from basically 0-18 skill (at 18 skill, great mace is 17/8, which is approximately equal to 13/6).

Comparing to axes is either hard or easy depending on whether you think cleave actually adds something (I will not take a position in this post). If you think cleaving adds nothing, the comparison is easy and axes are always bad; if you think cleaving is good then the comparison is not something you can do with just fsim.

[Sandman25]

I was trying to prove that it is hard to estimate damage without fsim. "Shields are bad and everyone knows it" is not a valid argument if it is not supported by real argument like fsim. Also numbers alone is not a proof, for instance, I did use that battle axe despite low damage because vampirism is awesome with huge defenses.

Spoiler: show

I have just switched to +7 morningstar of flaming and +2 shield of protection. I have just 5 levels in shields but already have 18 SH thanks to artefact with reflection and SH+4.
Sure, I lose some damage but AC 33/EV 21/SH 18 is much better than AC 30/EV 23/SH 0, previously I almost died and had to go explore another branch, now I comfortable evoked all 7 rods ignoring 2 adjacent monsters

Edit. Vs Stone Giant
+8 dire flail of draining

  Code:

       16 |     12.8 |     55 |      93% |  12.0 |    60  |  1.67 |     20.1
        17 |     12.6 |     57 |      92% |  11.7 |    60  |  1.67 |     19.5
        18 |     12.9 |     62 |      93% |  12.0 |    60  |  1.67 |     20.0
        19 |     13.1 |     59 |      93% |  12.3 |    60  |  1.67 |     20.5
        20 |     13.6 |     65 |      93% |  12.7 |    60  |  1.67 |     21.2


+7 Morningstar of flaming

  Code:

       16 |     12.1 |     55 |      94% |  11.4 |    70  |  1.43 |     16.3
        17 |     12.8 |     53 |      94% |  12.1 |    70  |  1.43 |     17.3
        18 |     13.0 |     57 |      93% |  12.2 |    70  |  1.43 |     17.4
        19 |     12.8 |     59 |      93% |  11.9 |    70  |  1.43 |     17.0
        20 |     13.3 |     57 |      93% |  12.4 |    70  |  1.43 |     17.7

Good enough, provided it gives me AC+3, rF+, Blink, extra 14 SH and hilarious suicides of Yaktaurs from reflection.

[crate]

I don't think drain brand really works with fsim. It's not just damage.

[WalkerBoh]

Personally I think shields are incredibly strong. I've won hundreds of pure melee characters and would use a shield every game, with the exception of an exceptionally strong early 2H. I haven't run fsim or whatever, but qualitatively the damage difference between a 1H and 2H does not feel that large; however the defensive impact of a shield is huge for damage taken.

Also the skill cost of getting a shield to 15 is minuscule compared to the additional skill cost of most 2H weapons.

[crate]

The main problem with shields is that you usually don't find one until after lair, especially if you're talking about just shield or large shield (check the objstat results: https://drive.google.com/folderview?id= ... sp=sharing). So there actually is a real xp cost, since you're not choosing between, say, 14 skill scimitar + shield vs 18 skill great sword. You're choosing between 18 skill scimitar + shield vs 18 skill great sword vs making your character worse until you do find a shield. I think the third option there is not very good, and it's not clear to me what is better between the other two options even if you ignore the xp cost of shields.

If you do find an early shield, then it's probably fine, but about half the time, maybe more, you do not. Bucklers are a bit more common but still rarer than the 2h weapons you'd prefer to be using in this part of the game (great sword, dire flail, halberd/glaive, battleaxe).

If your character has found good spells and is wearing light armour, then switching to using a shield lategame is spending 15 skill levels that could have been spell school levels, which is a pretty obvious cost. It's still a notable cost for plate characters if you have both good spells and wizardry, though that's not that common.

Lategame the problem with shields switches to them not working on what I find to be the most dangerous attacks, so I'd rather just kill things faster. Your mileage may vary, and if you find a great shield then that obviously makes shields better. If you found one early this is probably not enough for you to care about, but this makes me prefer 18 skill great sword to 18 skill scimitar + shield. Perhaps 18 skill demon blade + 0 skill shield is better, I have not done much 0-skill shield testing.

---

Personally I always use shields if I pick fi, because why else am I picking fi instead of something else?

[Sandman25]

It depends on aptitudes also. For example, Mu can have something like that.

  Code:

Skill      XL: |  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 |
---------------+-------------------------------------------------------------------+-----
Long Blades    |     3                                                             |  3.9
Maces & Flails |        1  2  3  5  7  8  9 11 12    14          16 17             | 17.8
Dodging        |                       2  4  5  6  7  8 10 11       12 13 14       | 14.1
Fighting       |                       4  5  7  8  9 10 12 13 15       16 17       | 17.6
Evocations     |                             1  3           5  7  8    10 11       | 11.0
Armour         |                                3  4  6  8  9 10       12 13       | 13.2
Slings         |                                2  5                               |  5.1
Shields        |                                                             10 12 | 12.3

As you can see, there is no need to train weapon higher than 14 (and 0 shields) until XL 16 which is all Lair, Orc and almost all Dungeon cleared:
36335 | D:14 | Reached XP level 16. HP: 131/132 MP: 19/19

[WalkerBoh]

So there actually is a real xp cost, since you're not choosing between, say, 14 skill scimitar + shield vs 18 skill great sword. You're choosing between 18 skill scimitar + shield vs 18 skill great sword vs making your character worse until you do find a shield.

I can safely say that I almost always find a shield before I would get my weapon skill to min delay for a 2H weapon. I'm also not convinced that it actively makes your character worse to use a scim + 14 skill rather than a great sword + 18 skill around that point of the game (end of Lair). Putting that much extra XP into weapon skill means a significant decrease in XP spent on fighting/armour/dodging(/invo), which is meaningful at that point.

Lategame the problem with shields switches to them not working on what I find to be the most dangerous attacks, so I'd rather just kill things faster.

Agreed that the most dangerous attacks make most defenses irrelevant, with the notable exception of HP. And shields help keep your HP from dropping into critical territory due to damage from other, less dangerous attacks. Unless you're an axe wielder, no 2H weapons really boost your damage to the point where you can just tear through mobs of enemies before taking significant damage. I would argue that shields extend the amount of time you spend in a "safe" hp range more than the damage output reduction from 2H to 1H extends how long it takes to defeat most groups of enemies.

[Shard1697]

Personally I always use shields if I pick fi, because why else am I picking fi instead of something else?

To start with identified Might, perhaps?

[crate]

And shields help keep your HP from dropping into critical territory due to damage from other, less dangerous attacks.

Yeah that is just not important for me personally, which is why the part you didn't quote (where I mention this is not true for all players) is something I said. I'm careful to use first-person when I'm talking about things that are good for me specifically but may not be good for all players.

Anyway the point of my post was mainly that this isn't even a real decision since most games the game just chooses for you and it's pretty obvious, but probably I should've been more clear about saying that.

When advising players I would still personally tell them to not use shields since that's simple and I think probably leads to the player asking for advice improving more quickly.

Personally I always use shields if I pick fi, because why else am I picking fi instead of something else?

To start with identified Might, perhaps?

I'd rather just pick artificer because it's more fun and just as good.

[Eyesburn]

All of this said, using shields outside of these situations won't make you unable to win, it will just make your character unnecessarily weak.

I've won many 1h+shield characters and for sure they were not weak.

Speaking about shields, I started to play 0.18 a bit, and have one question: how does amulet of reflection stack with shield of reflection ? Do I get better reflection if Im wearing both? (character is wearing the brooch of shielding and +1 shield of reflection).

[radinms]

good players tend to lure enemies, so they like shield??????
it's a joke??????
why do good players playing carefully need extra defense????
i think careful players don't need shield because they don't die without shield.

[Nebukadnezar]

Outside of comparing melee with(out) shields they do add something which I think is equally important: when you're not in melee.

A shield will help you against yaktaurs, stone giants, liches, etc.
Have reflection ego or amulet of reflection and see them killing themselves even when you haven't been able to get into melee. Even without reflection you have a % of blocking their ranged attacks.

For my style of play (luring enemies, trying to reduce the # of enemies who can damage me) a shield suits me because I'm taking less damage and have more time/turns to react before situations might become ugly.

A shield also gives you another chance of having equipment with resists which adds to reducing incoming damage in general. Again: this is what I'm looking for since I'm mainly concerned about survival and not that much about doing as much damage as possible against certain enemies in melee.

[Nebukadnezar]

All of this said, using shields outside of these situations won't make you unable to win, it will just make your character unnecessarily weak.

I've won many 1h+shield characters and for sure they were not weak.

Speaking about shields, I started to play 0.18 a bit, and have one question: how does amulet of reflection stack with shield of reflection ? Do I get better reflection if Im wearing both? (character is wearing the brooch of shielding and +1 shield of reflection).

Reflection is something you have or not and it works on blocked attacks.
Since you have that amulet: the +1 from your shield of reflection helps. The ego from your shield of reflection doesn't help.
Get another amulet or a different shield

[goodcoolguy]

I think the better player you are, the more you realize that strategic decisions like whether to go shields or not don't affect win rate, because with optimal tactics you win 100% of the time anyway. On the other hand, the better forum poster you are, the more you realize that this has nothing to do with what people are actually asking about when they make a request for advice like the OP's.

[njvack]

I was trying to prove that it is hard to estimate damage without fsim.

I think the larger point here is that the amount of damage your melee weapon does is not the most important issue.

Does it matter? Of course. But remember: folks played for quite a while without noticing all melee damage was doubled. Yes, knowing that even a highly-enchanted mace is worse than a dire flail once you have weapon skill is important, but damage in Crawl is super variable. Even with a +12 speed GSC you're going to need to know how to lure and reset fights and use consumables and understand monster movement and once you know those things you can win the game with a dire flail or a morningstar using the same tactics.

Shields are, IMO, OK. You're more likely to find a good 2H than you are to find a good 1H and a shield. Most characters can make either work. Shields are a bit better if you're gonna be using enhancer staves or have a hard time getting other defenses up or you're planning on not killing lots of dudes. 2H is better if you're mainly killing tough dudes with melee. Shields preclude some pretty great launchers.

Thinking about weapon damage and shields and fsimming it and stuff is fine if it's the game you want to play, but I think it's not a particularly good way to get more orbs on your mantlepiece.

[Sandman25]

I agree with your post except last sentence. I ran fsim for Orb of Fire with my current character, the Mummy cannot safely kill an orb of fire even with great mace (no might, resist, agility, berserk, no luck with haste source in that particular game), also I wizmoded to see how much XP I need for Animate Dead/Death Channel/Simulacrum/Shadow Creatures. So fsim does allow to get higher winrate.

Edit. Crawl punished me for defending shields in the thread by giving me a vampiric lajatang (Mummy can freely use it together with rods and I have 12 skills from crosstraining despite 0 in actual staves)

[njvack]

Sorry, didn't mean to say it's useless to fsim; just that it's not usually a thing worth stressing about. If you're playing marginal characters, yeah, it's more important to know the boundaries.