# Can objects be folded or otherwise compressed to satisfy Mending's requirements?

The cantrip Mending states:

This spell repairs a single break or tear in an object you touch, such as broken chain link, two halves of a broken key, a torn cloack, or a leaking wineskin. As long as the break or tear is no larger than 1 foot in any dimension, you mend it, leaving no trace of the former damage.

Assuming that I have a cloak with a tear longer than 1 foot, and I can physically fold the cloak so that it fits within a 1 foot cube (thus, no more than 1 foot in any dimension) - would that satisfy the requirements for repairing it with Mending?

• Is this an actual in game problem? Have you discussed this with your DM? – KorvinStarmast Mar 24 '18 at 17:12
• Related question: in one round, can my character run in a 5-foot-diameter circle an infinite number of times since the circumference is less than his Speed of 30 feet? – Asher Mar 24 '18 at 18:34
• @KorvinStarmast well, I am the DM, and I'm asking because I'm interested in the RAW. Otherwise, I'd probably rule that this is ok and allow it. – PixelMaster Mar 24 '18 at 18:57
• I'd suggest that you offer that perspective in your problem statement. Since you are the DM, I am not sure what problem you feel needs solving. Is your concern that there is a rules exploit that you may be overlooking? – KorvinStarmast Mar 24 '18 at 18:59
• No, I'm just curious what the RAW would be. While Mending isn't the most commonly chosen cantrip, it would be significantly more powerful if you could somehow compress the respective object so that it fits in a 1-foot-sphere (thanks @theDarkWanderer) – PixelMaster Mar 24 '18 at 19:28

# No

As long as the break or tear is no larger than 1 foot in any dimension, you mend it (...)

You have stated that the tear in your cloak is longer than 1 foot, so you can't use Mending on it. It doesn't matter if the cloak itself fits within a 1 foot cube, since at least one of the tear's dimensions (its length) is larger than 1 foot, even if you fold it.

You might, however, want to try something like sewing it in the middle so that you have two smaller tears, and Mending it then, but only if your DM is prone to allow such shenanigans. Or just ask them directly if you could Mend your cloak, Mending does not usually influence the plot or gameplay in a major way, so he might just waive the constraints on it this time.

# No

Your idea is good, but you are misunderstanding what 'any dimension' means.

When you think of a 1-foot cube, you are thinking that, since the cube is no longer than 1 foot to a side, it has a length of 1 foot in each direction. This is not true. In fact, the cube has an extent of no less than one foot in each direction and no more than √3.

In order for total extent of the cloak to be no longer than 1 foot in any dimension, you actually need, at best (in terms of volume), a 1-foot diameter sphere. If you are using a cube it would need to be a 1/√3-foot cube.

That said, even this is not enough.

Imagine your tear is a simple 2-foot line. Let's say you fold it in half, so now you have a line that goes 1 foot in a direction and then another 1 foot back to where it started. Being able to rely on the volumetric simplification described above means being able to rely on the folded tear being considered to be 1-foot in length. This is by no means the most likely interpretation of 'length'-- that would be that the total distance traveled along that axis is 2-feet, so the length of the path in that dimension is, similarly, 2 feet. Otherwise the intuitive theorem

$$\sum_{i=0}^n length_i = total \; length$$

is, in fact, false.

If the length of a line folded exactly upon itself does not change, then there is a maximum to the length of a single-line tear that can be mended since folding can only accomplish so much and travel in non-cardinal directions is not unrelated. Travel in any direction counts as travel in any other direction with the length reduced by multiplying by the cosine of the angle between them, which makes the math complicated.

You can mend a tear of pi/2 feet in length by folding it into a perfect circle, and this is the theoretical maximum in two dimensions. You can add more on for each extra dimension.

### But that's too hard/long for playing!

I mean, that depends on the group. But if it doesn't work, just remember wizards can use origami+complicated math to increase by a substantial but less-than-doubling amount how much they can mend, and if they can go to planes with more spatial dimensions then they can fix more with even more complicated folds and accompanying geometric analyses. The exact amount doesn't need to be calculated any more than how big the hole in the wizard's cloak needed to be when you put it there.

### Proof:

The theoretical maximum is achieved when the total length in each dimension exactly equals 1 foot, if such a curve exists.

An interesting property of circles is that they are perfectly identical regardless of how you rotate them; a circle is the same shape with the same relevant properties regardless of what orientation you pick to use as an axis. Because of this property, the 'length' of a circle in each and every dimension is the same.

Another helpful property is that, for any given dimension, we can break a circle up into two always-increasing or always-decreasing half-circles. This lets us sum the length for each and then combine them into a total length in that dimension, rather than having to take a line integral (it's an easy line integral though, and good practice if you are learning such things).

For always-increasing (or decreasing) curves, the 'length' as we have defined it is identical to the magnitude of the displacement because:

$$\Delta x = \sum_{i=0}^n dx_i$$ where dx is the signed infinitessimal displacement vector at x=i, and the length is $$s_x=\sum_{i=0}^n |dx_i|$$, but for always-positive dx in the always-increasing case we can remove the absolute value leaving $$s_x=\sum_{i=0}^n dx_i \cong \Delta x$$

So, the 'length' of a circle in a single dimension is twice its diameter. Since we can't exceed 1 foot, that gives $\pi d= \pi /2$ for the perimiter of the circle. We've know we've used up all our possible length in two dimenstions, so this result must be optimal.

"Wait!" You might think. "What if we used many circles? Each circle still uses the maximum amount of 'length' in each direction, so we should be able to combine them to make a thing, too!"

Yep! That's accurate, but the thing isn't any bigger-- it's just equally optimal. You can fold your tear into one big circle or a very circular infinity sign or the olympic rings or whatever you want, as long as it's all circles, and you'll still get exactly pi/2 length at best. Yay math!

You do get extra length in three dimensions (and additional length for each dimension you add after that), but the math is way more complicated so I need to work on making a good, simple explanation for that. Feel free to edit one in if so inclined!

## Yes to compressed, No to folded.

So for the first bit by strict RAW:

Use the Enlarge/Reduce spell to reduce an object to a smaller size, then cast Mending on the broken portion that is now significantly smaller. So in your case, reducing the cloak would allow you to mend it in one casting.

As for the second bit, this is an application of firm logic:

As a DM myself I allow players to close gaps and use Mending to help speed along the work. So for instance, if a boat gets torn open by a reef, deckhands can nail some planks in place under 1 foot apart, and the ship wizard can run about Mending the holes individually with the new wood that's been provided.

The logic I use to justify this is simply that a boat is made up of a great many single objects already, and the spell doesn't know how to differentiate between wood that was the boat, and wood that is now part of the boat. So as far as I am concerned, so long as the gap is less than 1 foot, it's not an issue for the spell.

So for your cloak example, all it would require is stitches being put in place to reduce the tear to less than 1 foot, and then casting Mending on each torn section until they were all repaired.

As for folding it and casting the spell, no. Mending is explicit in that the tear or break has to be less than 1 foot in any dimension. So folding it wouldn't reduce the tear, it would only reduce it in one dimension. When unfolded, it would still be longer than 1 foot, so Mending wouldn't work.

• Clever use of Enlarge/Reduce. That one is worth remembering. – keithcurtis Mar 25 '18 at 17:09

Holy smokes for overthinking everything! How long is the rip in the fabric? Less than 1 foot long? Good to go. More than 1 foot long? Doesn't work.

• +1 for the Common Sense ruling. The spell says nothing about folding. The intent should be obvious. It's only a cantrip. – keithcurtis Mar 25 '18 at 17:07

## What's a dimension, anyway?

The fundamental confusion here, as is amply evidenced by comments on other answers, lies in what is meant by "length in a dimension." There is, as far as I can no tell, no official word on what this phrase actually means in the context of the game rules. And mathematically speaking it's basically nonsense.

There is no single quanta of "dimension". Dimension is a numerical invariant which can be defined for various objects in various (and distinct) ways. For vector spaces it's the cardinality of a linearly independent set of spanning vectors. But there is, in general, no canonical way of constructing such a set of vectors. For a manifold it's the dimension of the Euclidean space which is homeomorphic to a neighborhood of each point in the manifold (conveniently, this Euclidean space is also a vector space and these two types of dimension are the same). But there is, again, no canonical way of picking the neighborhood (you could always pick a smaller one, for example). You probably don't even want to try to grasp what the (Hausdorff) dimension of a fractal is, or what Frobenius-Perron and categorical dimensions in (fusion) categories are. But suffice to say that these aren't even necessarily integers, or even rational numbers, and there's not necessarily an "ambient space" like the world we live in to try to make physical sense out of any of it.

Short of all that being: mathematically speaking the phrase "in a dimension" is ill-defined at best. We lack a definition, and there's no canonical justification for any particular definition. You can't really argue that one particular definition is clearly superior and preferable to another one; each one is equally valid, and will tend to yield different answers.

This lack of canonical choices is exactly the problem the OP poses: there are many potential ways of arranging a cloak into a 3-dimensional (or higher) space.

## So is there an answer here anywhere?

No, not really. I suggest you invoke one of the fundamental rules of DM'ing: when rules are unclear, adjudicate them with an eye towards preserving the fun of the game and preventing abuses. A major abuse, for example, might involve exploiting space-filling curves and/or the Banach-Tarsky paradox* to achieve ludicrous effects.

Allowing a player to use Mending by folding up a cloak with a standard but long tear in it is unlikely to be abusable. Disallowing it, on the other hand, may sap the fun of the player getting to use their spell to solve a problem.

You could, if desired, simply require that the spell be cast multiple times, with each cast repairing about a one foot length of tear (measured as a curve's arc length; not "in a dimension", which as I said is nonsense). This type of ruling would also inhibit space-filling curve craziness: if it's a true space-filler then it's infinitely long and needs infinite casts, or if it's an approximation to one it may need many days of work even for a high level player to cast the spell that many times. Needing multiple casts for repairing such major damages would also keep tailors in business: at some point they can just make a new one more quickly. This also makes it easier to ensure that your ruling doesn't make Mending better than any higher-level versions of it that might exist in your campaign.

*This paradox is really more of a demonstration that you can't be too loose with your axioms, and that it can be easy to be too loose. Try to let something do too much, and it will do way too much. If your players are trying to exploit this, you can pretty much just assert that it relies on assumptions are not valid here. And also deduce that you either have some very nerdy players or some very opportunistic ones.