 I respectfully acknowledge that I'm standing on the unceded traditional territory of the Comox and Qualicum First Nations. I thank them for the use of their lands and I want to express my sorrow for the victims of the residential schools. On a recent version of a raycast, we were asked to express our favorite tacit descriptions. Tacit is a way of showing a verb or a function without using variables. So it's also called point-free. You're just using the functions to express what you want to do. You're not worried so much about the variables. And by the way, you structure the functions, you can make them do different things. And that starts to get into the combinatorial logic and those kind of things. But to start with, this is just tacit. Now one of the things I have always found it difficult to do, and I'll just give an example. If I want to say double, this is the verb for double, double three, and then I want to link it to half of four. Now I don't have to put parentheses off this because execution in J is right to left. So it's going to say four, half of four, link to what's in the parentheses, double three, and I get it with six, two. So I've got two boxes, one with six and one with two in it. That's what I want to do. Now you can do this with a fork, however, it's a bit complicated. And the reason it's a bit complicated is because before you, and I'm actually going to do this tacitly, let's put a three in here to start with. So that's my left argument. And then what I want to do is I do want to double a three, but in order to see a three and not a four, I need to do the left verb. So what this part is doing now is it's looking at the three. This is looking to the left, and it's going to double it. The link stays in the middle, and then to the right, I have to do look to my right. And then that's where I'm going to put the four, this argument here. So what this is doing now, and this is a tacit form. So that means I can define this, and I will later, I'll define it as a variable. And then it would act just the way this verb does. I don't care about what these outside things are. What I'm caring about is I'm telling it to look to this side and look to this side, half this side, double this side, put it together. So I can do that. But I've got these extra little ceremonies in place to do what I wanted to do. And as I said before, what I'll do for fun, I'll do this, I'll go T equals, copy that. And now if I go three, T four, I get the same thing. So that's the advantage of having something tacit. Now that is tacit. That's all valid and everything. I could have stayed with that, but I would have been thinking about it. And I've always thought, I've always been really interested in hooks. And the way hooks work is if I do this, and what we're going to, we were going to half four, weren't we? Okay. So this together, just two verbs, not three, which is a fork, but two, which is two verbs, and it's dyadic hook. So this is a hook because it's got two verbs. And it's dyadic because it's got two arguments. What happens here is this verb will act only on the four. And then it takes the result and it acts on this verb with the three. So essentially what I end up doing here with this type of a hook is I've got three linked to half of two, half of four, which is two. So that's what it's going to do. But you can see it's only changing the four because hooks are not symmetric. They're one sided. What I really want to do is I want to go and change this side. In order to do that, I've got a thing called a left hook. And what a left hook does, if in place of this, I replace it with this, I'm effectively going to end up changing the three, but not the four, because I'm only changing the left side, just like that. So how is it doing that? Well, this is where it starts to get a bit more complicated. These adverbs here, this tilde is called passive. And when it's dyadic, it's got two arguments, its job is to switch the two arguments. So as soon as I see that, I'm putting the three over on this side and the four over on this side. So let's do that. I'm just going to do that again. And I'm going to get rid of this. So I'm getting rid of the thing that it did, which is a flip. Put the three over here and the four over here. So I have done the flip, and now I'll do it again. And I've got the same result. Why did I get the same result? Well, because now I'm just dealing with a standard hook, with one little twist over here, but a standard hook, it's going to double the three. And then it's going to link to the four from this side. But this is where this passive comes in, because I've got two arguments. I want to switch them back, because if I didn't switch them back, this is what would happen. Oh, I threw it in zero there just a second. That's just confusing. If I didn't switch them back, this is what would happen. I wouldn't get six, four. I would get four, six. So I have to do the switch here. Just do these guys again. I have to do the switch here. And that makes it double the three. And then the switch back again to put them in the right order. So I've got this left hook thing. And I've got a right hook, which I used up here. And that starts to give you a clue about how we can do a different operation on each of these at the same time. What I'm going to do is I'm going to substitute for this link here. Instead of doing just the link there, I am going to copy this. And I'm going to substitute in a hook here. So now I've got a hook within a hook. The outside hook is a left hook. The inside hook is the regular right hook. And what I've done is I've doubled six, and I've halved four, and I've linked them together. And I've done it with just the three original verbs. I've used two adverbs. And I've used a hook within a hook. But based on that, I've actually been able to create the effect that I want, which is this, because of this part here, this reverse, this works on the three. And then this part here works on the four. And then this is all reversed back again. So that's my explanation of what I was trying to do with a hook within a hook. And you may know now why it was almost impossible to describe just verbally. I'm hoping it's a little bit clear. I think I'm going to be doing another video in a short while, because as I did this, I realized there's a few different flavors of this, and they are interesting as well. So stay tuned for that. I hope you enjoyed this video.