This one is going to get some editing. For now, I'll explain what I'm on about, and add pictures and further tl:dr if it seems necessary.
I've been concentrating on the index finger and thumb for the first two parts of this roughly scienticious series. Because, those two are what control the blade and move it about. But what about the rest of the hand?
The aspect ratio roughly approximates how easy it is to keep the blade centered and cutting straight. But individual fingers have roles to play.
Your middle and ring fingers have a dual role to play in knife use. When you push cut, they're aligned with your arm, and provide resist the force of the material. Put another way, they are the "hard place", your arm is the "rock", and the knife itself is caught in the middle, forcing it to cut something. Or break, if you're doing it really super wrong.
In finer cutting tasks, these two middle fingers of your hand are what rotate the knife around your index finger. It's a subtle thing, but that's what's going on.
Your pinky finger is pretty much along for the ride, although it may provide some tertiary information about the orientation of the knife handle. It's interesting that a knife handle has to be long enough to let your pinky hold on to it, or it feels like you don't have as much control.
The more I pay attention to the finer details of knife design, the more I find these details in one place or another, but very few knives have more than two of them. As I feel motivated, I'll edit or add to this, to show what I'm talking about.
This is the Boker Kressler, an integral sub-hilt fighting knife, and a wonder of construction.
But mostly it's here so I can talk about surface area.
The little second strip of nicely shaped metal is there to poke between your index and middle fingers, and it gives a very sure grip on the knife. Trench knives have the same thing, with one or more finger holes to increase the amount of surface area that your paw can hold on to. Trench knives also use 'em as brass knuckles, but that's a whole different thing to write about.
So while these knives are very secure in the hand, there is a problem: you can't re-position as you cut.
Your hand moves along the knife handle as you cut. I don't have video of it or anything, but it's pretty easy to pay attention to how the pressure changes against your fingers, the next time you have your knife out for some work. Go ahead: I'll wait.
Somewhere in the middle is a handle shape which allows the necessary confidence in your hold on the knife, and the space for subtle rotation while cutting.
Once again, I'm going to pick on the Hinderer XM. And, I've added some shitty, hand-drawn arrows, to make this memorable.
The shitty, hand-drawn arrow on the left points at what some people call the "choil", with the "ch" pronounced like a "k". As if you needed to know that.
What's even worse, is that the exact size and shape of this feature is still up for debate: when it's tiny, some people will call it a "sharpening notch", while some people always call it a "choil". In the case of Hinderer's knives, it allows you to choke up on the blade, placing your index finger on the choil, and now you have the same kind of secure grip and control that you would with a sub-hilt.
The other arrow points to, roughly, the place where your index finger rests if you grip the knife normally. Actually though, it's the tiny little "peak", right behind, that does the heavy lifting.
This part:
Aw yeah, artist AF.
That small peak in the handle, and the pair of inward curves that go into making it, are just increasing the surface area that you get to touch with your fingers. If you felt like doing calculus, you could verify by maths, that this handle shape is "longer", than one without the peaks and valleys in it.
So why not go all the way?
Benchmade did, and they're not the first:
But now, there's another problem: if your fingers aren't the exact right size to fit those contours, the knife isn't going to feel as comfortable to use. So there has to be some kind of compromise between maximum surface area, and minimum "pointy bits to make hot spots".
I don't think there is one. It would be nice to be able to say that any knife between number X and number Y, will have a comfortable and secure grip, like I did earlier with the perimeter and aspect ratio.
But now we're talking about the design of the entire knife handle, so any single number will be an abstraction, which represents the surface area for a given length of handle, or some other blah blah blah that is really hard to explain without seven pages of pictures and context.
The Greeks had the "Golden Ratio", and it's somehow related to the Fibonacci Sequence. The only thing it proves is that if you wander off far enough the theoretical forest, you can use math to prove anything. I'm determined to find a number/ratio/theory that defines the phrase "great ergonomics!" (at least as far as knives go), but I haven't quite got there, yet.