Yo. I've been watching your TTYD TASing vods (due to timezone difference, which is also why I'm doing this in a dm rather than just telling you on stream), and there's something I want to ask/say about wallpushing and Fermat's principle. I'd first like to mention that I'm close to having my Bachelor's in math at a good uni, so there might at least be a little bit of value in what I'm saying. In your TTYD TAS you calculated the optimal angle with which to approach a wall using some trig and calc. You usually call this Fermat's principle, but I'll refrain from that name. You then later found out that this wall approach angle is (almost) the same as the wallpushing angle, ie the angle that gives you the greatest increase in X-speed when held while against a wall. Since then you've often said (first conjectured, then later stated as though you were convinced, my bad if this was a misinterpretation) that the wallpushing angle is the one it is BECAUSE of your derivation of the wallapproaching angle. In my opinion, there is quite a lot to fill between "this is the optimal angle to approach a wall given how fast you travel at a wall and how fast you travel without a wall" and "this is the optimal angle to get a speedboost from the wall". To begin with, there's the issue of WHY you get an X-speed increase pushing against the wall. I am aware that by pushing "against" the wall, you will be pushing inside it, and the wall will in return push you out. It is not however clear to me, why that should increase your X-speed at all. Given that you're only pushing into the wall in Z direction (as in, regardless of your X displacement, the same amount of Z displacement will put you into the wall, and as long as you fix your Z position, no X displacement will put you into a wall), one would intuitively expect that all the pushing happens only in Z direction, and that your X-speed is unchanged. You may have explained this part about it at some point (whether it be in one of your previous TASes or one of the earlier streams) and I just didn't catch it or forgot. Sorry if so. The next part is about why the angle is the one that it is. You say that the wallpushing angle is the one that it is, is because it is also the fastest angle to approach a wall. However you should consider that the wallapproaching angle depends on the wallpushing speed, which in turn again depends on the wallpush angle (this point is a little questionable, it's a little bit of a philosophical question whether the maximal wallpushing speed depends on the angle or not). To illustrate this, here's an example: Assume a situation similar to the one in TTYD, except for simplicity's sake, your XZ-speed is constant when walking (we we don't have to deal with upwards movement being faster than right movement). And assume that when pushing against a wall, your resulting X-speed will be your current X-speed (the one you would get when walking at the angle if there was no wall) plus half your current Z-speed (the one you would get when walking at the angle if there was no wall) and then that result cubed. Then the optimal wallpushing angle when facing a wall above you would be ~26.56° if 0° is straight right and the angle increases counterclockwise (I believe you typically put 90° to the right and say angle decreases counterclockwise, in which case the angle would be ~63.43°, which should be somewhat close to the actual wall pushing angle. However in this case your total X-speed will be sqrt(5)/2 your normal X-speed, so about a 12% increase. In which the optimal wallapproaching angle will be ~44.31° (or ~45.69 with the way you had it). A big difference. This example is to illustrate that as long as you don't know how exacty the wallpushing speedup-mechanic works, the approach angle and wallpush angle might be a lot different.

However, while messing around with this I found something somewhat interesting. Remember how I constructed my counter example, where you add half your Z speed to your X speed and then cube it. Now what if we leave out the cubing. Interestingly, if you take the construction above (which is simpler than the real TTYD construction, since there horizontal movement is slower than vertical movement), and assume that pushing against a wall adds p times your Z speed to your X speed, for some positive real p. Then the wallpush angle and wall approach angle will be exactly the same. (Since I'm working in an ideal model I'm not concerned with ingame limitations of calculations etc here). Now this is an indicator that it might not just be coincidence that the two angles do (almost) coincide in reality.

Err, short correction of the first message, the total increased speed is ~40%, 12% was without the cube.

Regardless, I think it's worth investigating whether the wallpushing works in the X+pZ way that I described above is indeed how much wallpushing speeds you up. If you want to test this, I would be expecting a p value of ~0.32 if the speed increase is ~5%. Hmm, I made this a lot more formal and wordy than I had originally intended, my bad for that.

Also sorry if you already knew all of this and I just misunderstood lol.

Oh, there are two more things I should mention.

The first is that if you want to test this, an easy way to do it is to pick an angle (we'll say only up-right type angles) and check your X-speed (call it x) and Z-speed (call it z) at that angle while not pushing against a wall, then use the same angle while pushing against a wall, and check your total X speed now (call it s), then solve s = x+pz for p. Repeat this with several different angles. If p is approximately constant for many angles, then odds are that that is indeed how it works.

The second thing I wanna mention is something that might be pointless to a mathematician, but you're a computer science major, so it might be relevant for you.

If the above is indeed how it works, then it isn't that the optimal wallpush angle is the same as the optimal wall approach angle (or in your words, the optimal wallpush angle is not determined by Fermat's principle). It's that the optimal wall approach angle is the same as the optimal wallpush angle, that is once the wallpush angle has already been established, Fermat's principle THEN tells you that the wall approach angle has to be the same.

That's super nitpicky though so you may disregard that.