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I finally got a look at the illustrations on the instruction card. It's not as tJ put it:
It's for practicing the feeling of rotating (without moving the elbow). You place one hand on top to steady it, place two fingers of the other hand on either side near the apex. You then rotate your wrist - eventually with no visible movement. Practice with different combinations of two fingers.
Can be done on the edge of a table but I don't think you have as much freedom of wrist.

What is the triangle actually for? I don't follow.

I mean specifically with regard to the movement. I'm intrigued by the device, but I'm really not clear as to what you're supposed to do with it or what it contributes to that, from your description.

If you are unaware what rotation is you need to read up. Matthay had to make very clear much of it happens without any visible movement - just pressure exerted in one or the other direction as below.

I'm aware of rotation is. I have absolutely no idea how this triangle might be applied to the act of doing it, or what purpose it might serve. Would you mind giving a slightly more detailed explanation?

Uhhhg - what utter nonsense. The idea of rotating on each note - I don't care how invisible - Matthay's genius was to convince so many pianist to follow such ridiculous nonsense.

You don't rotate on the notes, you rotate to them. It's the same basic idea pretty much everybody talks about, from Taubman to Sandor. (And you can see fairly obvious applications of it in the Brahms exercises and Chopin etudes.)

All due respect - to, from, on - still utter nonsense.

I don't rule out the notion of rotation, but the notion that you MUST always do it and that it's sometimes invisible? That's a different matter altogether. Why is rotation of benefit with every note? And what physiological evidence is there that is would always be of benefit or that it must occur?

I'm really confused. It's either because I'm blonde, or this isn't making much sense..

I don't believe anyone says you must. It's an important aspect of playing, but not the only one...

"The Matthay Triangle" made me think of the Bermuda Triangle. Not knowing anything of the topic, I imagined it to be a metaphor for some murky pedagogical Davy Jones' Locker.

Hey, maybe I was right.

I wonder if Tobias Matthay ever played the triangle? (I thought of that, too.)

If I remember rightly from kbk's post, Matthay claims everyone does this invisibly on every note, whether they know it or not. So does the Taubman approach. If there is any evidence to support this, I've never seen any.

I'm not near my Matthay books but he probably says something of the order of every good pianist does this all the time whether they are aware of it or not. By good pianist he will mean one whose body responds and acts in what is its natural relaxed way. Most pianists I observe go very much against how their body naturally wants to go about a task. So, it is a matter of does not must.

That's because it's invisible lol.

But the question remains as to why anyone should believe it. I haven't seen anything to so much as point towards the possibility that the theory about that which is invisible is grounded in anything, nevermind prove it. I haven't even seen the most basic explanation of why it would be desirable, nevermind essential, to good playing.

As for "natural", you see what that really entails in the majority of those who have never been trained at all. I think you are confusing "natural" technique and "efficient" technique- which are rarely the same thing. When they coincide, you are talking about genius (or arguably luck, in many ways), not the norm. Good piano playing requires a wealth of adaptations that are rarely present in natural instincts. Otherwise untrained pianists would thrive.

I think the opposite is equally ridiculous - that the playing of any finger is accompanied by a wrist/arm that does not rotate at all, and that the wrist/arm is either static or moves only vertically or laterally.

Put another way - if not rotation, then what else?

Sure, it should be free to respond to the motion. But that's a very different thing to initiating movement via muscular effort. Surely one could argue that if in the habit of creating motion, you are in the habit of using muscles that would be better off releasing? Why introduce an association with an addtional effort, rather than work on releasing it? Alternatively, if it really is a willful activity, what is it for? It still doesn't make any sense to me. To respond to an external force like the keys pushing back, you just need to be loose. For example, when you play chords, you practise following through. You wouldn't come away from the keyboard and practising slamming on the brakes at the point where the key would stop you. So why practise these minute rotations with willful simulation of what would be the product of an external force, not a willfully instigated motion?