I was fascinated by this 3b1b video about the Triangle of Powers and was tinkering around with it myself. Specifically, I noticed (after the timestamped part) that if I reflect the triangle while keeping the pivot vertex unchanged, then the numbers in the other vertices stay in the same place but change to their operative inverse.
For example, say the triangle has $a$ in the lower left, $b$ in the top, and $c$ in the lower right.
If I reflect the triangle over the $b$ vertex, then the lower left becomes $\frac{1}{a}$ and lower right becomes $\frac{1}{c}$.
$$\begin{align*} {}_a&\overset{b}{\triangle}_c &&\iff &{}_{(1/a)}&\overset{b}{\triangle}_{(1/c)}\\ a^b &= c &&\iff &\left(\frac 1a\right)^b &= \frac 1c \end{align*}$$
Similarly, if I reflect it over the $a$ vertex, then the top becomes $-b$ (since the assigned operation is a $+$) and the lower right becomes $\frac{1}{c}$. The same happens if I reflect over the $c$ vertex.
$$\begin{align*} &\ldots\iff &{}_{a}&\overset{-b}{\triangle}_{(1/c)} &&\iff &{}_{(1/a)}&\overset{-b}{\triangle}_{c}\\ &\ldots\iff &a^{-b} &= \frac 1c &&\iff &\left(\frac 1a\right)^{-b} &= c \end{align*}$$
BUT, if I attempt to do the opposite i.e. swap the other two vertices while making sure they're unchanged, then it only works for the reflection over the $b$ vertex. Specifically, the intuition I came up with is that since the other two vertices have the multiplication operation assigned to them, if I keep them unchanged then the pivot 'inherits' the operative inverse of the other two.
$$\begin{align*} {}_a&\overset{b}{\triangle}_c &&\iff &{}_{c}&\overset{1/b}{\triangle}_{a}\\ a^b &= c &&\iff &c^{1/b} &= a \end{align*}$$
Naturally, this intuition breaks apart when trying to apply it to the other ways of reflecting the triangle. Since, you know, how is the pivot supposed to inherit the operative inverse of the other two vertices if those vertices have different operations? One might think 'just make it become the negative reciprocal' but as we know, it doesn't work like that.
I'm looking for a better intuition for this, if this is at all possible. At first glance, it feels like there should be a way to make this work, but as 3b1b said here: "The asymmetries in the notation correspond to the actual asymmetries in the numerical relationship of $a^b=c$ itself" i.e. the fact that it is taught as $a$ being multiplied with itself $b$ number of times to get $c$.
It there a way to make this work or is this all just in vain?
