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Would it be tetrand, tetrant, or something else?

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    $\begingroup$ Not sure people care too much about this. Why do you need this? Perhaps just adopt a word and go with it. $\endgroup$ – Pedro Tamaroff Dec 13 '18 at 14:51
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    $\begingroup$ See Tetration : "The parameter $a$ may be called the base-parameter, while the parameter $n$ may be called the height-parameter." $\endgroup$ – Mauro ALLEGRANZA Dec 13 '18 at 14:53
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    $\begingroup$ @PedroTamaroff I don't need it, I find nomenclature itself interesting. $\endgroup$ – Griffin Dec 13 '18 at 14:57
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    $\begingroup$ FYI, I've called it the tetrated exponent in some early 2000s sci.math posts. $\endgroup$ – Dave L. Renfro Dec 13 '18 at 14:59
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    $\begingroup$ @PedroTamaroff: people who care about tetration probably do. Just like people interested in symplectic geometry care about symplectic forms. $\endgroup$ – Yves Daoust Dec 15 '18 at 11:56
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When I first (ca 2007) came in contact with (what I later called) "exponentialtower" and which is even in its un-generalized form commonly called "tetration", this occured under the name "powertower" (for instance see mathworld). It seemed very natural for me to use the term (iteration-) "height" for the extend of iteration of that imagination of increasing a "powertower". After that, to use the letter "h" as its default shortcut/variable is again natural. For instance in the "tetration-forum" (founded at 2007) the term "height" has become a much common use.
(By this I consequently don't like/use the term "super-log()" or "slog()" for the function which tells us the required height of iteration from one $z_0$ to $z_h$ but use simply $\text{hgh}()$ as that function in article and software)

disclaimer: this is only amateur-mathematics, no journal-publication or academic practice behind this


Additional remark: in a comment, if I recall correctly, there was a discussion whether exponentiation should be seen as "iterated multiplication" at all (as well as in its generalized to fractional iteration-"heights") but as a genuine operation (I think it was MSE/MO-user Qiaochu Yuan) and similarly for tetration, if at all, should also a more genuine operation be formulated - apart from the basic idea of iteration of an exponentiation. I like this idea much, but such a genuine operation has not been formulated yet. If it is formulated then the term height might be replaced by some more appropriate name - but I don't see any ansatz for this so far...

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