function growth pattern names for operations at the power-tier and beyond

The following chart shows names for patterns of growth exhibited by functions that are characterized by operations at the power-tier and beyond:

𝑓𝑘(𝑥)	↑	↑↑	↑↑↑
𝑥 _ 𝑘	power	tetrapower	pentapower
𝑘 _ 𝑥	exponential	tetrational	pentational
𝑥 _ 𝑥	metapower	metatetrapower	metapentapower

The names in yellow cells are already established, and the names in magenta cells are my proposals. Outside of my proposals, the typical way to refer to any of these types of growth is “super-exponential”.

The ↑ symbol is Knuth’s up-arrow notation. So a single ↑ represents exponentiation. I could have used the more familiar caret symbol ^ for this, but I decided to optimize for consistency with the higher-level operations.

To be clear, in 𝑓_𝑘(𝑥), 𝑘 is a constant and 𝑥 is the variable.

So, for a couple examples, the top-left data cell is showing how 𝑓_𝑘(𝑥) = 𝑥 ↑ 𝑘 exhibits power growth, and the bottom-right data cell is showing how 𝑓_𝑘(𝑥) = 𝑥 ↑↑↑ 𝑥 could be said to exhibit metapentapower growth.

For context, some well-known function growth pattern names include:

• constant for 𝑓_𝑘(𝑥) = 𝑘,
• linear for 𝑓_𝑘(𝑥) = 𝑥,
• quadratic for 𝑓_𝑘(𝑥) = 𝑥 ↑ 2,
• cubic for 𝑓_𝑘(𝑥) = 𝑥 ↑ 3,
• quartic for 𝑓_𝑘(𝑥) = 𝑥 ↑ 4,
• factorial for 𝑓_𝑘(𝑥) = 𝑥! , and
• logarithmic for 𝑓_𝑘(𝑥) = log₂𝑥 (or in my proposed reformed notation, ₂√𝑥).

I initially thought that a simpler and thus better name for the function growth pattern exhibited by 𝑓_𝑘(𝑥) = 𝑥 ↑ 𝑥 would be “double-exponential growth”, however, that is already established as the name for a different function growth pattern, one that does not appear in this table: 𝑓_{(𝑘₁,𝑘₂)}(𝑥) = 𝑘₁ ↑ (𝑘₂ ↑ 𝑥).

Another name I considered for 𝑓_𝑘(𝑥) = 𝑥 ↑ 𝑥 was “power-exponential”, but that name has the problem that it does not follow the same pattern as “logarithmic-linear”, 𝑓_𝑘(𝑥) = 𝑥 log 𝑥 = _𝑥√𝑥, because 𝑥 ↑ 𝑥 does not have an equivalent growth pattern to (𝑥 ↑ 𝑘)(𝑘 ↑ 𝑥).

So the idea is that we copy the hyperoperation numeric prefix and apply it to “power” for the names of growth patterns for functions characterized by higher-order operations following the same variable and constant scheme as that of power growth, so that these fucntion growth pattern names correspond by numeric prefix to the analogous functions following the variable and constant scheme of exponential growth.

And then we need a brand new name for the row where the variable occurs as both the base and the exponent of the power; I’ve proposed “metapower” for the simplest example of this, where the “meta-” prefix captures the self-reference here, i.e. a variable operated on by itself. I chose this prefix as opposed to “auto-” because “autopower” appears to be somewhat popular in the automobile industry, while “metapower” has only rare usage in social and political science.

Note that since 𝑥 ↑ 2 is quadratic growth, 𝑥 ↑ 3 is cubic growth, and 𝑥 ↑ 4 is quartic growth, and these are all specific types of power growth, we could also propose the names “tetraquadratic’, “tetracubic”, and “tetraquartic” growth for 𝑥 ↑↑ 2, 𝑥 ↑↑ 3, and 𝑥 ↑↑ 4, respectively. However, there’s a ton about this that bothers me:

• the mixing of Greek and Latin prefixes,
• having more than one numeric prefix at a time, and
• this is aggravated by the messiness of the quadratic, cubic, quartic progression, where quadratic refers to 2 despite evoking 4 which is due to historical reasons because quadratum is the Latin word for square, cubic successfully refers to 3 via the dimensionality of a cube, but then quartic refers to 4 directly. (Maybe it’d be better if we had square, cubic, and hypercubic growth.)

Tetraquadratic growth is of the most interest here, because 𝑥 ↑↑ 2 = 𝑥 ↑ 𝑥, and so we have tetraquadratic as another name for metapower growth. Similarly, since 𝑥 ↑↑↑ 2 =  𝑥 ↑↑ 𝑥, we have pentaquadratic growth as another name for metatetrapower, and we’d have hexaquadratic growth for metapentapower, and so on. But I recommend the meta- style names, which are simpler, more evocative of what they are, and correspond directly numerically with their analogous power and exponential growths.

Here is a more complete table, that will be of use or interest to far fewer people: