the hyperoperator series

The program:          

The pieces in this series are inspired by the concepts behind the recursive definition of the mathematical idea called the "hyperoperation sequence." Although the applications of this are complicated, the central idea is quite simple.

We are often taught that multiplication as essentially a "hyper" version of addition. Instead of writing out multiple addition operations $$ 3+3+3 $$ we can just write $$ 3 \times 3 $$ In this way of thinking about it, multiplication is just iterated addition: \(3 \times 3\) is basically just saying add \(3\) to itself \(3\) times. Admittedly, there's nothing especially difficult about writing out \(3+3+3\), but I think the vast majority of us would much prefer not to have to parse \(3+3+3+3+3+3+3+3+3+3+3+3+3+3\) when it could far more easily be written as \(3 \times 14\). Furthermore, expressing something like \(3 \times 10,000,000\) in terms of addition is virtually impossible, and entirely pointless. Critically, there are some numbers that are virtually inconceivable if we restrict our thinking to addition, but are easily conceived of and used in terms of multiplication.

The next step is to create a mathematical operation that is to multiplication, what multiplication was to addition. In fact, we use this all the time, and it's called exponentiation. For example $$ 3 \times 3 \times 3 $$ becomes simply $$ 3^3 $$ or "three to the third power." Everything I said about addition v. multiplication can be said here about multiplication v. exponentiation. It's far easier to think about \(3^9\) than to have to go through \(3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\), not to mention the truly hellacious strings of addition that we'd need. Further, we can quickly start discussing numbers easily in terms of exponentiation that would be virtually inconceivable in terms of addition or multiplication. Crucially, however, we could theoretically bring it all back to addition if we needed to; we know that the operation is no less well-founded than more basic operations.

Where things start to get truly interesting is when we take the next, next step of finding some new operation that is to exponentiation what exponentiation is to multiplication. This is an operator that almost no one gets taught in basic math called tetration, and it can convert, for example: $$ 3^{3^3} $$ into $$ 3 \uparrow \uparrow 3 $$ Immediately, we can start talking about truly colossal numbers like $$ 3^{3^{3^{3^{3^3}}}} $$ by just writing \(3 \uparrow \uparrow 6\). And of course we can continue this process indefinitely, finding pentation as the next iteration of tetration (\(3 \uparrow \uparrow 3 \uparrow \uparrow 3\) becomes simply \(3 \uparrow \uparrow \uparrow 3\) ), and then finding hexation after that (\(3 \uparrow \uparrow \uparrow 3 \uparrow \uparrow \uparrow 3\) becoming \(3 \uparrow^4 3\) ), and so on. These increasing up-arrows (which are a notation invented by the mathematician Donald Knuth) form the titles of these pieces. The first one was just \(\uparrow\)—technically, that piece is also the sixteenth member of my PortRait series, so the \(\uparrow\) is substituting for the 1 part of the number 16—and the second one was \(\uparrow \uparrow\). After that, I just use a superscript for the successive titles (as do mathematicians), so the next pieces are \(\uparrow^3\), \(\uparrow^4\), \(\uparrow^5\), etc.

What fascinates me about this is that we can have these inconceivably huge numbers like \(15 \uparrow \uparrow \uparrow \uparrow \uparrow 23\)—which, by the way, is still tiny compared to some numbers that are discussed regularly in mathematical circles—and yet still know that it really isn't expressing anything that couldn't theoretically be expressed as basic addition. I like the idea of taking an incredibly simple idea, and using it iteratively to build something virtually incomprehensible or vice-versa. These pieces explore that abstract idea.

The structure:          

Pieces in this series:          

PortRait of the ArTist,**NYC2001
Solo Cello and Recorded Electronics
Summer '01

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