Art is more than the sum of its parts.

*origami*can lead to expressive paper sculptures, but we can also create supremely complicated works, especially in

*representational origami*. This may not necessarily involve a lot of emotion, but it is still something interesting that the folder injects into the paper.

Art is the difference between the whole and the sum of its parts.

What's weird is that we can turn this into an

**algebraic metaphor**:

$$\text{Art}(\text{whole}) = \text{value}(\text{whole}) - (\text{value}(\text{part}_1) + \text{value}(\text{part}_2) + \dotsb)$$

where $\text{Art}(x)$ is the "amount of art" in object $x$, $\text{value}(x)$ is the "amount of experience derived from" $x$ (e.g. emotion, sensory stimulation, etc), and $\text{whole} = \text{part}_1 + \text{part}_2 + \dotsb$.

I hate making disclaimers, but I must assure you that I do not think art and experiences can be quantified by such a terribly lossy compression into a single number (but see the remark in the final section); this is merely a

*metaphor*.

## Measure of distributivity

In Abstract Algebra, we can have sets of objects and define how we can "multiply" them together to produce objects in the same set. That definition can be quite different from what we are used to; in fact, sometimes it is not commutative, i.e. $xy \neq yx$. A commutator is a binary operation that measures "how commutative multiplication is". Sometimes we define it as

$$[x, y] = x^{-1}y^{-1}xy$$

which is obviously equal to $1$ if $xy = yx$. Sometimes it may be useful to define it as

$$[x, y] = xy - yx$$

which also "vanishes" only if $xy = yx$.

However, we're not only interested about the times when $xy = yx$; sometimes, the "defect value" $xy - yx$ is useful. Apparently, this value is related to Quantum Physics, where it represents how well quantities $x$ and $y$ (e.g. momentum and position) can be measured simultaneously. This leads to the famous Heisenberg Uncertainty Principle.

Another property in Abstract Algebra that is as important as commutativity is

*distributivity*; in a general setting, a function $f(x)$ may distribute over a sum:

$$f(a + b + c + \dotsb) = f(a) + f(b) + f(c) + \dotsb$$

A familiar example would be how multiplication distributes over addition:

$$k \times (a + b + c + \dotsb) = ka + kb + kc + \dotsb$$

Readers familiar with Number Theory can also consider the example of multiplicative functions, which distribute over multiplication.

Statement 1 is then analogous to

$$\text{value}(\text{whole}) > \text{value}(\text{part}_1) + \text{value}(\text{part}_2) + \dotsb$$

If both sides were equal, the "$\text{value}$" function would distribute over "$\text{whole}$", and the situation would be artistically quite boring. Hence, the "defect value" is of interest, which brings us to statement 2:

$$\text{Art}(\text{whole}) = \text{value}(\text{whole}) - (\text{value}(\text{part}_1) + \text{value}(\text{part}_2) + \dotsb)$$

The "defect value" $\text{Art}(\text{whole})$ measures how well (or how badly) the $\text{value}(x)$ function distributes over a sum of parts. Consequently, the "greater" the defect, the more art was injected into the medium, and the better the art piece is.

Strange examples pop into my head and refuse to leave, so here's one:

\begin{multline}\text{Art}(\text{chicken rice}) = \text{value}(\text{chicken rice}) - \text{value}(\text{chicken}) - \text{value}(\text{rice})\\ - \text{value}(\text{chilli sauce}) - \text{value}(\text{oyster sauce veggies}) - \dotsb\end{multline}

Another example, this time from Chemistry:

$$\text{Art}(\text{Glucose}) = \text{value}(\text{C}_6\text{H}_{12}\text{O}_6) - 6 \cdot \text{value}(\text{C}) - 12 \cdot \text{value}(\text{H}) - 6 \cdot \text{value}(\text{O})$$

## A caveat

There's something about the plane, and that there's only so many ways that you can push or pull it around, that constraint is quite different from other mediums. There's no carving in wood, or in stone, or knots on a rug, that's gonna look anything near like what pleats do, when they express a line.