Ask a mathematician what is the most beautiful formula, and you are likely to get Euler's identity:

$$
e^{iπ}+1 = 0
$$

Is beauty fundamental or emergent? Is it a principle embedded in the fabric of existence, an axiom or brute fact that defies definition but governs all laws and structures that emerge in the universe? Or is it itself an emergent construct, a byproduct of a pattern matching mind evolved for survival? Some questions are worth asking, even if they are impossible to answer. This project is an effort to nibble at the edges of the question - a study in experimental mathematical aesthetics. For the purpose of the experiment, we create a simple construct with very few rules and study the structures that emerge. Let us consider Euler's identity above. It comes from Euler's formula:

$$
e^{it} = \cos(t) + i \sin(t)
$$

which turns into the famed identity when *t*=π. This is a relationship between complex numbers, which have their own fascinating story, but if we took the real and imaginary parts of these numbers and imagined *t* as something that ticks regularly, say time, we get a trajectory, a parametric function plot, in the shape of a circle. This may seem trivial on the surface, but there is a deeper hidden beauty (and truth) underneath. The numbers 0,1,𝑖,π and 𝑒 are not inventions. If mathematics exists, they must exist -- they are fundamental. Similarly, points equidistant from a point, i.e. circles, must exist in any mathematical universe. The above expression embodies all of these existential truths.

In the interest of our quest for beauty, let us consider the parametric functions *t → x(t),y(t)*. For *x(t)=cos(t)* and *y(t)=sin(t)* we get a circle, which we can think of as taking a line segment *t* ∈ {-π,π} and wrapping it around to get the shape. With the circle, we have reached an aesthetic maximum. Let us explore the space of possibilities around it. Imagine the maps as defined above, but allow mathematically viable statements composed solely of the operations *Ω* ∈ {+,-,\*,/, modulo(%), power( ^ ), sin, cos},  and the parameters π and *e*. For simplicity and elegance we will use Feynman trigonometric notation, denoting sin and cos with *σ* and *γ*. This is how all p-f\_p forms we created.

For example, the expression of a circle would be *γ(t),σ(t)* and if we modulate the parameter t by adjusting the frequencies we get familiar shapes known as Lissajous curves, e.g. *γ(πet),σ(et)* looks like this.&#x20;

![](ipfs://QmaGrc4sewc9m6yESJpCCZpHg4eSYro3cqdUGQbVDRJpts)

The map only generates a set of points. The platonic ideal curve defined by the form is continuous, but its manifestation is limited by our means to represented in reality. To emphasize this, the expression is sampled at different resolutions revealing different aspects of the construct. The formal grammar becomes a kind of poetic language translating between mathematical semiotics and geometry. As the sentences become more complex, chaos and uncomputability creep in -- smooth lines break up into points, points become error bars and movement shows the inherent dynamics embedded in a mathematical micro-universe.

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The set of possible forms, even within such a constrained system is potentially infinite. By inviting the audience to actively participate in the process of natural parameter selection by choosing from the possible, the project becomes a way to ask questions that may be impossible to answer.

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There are questions worth asking, even if they are impossible to answer.

Is Beauty Fundamental or Emergent?

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