A Nakayama algebra with almost separate relations is one where the overlap between any pair of relations is at most one arrow.
In this paper we give a derived equivalence between such Nakayama algebras and path algebras of quivers of a special form known as quipu quivers. Furthermore, we show how this derived equivalence can be used to produce a complete classification of linear Nakayama algebras with almost separate relations.

As an application, we include a list of the derived equivalence classes of all Nakayama algebras of length $\leq 8$ with almost separate relations.

§ INTRODUCTION

Throughout this paper $k$ will be a field and $A_n$ will refer to the linearly ordered quiver of Dynkin type $\mathbb{A}$ with $n$ vertices labelled $1,\ldots, n$.
We will denote by $A_{n,(n_0,\ldots,n_r)}^{(l_0,\ldots, l_r)}$ the linear Nakayama algebra $kA_n/I$ where $I$ is the ideal generated by a relation of length $l_i\geq 2$ beginning in vertex $n_i$ for each $0\leq i\leq r$. We assume that $n_i < n_{i+1}$ and $n_i+l_i< n_{i+1}+l_{i+1}$ for each $0\leq i \leq r-1$, and that $n_r+l_r\leq n$.

Our focus will be on Nakayama algebras where the overlap between any pair of relations is at most one arrow, meaning $n_{i+1}\geq n_i+l_i-1$ for all $i$, which we will refer to as the algebra having almost separate relations.

In order to investigate the derived equivalence classes of these algebras, we will employ the combinatorial quiver mutation developed in <cit.>.

We will show that there exist derived equivalences between linear Nakayama algebras with almost separate relations and so-called quipu algebras. These are path algebras of so-called quipu quivers, whose underlying graphs are what's known as open quipus.
An open quipu (or simply quipu) is a tree of maximum degree $3$, such that all vertices of degree $3$ lie on a path. They were introduced by Woo and Neumaier in <cit.> as an example of graphs with small spectral radius. The name comes from their shape, a line with straight cords going out of it, which is reminiscent of the knotted strings used by the Incas for record keeping purposes.

The main result of this paper is the existence of derived equivalences between quipu algebras and Nakayama algebras with almost separate relations, as stated in the following theorem. The notation is explained in <ref>.

Let $P_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$ be a quipu, and let

$$n_i = k_0 + \sum\limits_{j=1}^{i} (m_{j-1} + k_{j} + 1) \text{ for } 1\leq i\leq r+1.$$

If $D$ is a quipu quiver, then there exists a derived equivalence

$$kD\simeq A_{n_{r+1},(k_0,n_1,n_2,\ldots, n_{r} )}^{(m_{0}+2, m_{1}+2, \ldots, m_{r}+2)}$$

if and only if $D$ is $P_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$ with some orientation.

Using this theorem, we are able to give a complete classification up to derived equivalence of linear Nakayama algebras with almost separate relations. Each equivalence class is determined by the shape of a given quipu, with the cords of the quipu corresponding to the relations of the Nakayama algebras in a certian way. We will go through the details in <ref>.

Relations of length $2$ do not affect the derived equivalence class of a Nakayama algebra with almost separate relations, and the number of almost separate relations of length $\geq 3$ is a derived invariant of such algebras. The length and position of each relation of length $\geq 3$ is also a derived invariant, except for (possibly) the first and last such relations along the quiver.

The precise collection of Nakayama algebras included in each of these derived equivalence classes is stated in the following theorem. Notice that all Nakayama algebras with almost separate relations in a given class can be obtained from each other by a combination of a few simple operations. The operations are:

* Changing the length and end vertex (resp. start vertex of) the first (resp. last) relation by a certain amount.

* Adding or removing relations of length $2$.

* Reversing all arrows in the quiver.

This works because all Nakayama algebras in a class correspond to orientations of the same quipu, and the operations correspond to changing that orientation.

If $A_{n,(n_0,\ldots,n_r)}^{(l_0,\ldots, l_r)}$ is a linear Nakayama algebra with almost separate relations, where $l_i\geq 3$ for all $i$, then the following Nakayama algebras are derived equivalent.

\begin{align*}
&A_{n,(n_0,n_1,\ldots,n_{r-1},n_r)}^{(l_0, l_1, \ldots, l_{r-1}, l_r)}  &&A_{n,(n-n_r-l_r+1,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(l_r, l_{r-1}, \ldots, l_{1}, l_0)}\\
&A_{n,(n_0,n_1,\ldots,n_{r-1},n_r)}^{(l_0, l_1, \ldots, l_{r-1}, n-n_r-l_r+3)} &&A_{n,(l_r-2,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(n-n_r-l_r+3, l_{r-1}, \ldots, l_{1}, l_0)}    \\
&A_{n,(l_0-2,n_1,\ldots,n_{r-1},n_r)}^{(n_0+2, l_1, \ldots, l_{r-1}, l_r)}  && A_{n,(n-n_r-l_r+1,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(l_r, l_{r-1}, \ldots, l_{1}, n_0+l_0)}\\
&A_{n,(l_0-2,n_1,\ldots,n_{r-1},n_r)}^{(n_0+2, l_1, \ldots, l_{r-1}, n-n_r-l_r+3)} &&A_{n,(l_r-2,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(n-n_r-l_r+3, l_{r-1}, \ldots, l_{1}, n_0+l_0)}.
\end{align*}

Also, adding a set of almost separate relations of length $2$ to available positions in any of these algebras will not change the derived equivalence class.

The path algebra of the quipu $P_{(1,2,0,1)}^{(2,1,3)}$ with any orientation is derived equivalent to (among others) the following Nakayama algebras.

[column sep = tiny, row sep = tiny]
∙[r, no head]     ∙[r, no head]  [d, no head]     ∙[r, no head]     ∙[r, no head]     ∙[r, no head] [d, no head]     ∙[r, no head] [d, no head]     ∙                P_(1,2,0,1)^(2,1,3)

∙[d, no head]              ∙    ∙[d, no head]

∙                ∙[d, no head]

∙[r] [rrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r]      ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r] [rrrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r]     ∙    A_13,(1,6,8)^(4,3,5)

∙[r] [rrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r]      ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r]     ∙    A_13,(1,6,8)^(4,3,3)

∙[r]     ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]      ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r] [rrrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r]     ∙    A_13,(2,6,8)^(3,3,5)

∙[r]     ∙[r]  [rrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]      ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r]     ∙    A_13,(2,6,8)^(3,3,3)

∙[r] [rrrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r] [rrr, bend left, dashed]     ∙[r]     ∙[r]     ∙[r]     ∙[r] [rrrr, dashed, bend left]    ∙[r]     ∙[r]     ∙[r]     ∙    A_13,(1,5,9)^(5,3,4)

∙[r] [rrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r] [rr, bend left, dashed]     ∙[r] [rr, bend left, dashed]      ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r] [rrrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r]     ∙    A_13,(1,4,5,6,8)^(4,2,2,3,5)

These Nakayama algebras are all derived equivalent to the path algebra of the quipu $P_{(1,2,0,1)}^{(2,1,3)}$ with any orientation.

§ QUIPU QUIVERS

An (open) quipu is a tree of maximal degree $3$, such that all vertices of degree $3$ lie on a path.
This path is called the main string of the quipu, while the remaining paths going out of each of the vertices of degree $3$ are called cords. Both of these terms are taken from <cit.>.

In the notation of <cit.>, a quipu can be written as $P_{(k_0,k_1,\ldots, k_r,k_{r+1})}^{(m_0,m_1,\ldots, m_r)}$ where $r\geq 0$ and all $k_i,m_i\geq 0$.
We denote by $m_i$ the length of cord $i$, while $k_i$ denotes the number of vertices along the main string between the beginning of cord $i$ and cord $i+1$.

Note that quipus are not uniquely determined by this notation, and that we do allow trivial cords of length $m_i=0$.

[column sep = small, row sep = small]
∙[r, no head, dotted, "k_0"]     ∙[r, no head]     ∙[r, no head] [d, no head]     ∙[r, no head, dotted, "k_1"]     ∙[r, no head]     ∙[r, no head] [d, no head]     ∙[r, no head, dotted]     ∙[r, no head]     ∙[r, no head] [d, no head]     ∙[r, no head, dotted, "k_r"]     ∙[r, no head]     ∙[r, no head] [d, no head]     ∙[r, no head, dotted, "k_r+1"]     ∙

∙[d, no head, dotted, "m_0"]                                                                                ∙[d, no head, dotted, "m_1"]                                                                                ∙[d, no head, dotted, "m_r-1"]                                                                                ∙[d, no head, dotted, "m_r"]

∙                                                                          ∙                                                                          ∙                                                                          ∙

The general form of a quipu $P_{(k_0,k_1,\ldots, k_r,k_{r+1})}^{(m_0,m_1,\ldots, m_r)}$

[remember picture, column sep = tiny, row sep = tiny]
∙[r, no head]    ∙[r, no head]     ∙[r, no head] [d, no head]     ∙[r, no head] [d, no head]     ∙[r, no head]     ∙[r, no head] [d, no head]     ∙[r, no head]    ∙[r, no head] [d, no head]    ∙[r, no head] [d, no head]    ∙

∙[d, no head]     ∙[d, no head]         ∙[d, no head]         ∙[d, no head]     ∙[d, no head]

∙    ∙[d, no head]         ∙[d, no head]         ∙[d, no head]     ∙[d, no head]

∙[d, no head]         ∙        ∙[d, no head]     ∙

∙[d, no head]                 ∙

∙[d, no head]

Example of a quipu, this is $P_{(2,0,1,1,0,1)}^{(2,6,3,4,3)}$.

Because we want to work with quivers, we also need to define an oriented version of a quipu. For a quipu $P_{(k_0,k_1,\ldots, k_r,k_{r+1})}^{(m_0,m_1,\ldots m_r)}$ the quipu quiver $D_{(k_0,k_1,\ldots, k_r,k_{r+1})}^{(m_0,m_1,\ldots, m_r)}$ is a directed version of the same graph, where the main string is linearly oriented from left to right and each cord is linearly oriented away from the main string. We will call the path algebra of a quipu quiver a quipu algebra.

[column sep = small, row sep = small]
∙[r, no head, dotted, "k_0"]     ∙[r]     ∙[r] [d]     ∙[r, no head, dotted, "k_1"]     ∙[r]     ∙[r] [d]     ∙[r, no head, dotted]     ∙[r]     ∙[r] [d]     ∙[r, no head, dotted, "k_r"]     ∙[r]     ∙[r] [d]     ∙[r, no head, dotted, "k_r+1"]     ∙

∙[d, no head, dotted, "m_0"]                                                                                ∙[d, no head, dotted, "m_1"]                                                                                ∙[d, no head, dotted, "m_r-1"]                                                                                ∙[d, no head, dotted, "m_r"]

∙                                                                          ∙                                                                          ∙                                                                          ∙

The general form of a quipu quiver $D_{(k_0,k_1,\ldots, k_r,k_{r+1})}^{(m_0,m_1,\ldots, m_r)}$

By looking at <ref> we can immediately make the following two observations:

The number of vertices in the quipu $P_{(k_0,k_1,\ldots, k_r,k_{r+1})}^{(m_0,m_1,\ldots, m_r)}$ is equal to $r+1+\sum\limits_{i=0}^{r+1}k_i +  \sum\limits_{i=0}^r m_i$.

There exist graph isomorphisms

\begin{align*}
P_{(k_0,k_1,\ldots, k_{r-1},k_r,k_{r+1})}^{(m_0,m_1,\ldots ,m_{r-1},m_r)} &\simeq P_{(m_0,k_1,\ldots, k_{r-1}, k_r, k_{r+1})}^{(k_0,m_1,\ldots, m_{r-1}, m_r)}\\
&\simeq P_{(k_0,k_1,\ldots, k_{r-1}, k_r,m_{r})}^{(m_0,m_1,\ldots, m_{r-1}, k_{r+1})}\\
&\simeq P_{(k_{r+1},k_r,k_{r-1}\ldots, k_1,k_{0})}^{(m_r,m_{r-1},\ldots, m_1, m_0)}
\end{align*}

The second observation essentially says that a quipu is not uniquely defined in this notation. We can view the first cord as a part of the main string, making the first part of the main string a cord instead, and the same is true for the last cord and last part of the main string. Also reversing the order of the main string and the cords clearly doesn't change quipu.

By combining these operations we get a total of $8$ expressions for the same quipu, some of which may coincide depending on the shape of the ends of the quipu. Note that even if different expressions define the same quipu, their corresponding quipu quivers will in general not be isomorphic.

[remember picture, column sep = small, row sep = small]
∙[r]     ∙[r]     ∙[r] [d]     ∙     ∙[r]     ∙[r] [d]     ∙    ∙[r]     ∙[r] [d]     ∙[r]     ∙

∙            ∙[d]             ∙

Three different quipu quivers which are orientations of the same quipu: $D_{(2,1)}^{(1)}$, $D_{(1,1)}^{(2)}$ and $D_{(1,2)}^{(1)}$.

§ CORD/RELATION-SWAP

In this section we introduce the notion of a quipu quiver with almost separate relations, and we develop a technique for going between such quivers in a way that preserves the derived equivalence class of their path algebras. Later we will use this to find derived equivalences between quipu algebras with no relations and Nakayama algebras with almost separate relations.

We say that a quipu quiver has almost separate relations if it comes with a set of zero relations satisfying the following conditions.

* The relations must all be along the main string

* No cord may begin at an internal vertex of a relation

* No pair of relations may have an overlap of more than one arrow

We will now show a technique for generating derived equivalent quipus with almost separate relations. We do this by repeatedly applying the quiver mutation defined in <cit.>, which modifies quivers with relations in a way that preserves the derived equivalence class of their path algebras. We will not show the mutation procedure in detail here, see the paper for more information.

Let $Q$ be a quipu quiver with almost separate relations, and let $r$ be a relation in $Q$ with length $l\geq 2$, start vertex $s(r)$ and end vertex $t(r)$. Let $m\geq 0$ denote the length of the (possibly trivial) cord starting in $s(r)$. We construct a new quipu with almost separate relations $Q'$ from $Q$ as follows:

* Remove the relation $r$ and the cord of length $m$ starting in $s(r)$.

* Replace the path $s(r)\rightarrow \ldots \rightarrow t(r)$ of length $l$ by a path of length $m+2$. If there is a relation starting in the vertex before $t(r)$, keep it.

* If there is an arrow ending in $s(r)$, add a relation of length $m+2$ from the start vertex of this arrow to the vertex one step before $t(r)$ along the main string.

* Add a cord of length $l-2$ going out of the vertex before $t(r)$.

The quipu quiver $Q'$ with relations constructed in <ref> has a path algebra which is derived equivalent to that of the quipu quiver Q with relations.

This move can be viewed as the cord and relation swapping order and swapping lengths ($\pm\ 2$), hence we refer to it as cord/relation-swap (CR-swap for short).

Below is a visual representation of the CR-swap in action.

[remember picture, column sep = tiny, row sep = small]
[baseline,remember picture][anchor=base,inner sep=0pt] (node1) ;  ⋯[rr, bend left, dotted]     [-15] ∙[baseline,remember picture][anchor=base,inner sep=0pt] (node2) ; [r]     ∙[r] [d] [rrrrrr, dashed, bend left]     ∙[r]     ⋯[r]  [r]     ∙[r]    ∙[r]     ∙[r] [rrr, bend left, dotted]     [baseline,remember picture][anchor=base,inner sep=0pt] (node3) ; ∙[r]     ∙    [-15] ⋯ [baseline,remember picture][anchor=base,inner sep=0pt] (node4) ;

[baseline,remember picture][anchor=base,inner sep=0pt] (node5) ;  ⋯[rr, bend left, dotted]     [-15] ∙[baseline,remember picture][anchor=base,inner sep=0pt] (node6) ;  [r] [rrrrrr, dashed, bend left]     ∙[r]     ∙[r]     ⋯[r]  [r]     ∙[r]    ∙[r]     ∙[r] [d] [rrr, dotted, bend left]     [baseline,remember picture][anchor=base,inner sep=0pt] (node7) ;  ∙[r]     ∙    [-15] ⋯ [baseline,remember picture][anchor=base,inner sep=0pt] (node8) ;

[overlay, remember picture]
[decorate, decoration = calligraphic brace] (-2.8,4.4) –  (-2.8,6.7) node[midway,left] $m$;
[decorate, decoration = calligraphic brace] (2.75,7.3) –  (-2,7.3) node[midway,below] $l$;
[cyan,-,double=cyan, double distance = 16, opacity=0.1] (node1) – (node2);
[cyan,-,double=cyan, double distance = 16, opacity=0.1] (node3) – (node4);
[decorate, decoration = calligraphic brace] (1.6,0.21) –  (1.6,2.5) node[midway,left] $l-2$;
[decorate, decoration = calligraphic brace] (1.95,3.1) –  (-2.8,3.1) node[midway,below] $m+2$;
[cyan,-,double=cyan, double distance = 16, opacity=0.1] (node5) – (node6);
[cyan,-,double=cyan, double distance = 16, opacity=0.1] (node7) – (node8);

Relevant part of the quipu quiver before and after the CR-swap. The dotted arrows are optional relations and the shaded parts of the quiver remain unchanged.

Starting from a quipu quiver with almost separate relations, we will mutate through a sequence of other quivers, until we arrive at another quipu quiver with almost separate relations.
We will explicitly state which vertices to mutate at in order to produce the desired change in the quiver. We begin by numbering the vertices like this

\begin{equation*}
\hspace{-1cm}
\begin{tikzcd}[remember picture, column sep = small, row sep = small]
\hspace{1cm} \cdots \ar[rr, bend left, dotted] & [-20] {\scriptstyle -1} \ar[r] & {\scriptstyle 0} \ar[r] \ar[d] \ar[rrrrrr, dashed, bend left] & {\scriptstyle 1} \ar[r] & \cdots \ar[r]  \ar[r] & {\scriptstyle l-3} \ar[r] & {\scriptstyle l-2} \ar[r] & {\scriptstyle l-1} \ar[r] \ar[rrr, bend left, dotted] & {\scriptstyle l} \ar[r] & {\scriptstyle l+m+1} & [-20] \cdots \hspace{1cm} \\
& & {\scriptstyle l+1} \ar[d] & & & & & & &\\
& & \vdots \ar[d] & & & & & & &\\
& & {\scriptstyle l+m} & & & & & & &
\end{tikzcd}
\end{equation*}

The dotted arrows represent optional zero relations to/from some vertices in other parts of the quiver. We include these relations to show that they remain unchanged after the CR-swap. If there are no such relations, the CR-swap still works the same, which is easily seen by repeating the following proof without the dotted relations.

With this numbering, we will mutate at the following sequence of vertices:

\begin{equation*}
{\scriptstyle l-2},\ {\scriptstyle l-3},\ \ldots,\ {\scriptstyle 1},\ {\scriptstyle 0},\ {\scriptstyle l+1},\ {\scriptstyle l+2},\ \ldots,\ {\scriptstyle l+m}
\end{equation*}

In other words, we begin by mutating on the second vertex from the end of the relation. Then we move one vertex back each time, until we hit the start of the relation, and then we mutate along the cord.

Now, let us mutate the quiver at vertex ${\scriptstyle l-2}$. We do this by applying the mutation rules, which in this case just means adding an arrow from ${\scriptstyle l-3}\rightarrow {\scriptstyle l-1}$, flipping the arrow ${\scriptstyle l-2}\rightarrow {\scriptstyle l-1}$, and replacing the arrow ${\scriptstyle l-3}\rightarrow {\scriptstyle l-2}$ by a relation. Thus, we end up with the quiver

\begin{equation*}
\begin{tikzcd}[remember picture, column sep = small, row sep = small]
\hspace{1cm} \cdots \ar[rr, bend left, dotted] & [-20] {\scriptstyle -1} \ar[r] & {\scriptstyle 0} \ar[r] \ar[d] \ar[rrrrr, dashed, bend left] & {\scriptstyle 1} \ar[r] & \cdots \ar[r]  \ar[r] & {\scriptstyle (l-3)} \ar[r] \ar[dr, dashed] & {\scriptstyle (l-1)} \ar[r] \ar[d] \ar[rrr, bend left, dotted] & {\scriptstyle l} \ar[r] & {\scriptstyle (l+m+1)} & [-20] \cdots \hspace{1cm} \\
& & {\scriptstyle (l+1)} \ar[d] & & & & {\scriptstyle (l-2)} & & &\\
& & \vdots \ar[d] & & & & & & &\\
& & {\scriptstyle (l+m)} & & & & & & &
\end{tikzcd}
\end{equation*}

Mutation at vertex ${\scriptstyle l-3}$ is now done similarly, except we also turn the relation ${\scriptstyle l-3}\dashrightarrow {\scriptstyle l-2}$ into an arrow, and add a relation which makes the new composition ${\scriptstyle l-1}\rightarrow {\scriptstyle l-3}\rightarrow {\scriptstyle l-2}$ equal to the arrow ${\scriptstyle l-1}\rightarrow {\scriptstyle l-2}$. All together, these changes result in the quiver

\begin{equation*}
\begin{tikzcd}[remember picture, column sep = small, row sep = small]
\hspace{1cm} \cdots \ar[rr, bend left, dotted] & [-20] {\scriptstyle -1} \ar[r] & {\scriptstyle 0} \ar[r] \ar[d] \ar[rrrrr, dashed, bend left] & {\scriptstyle 1} \ar[r] & \cdots \ar[r]  \ar[r] & {\scriptstyle l-4} \ar[r] \ar[dr, dashed] & {\scriptstyle l-1} \ar[r] \ar[d] \ar[rrr, bend left, dotted] & {\scriptstyle l} \ar[r] & {\scriptstyle l+m+1} & [-20] \cdots \hspace{1cm} \\
& & {\scriptstyle l+1} \ar[d] & & & & {\scriptstyle l-3} \ar[d] & & &\\
& & \vdots \ar[d] & & & & {\scriptstyle l-2} & & &\\
& & {\scriptstyle l+m} & & & & & & &
\end{tikzcd}
\end{equation*}

This pattern continues as we mutate along ${\scriptstyle l-4},\ \ldots, {\scriptstyle 1}$. With each mutation the relation on top becomes one arrow shorter, and the path going out of vertex ${\scriptstyle l-1}$ becomes one arrow longer. This is easily seen using the mutation rules. Eventually, after mutating at each vertex from ${\scriptstyle l-2}$ down to and including ${\scriptstyle 1}$, we have this quiver

\begin{equation*}
\begin{tikzcd}[remember picture, column sep = small, row sep = small]
\hspace{1cm} \cdots \ar[rr, bend left, dotted] & [-20] {\scriptstyle -1} \ar[r] & {\scriptstyle 0} \ar[r] \ar[d] \ar[rr, dashed, bend left] \ar[dr, dashed] & {\scriptstyle l-1} \ar[r] \ar[d] \ar[rrr, bend left, dotted] & {\scriptstyle l} \ar[r] & {\scriptstyle l+m+1} & [-20] \cdots \hspace{1cm} \\
& & {\scriptstyle l+1} \ar[d] & {\scriptstyle 1} \ar[d] & & &\\
& & \vdots \ar[d] & \vdots \ar[d] & & &\\
& & {\scriptstyle l+m} & {\scriptstyle l-2} & & &
\end{tikzcd}
\end{equation*}

Mutating at vertex ${\scriptstyle 0}$ will now yield, after some rearranging and simplifying, the following quiver. Note that the dotted relation on the left side gives rise to two new relations, both starting at the same vertex.

\begin{equation*}
\begin{tikzcd}[remember picture, column sep = small, row sep = small]
\hspace{1cm} \cdots \ar[rrr, bend left, dotted] \ar[drr, dotted] & [-20] {} \ar[r] & {\scriptstyle -1} \ar[r] \ar[d] \ar[dr, dashed] & {\scriptstyle l-1} \ar[d] & & {\scriptstyle l} \ar[r] & {\scriptstyle l+m+1} & [-20] \cdots \hspace{1cm} \\
& & {\scriptstyle l+1} \ar[r] \ar[d] \ar[urrr, dashed] \ar[dr, dashed] & {\scriptstyle 0} \ar[d] \ar[urr] \ar[urrrr, dotted, bend right, looseness=0]& & & &\\
& & {\scriptstyle l+2} \ar[d] & {\scriptstyle 1} \ar[d] & & & &\\
& & \vdots \ar[d] & \vdots \ar[d] & & & &\\
& & {\scriptstyle l+m} & {\scriptstyle l-2} & & & &
\end{tikzcd}
\end{equation*}

Mutating at ${\scriptstyle l+1}$ now yields

\begin{equation*}
\begin{tikzcd}[remember picture, column sep = small, row sep = small]
\hspace{1cm} \cdots \ar[rrr, bend left, dotted] \ar[drrr, dotted] & [-20] {} \ar[r] & {\scriptstyle -1} \ar[r] \ar[dr] \ar[drr, dashed] & {\scriptstyle l-1} \ar[r] & {\scriptstyle 0} \ar[d] & & {\scriptstyle l} \ar[r] & {\scriptstyle l+m+1} & [-20] \cdots \hspace{1cm} \\
& & & {\scriptstyle l+2} \ar[r] \ar[d] \ar[urrr, dashed] \ar[dr, dashed] & {\scriptstyle l+1} \ar[d] \ar[urr] \ar[urrrr, dotted, bend right, looseness=0]& & & &\\
& & & {\scriptstyle l+3} \ar[d] & {\scriptstyle 1} \ar[d] & & & &\\
& & & \vdots \ar[d] & \vdots \ar[d] & & & &\\
& & & {\scriptstyle l+m} & {\scriptstyle l-2} & & & &
\end{tikzcd}
\end{equation*}

Notice that the structure of the quiver is mostly unchanged, except ${\scriptstyle l+1}$ has taken the place of ${\scriptstyle 0}$, ${\scriptstyle l+2}$ has taken the place of ${\scriptstyle l+1}$, and ${\scriptstyle 0}$ has been inserted into the upper path of the commutative diagram. So the difference is that the cord going out of the lower path of the commutativity relation has become shorter by one arrow, and the upper path has become longer by one arrow.

This pattern continues as we keep mutating along the cord ${\scriptstyle l+2},\ {\scriptstyle l+3},\ \ldots$, and eventually, after mutating at ${\scriptstyle l+m-1}$, we have the quiver

\begin{equation*}
\begin{tikzcd}[remember picture, column sep = small]
\hspace{1cm} \cdots \ar[rrr, bend left, dotted] \ar[drrrrr, dotted] & [-20] {} \ar[r] & {\scriptstyle -1} \ar[r] \ar[drrr] \ar[drrrrr, dashed] & {\scriptstyle l-1} \ar[r] & {\scriptstyle 0} \ar[r] & {\scriptstyle l+1} \ar[r] & \cdots \ar[r] & {\scriptstyle l+m-2} \ar[d] & & {\scriptstyle l} \ar[r] & {\scriptstyle l+m+1} & [-20] \cdots \hspace{1cm} \\
& & & & & {\scriptstyle l+m} \ar[rr]  \ar[urrrr, dashed] \ar[drr, dashed] & & {\scriptstyle l+m-1} \ar[d] \ar[urr] \ar[urrrr, dotted, bend right, looseness=0] & & &  &\\
& & {}  & & & & & {\scriptstyle 1} \ar[d] & & & &\\
& &   & & & & & \vdots \ar[d] &  & & &\\
& & {} & & & & & {\scriptstyle l-2}  & & & &
\end{tikzcd}
\end{equation*}

Finally, if we now mutate at ${\scriptstyle l+m}$, and do a bit of rearranging, we end up with the following quiver

\begin{equation*}
\begin{tikzcd}[remember picture, column sep = tiny, row sep = small]
\hspace{1cm} \cdots \ar[rr, dotted, bend left] & [-15] {\scriptstyle -1} \ar[r] \ar[rrrrrr, bend left, dashed, looseness=0.75] & {\scriptstyle l-1} \ar[r] & {\scriptstyle 0} \ar[r] & {\scriptstyle l+1} \ar[r] & \cdots \ar[r] & {\scriptstyle l+m-1} \ar[r] & {\scriptstyle l+m} \ar[r] \ar[d] \ar[rrr, bend left, dotted] & {\scriptstyle l} \ar[r] & {\scriptstyle l+m+1} & [-15] \cdots \hspace{1cm} \\
& & & & & & & {\scriptstyle 1} \ar[d] & & & \\
& & & & & & & {\scriptstyle 2} \ar[d] & & & \\
& & & & & & & \vdots \ar[d] & & & \\
& & & & & & & {\scriptstyle l-2} & & &
\end{tikzcd}
\end{equation*}

This is what we wanted to show. The cord and relation have swapped places, and it is easy to see that the length of the cord is $l-2$, and the length of the relation is $m+2$. Notice that relatively to the rest of the quiver, the dotted relations remain unchanged. Also note that if there is no arrow ${\scriptstyle -1}\rightarrow {\scriptstyle 0}$, then the mutation simplifies, and we get no relation after the swap. This completes the proof.

The CR-swap is a valuable tool for investigating derived equivalence classes of quipu algebras. We now give a corollary for some special cases of using the CR-swap.

Let $Q$ be a quipu quiver with almost separate relations, with a relation of length $l$ and a cord of length $m$ starting at the same vertex.
If we apply the CR-swap to this relation and cord, then the following holds:

* If $l=2$, then the resulting cord obtained by the CR-swap has length $0$.

* If $m=0$, then the resulting relation obtained by the CR-swap has length $2$

Point $1$ means the resulting quipu quiver after the CR-swap in that case just has a relation of length $m+2$, shifted one vertex back along the main string, with no cord going out of its last vertex. From point $2$, we see that if we swap at a relation which has no cord going out of its first vertex, then the resulting quipu quiver has a relation of length $2$, with a cord of length $l-2$ going out of its last vertex.

By combining these two points, we observe that CR-swapping a lone relation of length $2$ with no cord, will result in a new relation of length $2$ with no cord, shifted one vertex back in the quiver.

As presented here, the CR-swap uses right tilting mutation. There is an entirely dual result using left mutation, meaning we can swap cords and relations in both directions.

§ QUIPUS AND NAKAYAMA ALGEBRAS

In this section we will show that any linear Nakayama algebra with almost separate relations is derived equivalent to the path algebra of some quipu quiver. Furthermore we use this to develop a method to classify all Nakayama algebras with almost separate relations into derived equivalence classes.

The main idea is to find a way to mutate between a quiver with a relation of a certain length at a certian position, and one with a cord of a corresponding length at a corresponding position. By combining this with the CR-swap, we will be able to construct a sequence of quiver mutations between a quipu quiver and the quiver of a Nakayama algebra.

We will show that there exists a sequence of quiver mutations between the quipu quiver $D_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$ and the quiver $A_{n}$ with $(r+1)$ relations, where the $i$-th relation has length $(m_i+2)$ and its starting vertex is determined by counting the number of vertices appearing before the beginning of the $i$-th cord in $D_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$.
So if the vertices in $A_n$ are labelled from $1$ to $n$, the $0$-th relation will begin at vertex $k_0$ and have length $m_0$, the $1$-st relation will begin at vertex $(k_0+m_0+k_1+1)$ with length $m_1$, and so on. In general relation $i$ will have length $m_i$ and begin at vertex $k_0 + \sum\limits_{j=1}^{i} (m_{j-1} + k_{j} + 1)$ for $1\leq i\leq r$.
See <ref> for an example of this correspondence.

[column sep = tiny, row sep = tiny]
∙[r]     ∙[r] [rrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]      ∙[r]     ∙[r] [rrrrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r]     ∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r]     ∙

∙[r]     ∙[r]     ∙[r] [d]     ∙[r]     ∙[r]     ∙[r] [d]    ∙[r] [d]     ∙

∙[d]              ∙[d]     ∙

∙              ∙[d]

There is a derived equivalence between the algebras $A_{14,(2,7,11)}^{(4,5,3)}$ and $kD_{(2,2,0,1)}^{(2,3,1)}$.

Let $Q$ be a quipu quiver with almost separate relations, and let $s,\ t$ and $l$ denote resp. the source, target and length of the first relation along the main string of $Q$. Let $Q'$ be identical to $Q$, except the following changes:

* The relation $s\dashrightarrow t$ is removed.

* The path $s\rightarrow \ldots \rightarrow t$ is replaced by a path $s\rightarrow t'\rightarrow t$.

* There is a cord of length $l-2$ going out of vertex $t'$.

Then $Q$ and $Q'$ have derived equivalent path algebras.

The idea behind this proof is to repeatedly apply the CR-swap to the first relation along the main string in $Q$. Each time we do so, what constitutes the first relation is moved one step towards the beginning of the main string, until eventually it is removed entirely. Along the way the cords to the left of the first relation will be turned into relations and then back into cords, leaving everything except the first relation unchanged in the end. What was the first relation has then been replaced by a single arrow with a cord going out of its target vertex.

Let $m_i\geq 0$ be the length of the (possibly trivial) cord starting in vertex number $i$ of the main string for each $1\leq i\leq n_0$. If the first relation of $Q$ begins in vertex $n_0$ and has length $l_0$, we can apply the CR-swap to it with $l=l_0$ and $m=m_{n_0}$.
This will yield a quiver which is the same as before, except the given cord and relation have been removed, and replaced by a relation of length $m_{n_0}+2$ starting in vertex $n_0-1$ with a cord of length $l_0-2$ going out of its last vertex.

Since vertex $n_0-1$ by assumption also has a cord of length $m_{n_0-1}$ going out of it, we can now apply the CR-swap to this new relation with $l=m_{n_0}+2$ and $m=m_{n_0-1}$. This will yield a relation of length $m_{n_0-1}+2$ starting in vertex $n_0-2$, with a cord of length $(m_{n_0}+2)-2=m_{n_0}$ going out of its last vertex, which is $n_0$. In other words, we have recreated the cord of length $m_{n_0}$ going out of vertex $n_0$, but the first relation on the main string now appears before this cord, rather than after. And from vertex $n_0$ there is now a single arrow to the start vertex of the cord of length $l_0-2$ we constructed in the first step.

Because each vertex from $1$ to $n_0$ has a cord of length $\geq 0$ we can repeat this process. Each time we apply the CR-swap to the current first relation on the main string, it is removed and replaced by a new relation starting one vertex further back. Also, the cord we removed in the previous step is reformed just after this new relation. After $n_{1}$ CR-swaps the relation is removed from the first relation of the main string, and all the cords have been restored. This leaves us with a quiver isomorphic to the one we started with, except the relation of length $l_0$ starting in vertex $n_0$ has been replaced by a cord of length $l_0-2$ starting in vertex $n_0+1$

Let $A_{n,(n_0,n_1,\ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_{r})}$ be a linear Nakayama algebra with almost separate relations. Then there exists a derived equivalence between $A_{n,(n_0,n_1,\ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_{r})}$ and the path algebra of the quipu quiver

$$D_{(\hspace{0.35em} n_0 \hspace{0.35em},\ (n_1+1-n_0-l_0),\ \ldots\ ,\ (n_r + 1 - n_{r-1}-l_{r-1}),\ (n + 1 - n_r-l_r))}^{(l_0-2,\hspace{1.75em}  l_1-2 \hspace{1.75em},\ \ldots\ , \hspace{2.6em}  l_r-2 \hspace{2.6em})}$$

Since $A_{n,(n_0,n_1,\ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_{r})}$ is a has almost separate relations, its quiver can be considered a quipu quiver with almost separate relations. This means that the assumptions in <ref> are satisfied, and applying the CR-swap $n_0$ times to the relation starting in vertex $n_0$ will replace that relation by a cord of length $l_0-2$ going out of vertex $n_0+1$.

Because the internal vertices of the removed relation have been moved out to form the cord, all subsequent vertices in the main string have had their distance from the beginning of the main string reduced by $l_0-2$. As a consequence the next relation along the main string, whose start vertex was $n_1$ before the swaps, now starts in vertex $n_1-l_0+2$ along the main string. Again, applying the CR-swap $n_1-l_0+2$ times to this relation will remove it and replace it by a cord of length $l_1-2$ starting in vertex $n_1-l_0+3$. Notice that the number of vertices strictly between the starting vertices for these two cords is

which is equal to one plus the number of vertices between the end of the first and the beginning of the second relation.

We can repeat this process for the remaining relations in the quiver, each time replacing whichever relation is first by a corresponding cord. Note that this will change the numbering of the main string, but not the total number of vertices that appear before the start of a relation, as we work our way through the quiver.

The relation which begins in what was originally vertex $n_i$ will be replaced by a cord of length $l_i-2$, and the total number of vertices (counting the main string and cords) before the start of this cord is $n_i$. The preceding cord has length $l_{i-1}-2$ and a total of $n_{i-1}$ vertices before it. So the number of vertices strictly between these two cords is

Finally, when all relations have been replaced by cords, the last cord has $n_r$ vertices before it and contains $l_r-1$ vertices, including the beginning. Since the total number of vertices in the quiver is $n$, this means that the number of vertices after the last cord is $n+1-n_r-l_r$.

In other words, the quiver we end up with is precisely the quipu quiver

$$D_{(\hspace{0.35em} n_0 \hspace{0.35em},\ (n_1+1-n_0-l_0),\ \ldots\ ,\ (n_r + 1 - n_{r-1}-l_{r-1}),\ (n + 1 - n_r-l_r))}^{(l_0-2,\hspace{1.75em}  l_1-2 \hspace{1.75em},\ \ldots\ , \hspace{2.6em}  l_r-2 \hspace{2.6em})},$$

and since there is a sequence of mutations between these quivers, their path algebras are derived equivalent.

In <ref> we have included a concrete example of the approach used in this proof, which step by step transforms the quiver of a Nakayama algebra into a quipu quiver.

[column sep = tiny, row sep = tiny]
∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r] [rrrr, dashed, bend left]     ∙[r]      ∙[r]     ∙[r]     ∙[r] [rr, dashed, bend left]     ∙[r]     ∙            ∙[r]     ∙[r] [d]     ∙[r] [d] [rr, dashed, bend left]     ∙[r]     ∙[r]     ∙

[dd, squiggly]                                   ∙    ∙[d]     [dd, squiggly]

∙[r]     ∙[r] [d] [rrrr, dashed, bend left]     ∙[r]      ∙[r]     ∙[r] [dd, squiggly]     ∙[r] [rr, dashed, bend left]     ∙[r]     ∙            ∙[r]     ∙[r] [d] [rrrr, bend left, dashed]     ∙[r]     ∙[r]     ∙[r] [dd, squiggly]      ∙[r]     ∙[r]     ∙

∙                                            ∙

∙[r] [rrr, dashed, bend left]     ∙[r]     ∙[r]     ∙[r] [d]     ∙[r] [rr, dashed, bend left]     ∙[r]     ∙                 ∙[r] [rrr, bend left, dashed]    ∙[r]     ∙[r]     ∙[r] [d]     ∙[r]     ∙[r]     ∙

∙[d]      [dd, squiggly]                                       ∙[d]      [dd, squiggly]

∙                                          ∙

∙[r]     ∙[r] [d]     ∙[r] [d]     ∙[r] [rr, dashed, bend left]     ∙[r]     ∙[uuuuuuuuuuurrrrrr, squiggly, out=30, in=200]                    ∙[r]     ∙[r] [d]     ∙[r] [d]     ∙[r]     ∙[r]     ∙

∙    ∙[d]                                       ∙    ∙[d]

∙                                            ∙

Going from the quiver of $A_{9,(1,3,7)}^{(3,4,2)}$ to $D_{(1,0,3)}^{(1,2)}$. Each squiggly arrow indicates that we apply the CR-swap to the leftmost relation in the current quipu quiver.

If $A_{n,(n_0,n_1, \ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_{r})}$ is a linear Nakayama algebra with almost separate relations, and if $l_i=2$ for some $0\leq i\leq r$, then $A_{n,(n_0,n_1,\ldots, n_i,\ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_i, \ldots, l_{r})}$ is derived equivalent to $A_{n,(n_0,n_1,\ldots, n_{i-1}, n_{i+1},\ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_{i-1},l_{i+1}, \ldots, l_{r})}$.

Both these algebras satisfy the conditions in <ref>, and it's easy to see that the resulting quipus will be identical. This is because the only potential difference is the cord corresponding to the relation starting in $n_i$, but that cord has length $l_i-2=0$, meaning there is no cord there in both cases. So the two Nakayama algebras are derived equivalent to the same quipu algebra, and hence to each other.

<Ref> implies that relations of length $2$ have no effect on the derived equivalence class of a linear Nakayama algebra with almost separate relations.

Two quipu quivers without relations have derived equivalent path algebras if and only if they are different orientations of the same underlying quipu.

From theorem $1.2$ in <cit.> we know that if $Q$ is a quiver whose underlying graph is a tree, and $Q'$ is a different orientation of the same tree, then $kQ$ and $kQ'$ are derived equivalent. Because quipus are trees, one direction of the lemma follows.

The other direction follows from the proof of corollary 4 in <cit.>.

Let $P_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$ be a quipu, and let

$$n_i = k_0 + \sum\limits_{j=1}^{i} (m_{j-1} + k_{j} + 1) \text{ for } 1\leq i\leq r+1.$$

If $D$ is a quipu quiver, then there exists a derived equivalence

$$kD\simeq A_{n_{r+1},(k_0,n_1,n_2,\ldots, n_{r} )}^{(m_{0}+2, m_{1}+2, \ldots, m_{r}+2)}$$

if and only if $D$ is $P_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$ with some orientation.

We will show that the derived equivalence exists for one orientation of the quipu, namely $D_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$. The rest of the theorem follows from <ref>.
If we set

\begin{align*}
m_i &= l_i - 2 \\
k_0 &= n_0 \\
k_i &= n_i + 1 - n_{i-1} - l_{i-1}, \quad 1\leq i\leq r+1,
\end{align*}

then we know from <ref> that there is a derived equivalence

$$kD_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}\simeq A_{n,(n_0,n_1,\ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_{r})}$$

So we get that $n_0=k_0$ and $l_i=m_i+2$ for all $i$, and solving the last equation for $n_i$ gives $n_i=n_{i-1}+l_{i-1}+k_{i}-1$ for $1\leq i\leq r+1$. This implies that $n_1=k_0+m_{0}+k_1+1$, and an easy induction then shows that

$$n_i = k_0 + \sum\limits_{j=1}^{i} (m_{j-1} + k_{j} + 1) \text{ for } 1\leq i\leq r+1.$$

In particular, we have that

$$n_{r+1} = k_0 + \sum\limits_{j=1}^{r+1} (m_{j-1} + k_{j} + 1) = r+1+\sum\limits_{i=0}^{r+1}k_i + \sum\limits_{i=0}^r m_i,$$

which is equal to the number of vertices in $D_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$.
If we now plug these values into $A_{n,(n_0,n_1,n_2,\ldots, n_{r} )}^{(l_{0}, l_{1}, \ldots, l_{r})}$, we get that there is a derived equivalence

$$kD_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}\simeq A_{n_{r+1},(k_0,n_1,n_2,\ldots, n_{r} )}^{(m_{0}+2, m_{1}+2, \ldots, m_{r}+2)},$$

which is what we wanted to show. The above Nakayama algebra is derived equivalent to the path algebra of one orientation of the quipu $P_{(k_0,k_1,\ldots,k_r,k_{r+1})}^{(m_0,m_1,\ldots,m_r)}$, and hence, by <ref> it is derived equivalent to the path algebras of all possible orientations of the quipu.

With <ref> and <ref> we have explicit formulas to translate between quipu algebras and linear Nakayama algebras with almost separate relations, in a way that preserves derived equivalence.

If $A_{n,(n_0,\ldots,n_r)}^{(l_0,\ldots, l_r)}$ is a linear Nakayama algebra with almost separate relations, where $l_i\geq 3$ for all $i$, then the following Nakayama algebras are derived equivalent.

\begin{align*}
&A_{n,(n_0,n_1,\ldots,n_{r-1},n_r)}^{(l_0, l_1, \ldots, l_{r-1}, l_r)}  &&A_{n,(n-n_r-l_r+1,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(l_r, l_{r-1}, \ldots, l_{1}, l_0)}\\
&A_{n,(n_0,n_1,\ldots,n_{r-1},n_r)}^{(l_0, l_1, \ldots, l_{r-1}, n-n_r-l_r+3)} &&A_{n,(l_r-2,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(n-n_r-l_r+3, l_{r-1}, \ldots, l_{1}, l_0)}    \\
&A_{n,(l_0-2,n_1,\ldots,n_{r-1},n_r)}^{(n_0+2, l_1, \ldots, l_{r-1}, l_r)}  && A_{n,(n-n_r-l_r+1,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(l_r, l_{r-1}, \ldots, l_{1}, n_0+l_0)}\\
&A_{n,(l_0-2,n_1,\ldots,n_{r-1},n_r)}^{(n_0+2, l_1, \ldots, l_{r-1}, n-n_r-l_r+3)} &&A_{n,(l_r-2,n-n_{r-1}-l_{r-1}+1,\ldots,n-n_0-l_0+1)}^{(n-n_r-l_r+3, l_{r-1}, \ldots, l_{1}, n_0+l_0)}.
\end{align*}

Also, adding a set of almost separate relations of length $2$ to available positions in any of these algebras will not change the derived equivalence class.

We can translate $A_{n,(n_0,\ldots,n_r)}^{(l_0,\ldots, l_r)}$ into a quipu using <ref>. Then, the other Nakayama algebras are obtained by applying  <ref> to the other orientations of this quipu, given by <ref>. The last part follows from <ref>.

Any quipu quiver with almost separate relations can be viewed as an intermediate step between a Nakayama algebra with almost separate relations and a quipu quiver with no relations. Hence, this derived classification also covers all quipu quivers with almost separate relations, although the combinatorics for describing these become much more involved.

§ COMPLETE CLASSIFICATION FOR $N\LEQ 8$

We will now explicitly write out the complete classification up to derived equivalence of all linear Nakayama algebras of length $n\leq 8$ with almost separate relations.
From <ref> we know that we can limit our considerations to algebras $A_{n,(n_0,\ldots,n_r)}^{(l_0,\ldots, l_r)}$ where $l_i\geq 3$ for all $i$.

We now make a list of every quipu of order $\leq 8$. For each quipu we choose an orientation to get a quipu quiver, and we apply <ref> to get the corresponding linear Nakayama algebra. Finally we apply <ref> to get the other Nakayama algebras in the same class.

We stopped at $n=8$ for space concerns, but we could easily continue the list for as large $n$ as we need. The following observation is an easy consequence of <ref>

In each derived equivalence class there are at most $8$ different such algebras whose relations all have length $\geq 3$.

These derived equivalence classes will also contain the same algebras decorated with all possible sets of relations of length $2$.

Note that most of these classes will contain other linear Nakayama algebras, with bigger overlap between their relations, but in this paper we focus only on Nakayama algebras with almost separate relations.

In the following table, the quipus that are equal to (extended) Dynkin diagrams are labelled as such, for the rest we use our notation for quipus.

2|c|Quipu     Corresponding Nakayama algebras

gray!20 $\mathbb{A}_1$
[xshift=0 cm,thick,fill=black] (0 cm,0) circle (.3cm);

in 0,...,1
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
[xshift=0.15 cm,thick] (0.15 cm,0) – +(1.4 cm,0);

gray!20 $\mathbb{A}_3$
in 0,...,2
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
in 0.15,...,1.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);

in 0,...,3
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
in 0.15,...,2.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);

in 0,...,2
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,1
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);

gray!20 $\mathbb{A}_5$
in 0,...,4
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
in 0.15,...,3.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);

gray!20 $\mathbb{D}_5$
in 0,...,3
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,2
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);

$A_{5,(1)}^{(3)}$, $A_{5,(1)}^{(4)}$, $A_{5,(2)}^{(3)}$

in 0,...,5
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
in 0.15,...,4.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);

in 0,...,4
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);

$A_{6,(1)}^{(3)}$, $A_{6,(1)}^{(5)}$, $A_{6,(3)}^{(3)}$

in 0,...,4
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);

$A_{6,(1)}^{(4)}$, $A_{6,(2)}^{(3)}$, $A_{6,(2)}^{(4)}$

in 0,...,3
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,2
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);

gray!20 $\mathbb{A}_7$
in 0,...,6
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
in 0.15,...,5.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);

gray!20 $\mathbb{D}_7$
in 0,...,5
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);

$A_{7,(1)}^{(3)}$, $A_{7,(1)}^{(6)}$, $A_{7,(4)}^{(3)}$

gray!20 $\mathbb{E}_7$
in 0,...,5
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);

$A_{7,(1)}^{(4)}$, $A_{7,(1)}^{(5)}$, $A_{7,(2)}^{(3)}$, $A_{7,(2)}^{(5)}$, $A_{7,(3)}^{(3)}$, $A_{7,(3)}^{(4)}$

gray!20 $\tilde{\mathbb{D}}_6$
in 0,...,4
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (6 cm,1 cm) circle (3 mm);
[thick] (6 cm, 3mm) – +(0, 0.7 cm);

gray!20 $\tilde{\mathbb{E}}_6$
in 0,...,4
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick, fill=black] (4 cm,2 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 1.7 cm);

gray!20 $P_{(1,0,2)}^{(1,1)}$
in 0,...,4
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);

$A_{7,(1,3)}^{(3,3)}$, $A_{7,(1,3)}^{(3,4)}$, $A_{7,(1,4)}^{(4,3)}$, $A_{7,(2,4)}^{(3,3)}$

in 0,...,7
[xshift=cm,thick,fill=black] (cm,0) circle (.3cm);
in 0.15,...,6.15
[xshift=cm,thick] (cm,0) – +(1.4 cm,0);

in 0,...,6
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,5
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);

$A_{8,(1)}^{(3)}$, $A_{8,(1)}^{(7)}$, $A_{8,(5)}^{(3)}$

in 0,...,6
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,5
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);

$A_{8,(1)}^{(4)}$, $A_{8,(1)}^{(6)}$, $A_{8,(2)}^{(3)}$, $A_{8,(2)}^{(6)}$, $A_{8,(4)}^{(3)}$, $A_{8,(4)}^{(4)}$

in 0,...,5
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (8 cm,1 cm) circle (3 mm);
[thick] (8 cm, 3mm) – +(0, 0.7 cm);

in 0,...,6
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,5
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (6 cm,1 cm) circle (3 mm);
[thick] (6 cm, 3mm) – +(0, 0.7 cm);

$A_{8,(1)}^{(5)}$, $A_{8,(3)}^{(3)}$, $A_{8,(3)}^{(5)}$

in 0,...,5
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (6 cm,1 cm) circle (3 mm);
[thick] (6 cm, 3mm) – +(0, 0.7 cm);

$A_{8,(1,4)}^{(3,3)}$, $A_{8,(1,4)}^{(3,4)}$, $A_{8,(1,5)}^{(4,3)}$, $A_{8,(2,5)}^{(3,3)}$

in 0,...,5
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);

$A_{8,(1,3)}^{(3,3)}$, $A_{8,(1,3)}^{(3,5)}$, $A_{8,(1,5)}^{(5,3)}$, $A_{8,(3,5)}^{(3,3)}$

in 0,...,4
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (6 cm,1 cm) circle (3 mm);
[thick] (6 cm, 3mm) – +(0, 0.7 cm);

in 0,...,5
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (6 cm,1 cm) circle (3 mm);
[thick] (6 cm, 3mm) – +(0, 0.7 cm);

$A_{8,(1,4)}^{(4,3)}$, $A_{8,(1,4)}^{(4,4)}$, $A_{8,(2,4)}^{(3,3)}$, $A_{8,(2,4)}^{(3,4)}$

in 0,...,4
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,3
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (2 cm,1 cm) circle (3 mm);
[thick] (2 cm, 3mm) – +(0, 0.7 cm);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick, fill=black] (4 cm,2 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 1.7 cm);

$A_{8,(1,3)}^{(3,4)}$, $A_{8,(2,5)}^{(4,3)}$

in 0,...,5
[thick,xshift=cm, fill=black] (cm,0) circle (3 mm);
in 0,...,4
[thick,xshift=cm] (cm,0) ++(.3 cm, 0) – +(14 mm,0);
[thick, fill=black] (4 cm,1 cm) circle (3 mm);
[thick, fill=black] (4 cm,2 cm) circle (3 mm);
[thick] (4 cm, 3mm) – +(0, 1.7 cm);

$A_{8,(2)}^{(4)}$, $A_{8,(2)}^{(5)}$, $A_{8,(3)}^{(4)}$

All quipus of order $\leq 8$, with corresponding derived equivalent Nakayama algebras (not including relations of length $2$).

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press (CUP)

journaltitleNagoya Mathematical Journal
titleAlgebras of acyclic cluster type: Tree type and type $\tilde{A}$

journaltitleAnnales scientifiques de l'École Normale Supérieure
titleFrom triangulated categories to cluster algebras II
volumeSer. 4, 39

titleA combinatorial procedure for tilting mutation

journaltitleApplied Mathematics and Computation
titleOpen quipus with the same Wiener index as their quadratic line graph

journaltitleLinear Algebra and its Applications
titleGraphs of order n and diameter $2(n-1)/3$ minimizing the spectral radius

journaltitleGraphs and Combinatorics
titleOn Graphs Whose Spectral Radius is Bounded by $\frac{3}{2}\sqrt{2}$


# Modern Evolution Strategies for Creativity:
Fitting Concrete Images and Abstract Concepts

Yingtao Tian
Google Brain
<EMAIL_ADDRESS>
&David Ha
Google Brain
<EMAIL_ADDRESS>

###### Abstract

Evolutionary algorithms have been used in the digital art scene since the
1970s. A popular application of genetic algorithms is to optimize the
procedural placement of vector graphic primitives to resemble a given
painting. In recent years, deep learning-based approaches have also been
proposed to generate procedural drawings, which can be optimized using
gradient descent. In this work, we revisit the use of evolutionary algorithms
for computational creativity. We find that modern evolution strategies (ES)
algorithms, when tasked with the placement of shapes, offer large improvements
in both quality and efficiency compared to traditional genetic algorithms, and
even comparable to gradient-based methods. We demonstrate that ES is also well
suited at optimizing the placement of shapes to fit the CLIP model, and can
produce diverse, distinct geometric abstractions that are aligned with human
interpretation of language. Videos and demo: https://es-clip.github.io/.

Figure 1: Our proposed painting synthesization places transparent triangles
using evolution strategy (ES). Each concept represented as a text prompt is
accompanied by its corresponding synthesized image. Here, the fitness is
defined as the cosine distance between rendered canvas and text, both embedded
by CLIP [41], and we optimize the position and the color of triangles using
ES.  Figure 2: The architecture of our method. Our proposed method synthesizes
painting by placing transparent triangles using Evolution Strategy (ES). After
rendering the parameters on a canvas, we calculate the fitness, which measures
how well the canvas fits either a target image, or a concept in the form of a
text prompt. The fitness, in turn, guides the evolution process to find better
parameters.

##  1 Introduction

Staring from early 20th-century in the wider context of modernism [29], a
series of avant-garde art abandoned the depiction of objects from tradition
rules of perspective and instead picking revolutionary, abstract point of
views. The Cubism art movement [42], popularized by influential artists
including Pablo Picasso, proposed that objects are analyzed by the artist,
broken up, and reassembled in an abstract form consisting of geometric
representations. This naturally develops into the geometric abstraction [8],
where pioneer abstractionists like Wassily Kandinsky and Piet Mondrian
represented the world using composed primitives that are either purely
geometric or elementary. The impact is far-reaching: The use of simple
geometry can be seen as one of styles found in abstract expressionism [40]
where artists expressed their subconscious or impulsive feelings. It also
helped shape the minimalist art [37] and minimalist architecture [43]
movements, in which everything is stripped down to its essential quality to
achieve simplicity [3].

The idea of minimalist art has also been explored in computer art with a root
in mathematical art [35]. Schmidhuber [46] proposed an art form in the 1990s,
called low-complexity art, as the minimal art in the computer age that
attempts to depict the essence of an object by making use of ideas from
algorithmic complexity [28]. Similarly, algorithmic art [53] proposed to
generate arts using the algorithm designed by the artist. In a broad sense,
algorithmic art could be said to include genetic algorithm where the artist
determines the rules governing how images evolves iteratively, which are a
popular method applied to approximate images using simple shapes, often
producing abstract art style. As one example, a basic genetic algorithm using
evolution has been proposed [25, 1] to represent a target image using semi-
transparent, overlapping triangles. This approach has gained popularity over
the years with the creative coding community, resulting in many sophisticated
extensions [11, 6, 2, 39, 48]. These methods are iterative, enabling the
creation process [50] to be captured.

Figure 3: Our method leverages modern ES (PGPE with ClipUp), with $50$
triangles and runs for $10,000$ steps to fit the target image “Mona Lisa”
(left). Here it is followed by the evolved results (middle) and the evolution
process (right).

With the recent resurgence of interest in evolution strategies (ES) in the
machine learning community [44, 17], in this work, we revisit the use of ES
for creativity applications as an alternative to gradient-based methods. For
approximating an image with shapes, we find that modern ES algorithms offer
large improvements in both quality and efficiency when compared to traditional
genetic algorithms, and as we will also demonstrate, even comparable to state-
of-the-art differentiable rendering methods [30]. We show that ES is also well
suited at optimizing the placement of shapes to fit the CLIP [41] model, and
can produce diverse, distinct geometric abstractions that are aligned with
human interpretation of language. Such an alignment is due to the use of CLIP
model that are trained on aligned real-world text-image dataset.
Interestingly, the results produced by our method resemble abstract
expressionism [40] and minimalist art [37, 43]. We provide a reference code
implementation of our approach online so that it can be a useful tool in the
computational artist’s toolbox.

##  2 Background

Related Work In recent years, deep learning has also been applied to methods
that can generate procedural drawings, which can be optimized with gradient
descent. A growing list of works [57, 38, 23, 33] also tackle the problem of
approximating pixel images with simulated paint medium, and differentiable
rendering [26, 30] methods enable computer graphics to be optimized directly
using gradient descent. To learn abstract representations, probabilistic
generative models [16, 18, 14, 36, 34] are used to sample procedurally
drawings directly from a latent space, without any given input images, similar
to their pixel image counterparts. To interface with natural language, methods
have been proposed to procedurally generate drawings of image categories [56],
and word embeddings [21, 22], enabling an algorithm to draw what’s written.
This combination of NLP and pixel image generation is explored at larger scale
in CLIP [41], and its procedural sketch counterpart CLIPDraw [12].

Perhaps among the related works, the closest to our approach is [10], which,
similar to our work, uses a CLIP-like dual-encoder model pre-trained on the
ALIGN [24] dataset to judge the similarity between generated art and text
prompt, and leverages evolutionary algorithms to optimize a non-differentiable
rendering process. However, there are several key differences: [10]
parameterizes the rendering process with a hierarchical neural Lindenmayer
system [32] powered by multiple-layer LSTM [20] and, as a result, it models
well patterns with complex spatial relation, whereas our work favors a
drastically simpler parameterization which just puts triangles individually on
canvas to facilitate a different, minimalist art style that is complementary
to theirs [9]. Moreover, while [10] uses a simple binary-tournament genetic
algorithm [19], we opt for a modern state-of-the-art evolution strategy, PGPE
[47] with ClipUp [52], performing well enough to produce interesting results
within a few thousand steps.

Evolution Strategies (ES) [4, 5] has been applied to optimization problems for
a long period of time. A straightforward implementation of ES can be
iteratively perturbing parameters in a pool and keeping those that are most
fitting, which is simple yet inefficient. As a consequence, applying such a
straightforward algorithm can lead to sub-optimal performance for art
creativity [1]. To overcome this generic issue in ES, recent advances have
been proposed to improve the performance of ES algorithms. One such
improvement is Policy Gradients with Parameter-Based Exploration (PGPE) [47],
which estimates gradients in a black-box fashion so the computation of fitness
does not have to be differentiable _per se_. Since PGPE runs linear to the
number of parameters for each iteration, it is an efficient and the go-to
algorithm in many scenarios. With the estimated gradients, gradient-based
optimizers such as Adam [27] can be used for optimization, while there are
also work such as ClipUp [52] offering a simpler and more efficient optimizer
specifically tailored for PGPE. Another representative ES algorithm is
Covariance matrix adaptation evolution strategy (CMA-ES), which in practice is
considered more performant than PGPE. However, it runs in the quadratic time
w.r.t. the number of parameters for each iteration, which limits its use in
many problems with larger numbers of parameters where PGPE is still feasible.

Language-derived Image Generation has been seeing very recent trends in
creativity setting, where there are several directions to leverage CLIP [41],
a pre-trained model with two encoders, one for image and one for text, that
can convert images and text into the same, comparable low-dimensional
embedding space. As the image encoder is a differentiable neural network, it
can provide a gradient to the output of a differentiable generative model. The
gradient can be further back-propagated through the said model till its
parameters. For example, one direction of works uses CLIP’s gradient to guide
a GAN’s generator, such as guiding BigGAN [54], guiding VQGAN [45], guiding
Siren [55], or a GAN with genetic algorithm-generated latent space [13].
Another direction of work applies CLIP to differentiable renderers. CLIPDraw
[12] proposes to generate the images with diffvg [31], a differentiable SVG
renderer. Although all these methods use the same pre-trained CLIP model for
guidance, they show a drastically different artistic property, for which we
hypothesize that the art style is determined by the intrinsic properties of
“painter”, i.e., the GAN generator or renderer.

##  3 Modern Evolution Strategies for Creativity

The architecture of our proposed pipeline is shown in Figure 2. Our proposed
method synthesizes painting by placing transparent triangles using evolution
strategy (ES). Overall, we can represent a configuration of triangles in a
parameter space which composes of positions and colors of triangles, render
such configuration onto a canvas, and calculate its fitness based on how well
the rendered canvas fits a target image or an concept in the form of a text
prompt. The ES algorithm keeps a pool of candidate configurations and uses
mutations to evolves better ones measured by the said fitness. To have better
creative results, we use a modern ES algorithm, PGPE [47] optimized by ClipUp
[52] optimizer. Engineering-wise we use the pgpelib [51] implementation of
PGPE and ClipUp.

As we choose to follow the spirit of minimalist art, we use transparent
triangles as the parameter space. Concretely, a configuration of $N$ triangles
is parameterized by a collection of
$(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3},r,g,b,a)$ for each of the triangles,
which are vertex coordinates and the RGBA (Red, Green, Blue, and Alpha a.k.a.
transparency channel) color, totally making $10N$ parameters. In the ES, we
update all parameters and use a fixed hyper-parameter, the number of triangles
$N$. Note that $N$ is better understood as the upper bound of number of
triangles to use: although $N$ is fixed, the algorithm is still capable of
effectively using “fewer” triangles by making unwanted ones transparent.

Target Image | 10 Triangles | 25 Triangles | 50 Triangles | 200 Triangles
---|---|---|---|---
|  |  |  |
“Darwin” |  Fitness = $96.82\%$ |  Fitness = $99.25\%$ |  Fitness = $99.51\%$ |  Fitness = $99.75\%$
|  |  |  |
“Mona Lisa” |  Fitness = $98.02\%$ |  Fitness = $99.30\%$ |  Fitness = $99.62\%$ |  Fitness = $99.80\%$
|  |  |  |
“Anime Face” |  Fitness = $94.97\%$ |  Fitness = $98.17\%$ |  Fitness = $98.80\%$ |  Fitness = $99.07\%$
|  |  |  |
“Velociraptor” |  Fitness = $95.76\%$ |  Fitness = $98.03\%$ |  Fitness = $99.00\%$ |  Fitness = $99.25\%$
|  |  |  |
“Landscape” |  Fitness = $97.07\%$ |  Fitness = $98.83\%$ |  Fitness = $99.08\%$ |  Fitness = $99.25\%$
|  |  |  |
“Impressionism” |  Fitness = $98.82\%$ |  Fitness = $99.23\%$ |  Fitness = $99.34\%$ |  Fitness = $99.48\%$
Figure 4: Qualitative and quantitative results from fitting several targets
with $10$, $25$, $50$, and $200$ triangles, each running for 10,000 steps.
Images credits: Darwin, Mona Lisa, Velociraptor are from [1]. Anime Face is
generated by Waifu Labs [49]. Landscape is from Wikipedia [7]. Impressionism
is A May Morning in Moret by Alfred Sisley, collected by [15]. Fitness is
$100\%-$ L2 Loss.

As the ES is orthogonal to the concrete fitness evaluation, we are left with
many free choices regarding what counts as fitting. Particularly, we consider
two kinds of fitness, namely, fitting a concrete image (the lower branch in
Figure 2) and fitting a concept (the upper branch in Figure 2). Fitting a
concrete image is straightforward, where we can simply use the pixel-wise L2
loss between the rendered canvas and the target image as the fitness. Fitting
a concept requires more elaboration. We represent the concept as a text prompt
and embed the text prompt using the text encoder in CLIP [41] which we discuss
in detail in Section 2. Then we embed the rendered canvas using the image
encoder also available in CLIP. Since the CLIP models are trained so that both
embedded images and texts are comparable under Cosine distance for similarity,
we use such distance as the fitness. We note that since the ES algorithm
provides black-box optimization, the renderer, like fitness computation, does
not necessarily need to be differentiable.

We find in practice a few decisions should be made so the whole pipeline can
work reasonably well. First, we augment the rendered canvas by random cropping
in calculating the fitness and average the fitness on each of the augmented
canvas, following the practice of [45, 12]. This would prevent the rendered
canvas from overfitting and increase the stability in the optimization.
Second, we render the triangles on top of a background with a uniform
distribution noise. Mathematically, this equals to modeling the uncertainty of
parts in the canvas not covered by triangles with a max-entropy assumption,
and using Monte Carlo method for approximation. Finally, we limit the maximal
alpha value for each triangle to $0.1$, which prevents front triangles from
(overly) shadowing the back ones.

##  4 Fitting Concrete Target Image

In this section, we show the performance of our proposed work on fitting a
concrete target image. In doing so, the model takes the lower branch in Figure
2. We fit the famous painting “Mona Lisa” with $50$ triangles by running
evolution for $10,000$ steps in Figure 3. Our result is a distinctive art
style represented by well-placed triangles that care both fine-grained
textures and large backgrounds. The evolution process also displays the
coarse-to-fine adjustments of the shapes’ positions and colors.

Target Image |  Ours (10,000 iteration) |  Basic (10,000 iteration) |  Basic (560,000 iteration)
---|---|---|---
|  |  |
“Mona Lisa” | Fitness = $\mathbf{99.62\%}$ | Fitness = $97.23\%$ | Fitness = $99.58\%$

---

Figure 5: Compare choices of evolution algorithm: Ours (PGPE with ClipUp) vs.
basic evolution algorithm (mutation with simulated annealing) [1]. Both
settings fit 50 triangles and all choices except for the EA are the same. We
show the results of ours and the basic algorithm at the end of $10,000$
iterations, and the result of the basic algorithm after running $56$ times
more iterations. The details of evolution process until $10,000$ iterations
are shown in the bottom half, where the upper group is for ours and the lower
group is for the basic algorithm. Fitness is $100\%-$ L2 Loss.

Number of triangles and parameters. Our proposed pipeline is able to fit any
target images and could handle a wide range of number of parameters, since
PGPE runs efficiently, i.e., linear to the number of parameters. This is
demonstrated by applying our method to fit several target images with $10$,
$25$, $50$, $200$ triangles, which corresponds to $100$, $250$, $500$ and
$2000$ parameters respectively. As shown in Figure 4, our proposed pipeline
works well for a wide range of target images, and the ES algorithm is capable
of using the number of triangles as a “computational budget” where extra
triangles could always be utilized for gaining in fitness. This allows a human
artist to use the number of triangles in order to find the right balance
between abstractness and details in the produced art.

Choice of ES Algorithm. We compare two choices of evolution algorithm: ours,
which uses the recent PGPE with ClipUp, and a basic, traditional one, which
consists of mutation and simulated annealing adopted earlier [25, 1]. As shown
in Figure 5, our choice of more recent algorithms leads to better results than
the basic one under the same parameter budget. Subjectively, our final results
are more visually closer to the target image with a smoother evolution
process, and quantitatively, our method leads to much better fitness
($99.62\%$ vs. $97.23\%$). Furthermore, even allowing $56$ times more
iterations for the basic algorithm does not lead to results better than ours.

Comparison with Gradient-based Optimization. While our proposed approach is
ES-based, it is interesting to investigate how it compares to gradient-based
optimization since the latter is commonly adopted recently (See Section 2).
Therefore we conduct a gradient-based setup by implementing rendering of
composed triangles using nvdiffrast [30], a point-sampling-based
differentiable renderer. We use the same processing as mentioned in Section 3.
As shown in Figure 6, our proposed ES-based method can achieve similar yet
slightly higher fitness than results compared with the gradient-optimized
differentiable renderer. Furthermore and perhaps more interestingly, two
methods produce artworks with different styles: our proposed method can
adaptive allocating large triangles for background and small ones for detailed
textures, whereas the differentiable renderer tends to introduce textures
unseen in the target image (especially in the background). We argue that due
to the difference in the optimization mechanism, our method focuses more on
the placement of triangles while the differentiable renderer pays attention to
the compositing of transparent colors.

Target Image |  Evolution Strategy (Non-gradient) |  Differentiable Renderer (Gradient-based)
---|---|---
|  |
“Mona Lisa” | Fitness = $\mathbf{99.80\%}$ | Fitness = $99.68\%$

---

Figure 6: Evolution strategies (non-gradient method) vs. differentiable renderer (gradient based method) fitting target image with $200$ triangles. The upper half shows the final results and the bottom half shows the details of optimization. Fitness is $100\%-$ L2 Loss. Prompt | Evolved Results | Evolution Process
---|---|---
“Self” |  |
“Human” |  |
“Walt Disney World” |  |
“The corporate headquarters complex of Google located at 1600 Amphitheatre Parkway in Mountain View, California.” |  |
Figure 7: ES and CLIP fit the concept represented in text prompt, using 50
triangles and running evolution for $2,000$ steps. Each row shows the text
prompt followed by the finally evolved results and the evolution process. We
show exemplary prompts for 3 kinds of text, ranging from a single word (“Self”
and “Human”), a phrase (“Walt Disney Land”), and a long sentence (“The
corporate headquarters complex of Google located at 1600 Amphitheatre Parkway
in Mountain View, California.”).

##  5 Fitting Abstract Concept with CLIP

In this section, we show the performance of our method configured to fit an
abstract concept represented by language. In doing so, the model takes the
upper branch in Figure 2. Formally, the parameter space remains the same, but
the fitness is calculated as the cosine distance between the text prompt and
the rendered canvas, both encoded by CLIP. Since the model is given more
freedom to decide what to paint, this problem is arguably a much harder yet
more interesting problem than fitting concrete images in the previous section.

In Figure 7, we show the evolution result and process of fitting abstract
concept represented as text prompt, using 50 triangles and running evolution
for $2,000$ steps. We found that unlike fitting a concrete images, $2,000$
steps is enough for fitting a concept to converge. Our method could handle
text prompts ranging from a single word to a phrase, and finally, to a long
sentence, even though the task itself is arguably more challenging than the
previous one. The results show a creative art concept that is abstract, not
resembling a particular image, yet correlated with humans’ interpretation of
the text. The evolution process also demonstrates iterative adjustment, such
as the human shape in the first two examples, the shape of castles in Disney
World, as well as in the final example, the cooperate-themed headquarters.
Also, compared to fitting concrete images in the previous section, our method
cares more about the placement of triangles.

Prompt | 10 Triangles | 25 Triangles | 50 Triangles | 200 Triangles
---|---|---|---|---
“Self” |  |  |  |
“Human” |  |  |  |
“Walt Disney World” |  |  |  |
“A picture of Tokyo” |  |  |  |
“The corporate headquarters complex of Google located at 1600 Amphitheatre Parkway in Mountain View, California.” |  |  |  |
“The United States of America commonly known as the United States or America is a country primarily located in North America.” |  |  |  |
Figure 8: Qualitative results from ES and CLIP fitting several text prompt
with 10,25,50, and 200 triangles, each running for 2,000 steps. We show
exemplary prompts for 3 kinds of text, ranging from a single word (“Self” and
“Human”), to phrases (“A picture of Tokyo”), to long sentences.

Number of triangles and parameters. Like fitting a concrete image, we can also
fit an abstract concept with a wide range of number of parameters since the
PGPE algorithm and the way we represent canvas remains the same. In Figure 8
we apply our method to fit several concept (text prompt) with $10$, $25$,
$50$, $200$ triangles, which corresponds to $100$, $250$, $500$ and $2000$
parameters respectively. It is shown that our proposed pipeline is capable of
leveraging the number of triangles as a “budget for fitting” to balance
between the details and the level of abstraction. Like in the previous task,
this allows a human artist to balance the abstractness in the produced art.

We observe that while the model could comfortably handle at least up to $50$
triangles, more triangles ($200$) sometimes poses challenges: For example,
with $200$ triangles, “corporate headquarters …” gets a better result while “a
picture of Tokyo” leads to a poor one. This may be due to the difficulties
composing overly shadowed triangles, and we leave it for future study.

Prompt | 4 Individual Runs
---|---
“Self” |  |  |  |
“Human” |  |  |  |
“Walt Disney World” |  |  |  |
“A picture of Tokyo” |  |  |  |
“The corporate headquarters complex of Google located at 1600 Amphitheatre Parkway in Mountain View, California.” |  |  |  |
“The United States of America commonly known as the United States or America is a country primarily located in North America.” |  |  |  |
Figure 9: Qualitative results from ES and CLIP fitting several text prompt
with 50 triangles, each running for 2,000 steps. The text prompts follow that
of Figure 8. Results from four runs are shown.

Prompt |  Evolution Strategy (Non-gradient) |  Differentiable Renderer (Gradient-based)
---|---|---
“Self” |  |
“Walt Disney World” |  |

---

Figure 10: Evolution Strategies (non-gradient method) v.s. Differentiable
renderer (gradient-based method) with fitting text with CLIP. Both settings
are fitting 200 triangles to the target prompts. The details of evolution
process is shown in the bottom half, where the upper group is for evolution
strategy and the lower group is for differtiable renderer.

Multiple Runs. Since the target is an abstract concept rather than a concrete
image, our method is given much freedom in arranging the configuration of
triangles, which means random initialization and noise in the optimization can
lead to drastically different solutions. In Figure 9, we show 4 separate runs
of our method on several text prompts, each using $50$ triangles with $2,000$
iterations, which is the same as previous examples. As shown, our method
creates distinctive abstractions aligned with human interpretation of language
while being capable of producing diverse results from the same text prompt.
This, again, is a desired property for computer-assisted art creation, where
human creators can be put “in the loop”, not only poking around the text
prompt but also picking the from multiple candidates produced by our method.

Comparison with Gradient-based Optimization. With CLIP in mind, we are also
interested in how our ES-based approach compares to the gradient-based
optimization, especially since many existing works [54, 45, 55, 12] have
proposed to leverage CLIP to guide the generations using gradients. Arguably,
this is a more challenging task due to the dynamic presented by two
drastically different gradient dynamics by renderer and CLIP. Usually, to make
such kind of combination work ideally, more studies are required, which
warrant a manuscript itself like [45, 12]. Nonetheless, we have made a
reasonably working version for comparison. Like fitting a target image, we
implement the rendering process of composing triangles using nvdiffrast [30].
In the forward pass, we render the canvas from parameters, feed the canvas to
CLIP image encoder, and use Cosine distance between encoded image and encoded
text prompt as a loss. Then we back-propagate all the way til the parameters
of triangles to allow gradient-based optimization. We use the same processing
as mentioned in Section 3.

As shown in Figure 10, while both our ES method and the differentiable method
produce images that are aligned with human interpretation of the text prompt,
ours produces more clear abstraction and clear boundaries between shapes and
objects. More interestingly, since ours represents an art style closely
resembling abstract expressionism art, the difference between ours and the
differentiable rendered is similar to that between post-impressionism and
impressionism, where bolder geometric forms and colors are used. Like the
counterpart comparison in fitting a concrete image, we argue that such results
are intrinsically rooted in the optimization mechanism, and our proposed
method leads to a unique art style through our design choices.

##  6 Discussion and Conclusion

In this work, we revisit evolutionary algorithms for computational creativity
by proposing to combine modern evolution strategies (ES) algorithms with the
drawing primitives of triangles inspired by the minimalism art style. Our
proposed method offers considerable improvements in both quality and
efficiency compared to traditional genetic algorithms and is comparable to
gradient-based methods. Furthermore, we demonstrate that the ES algorithm
could produce diverse, distinct geometric abstractions aligned with human
interpretation of language and images. Our finds suggests that ES method
produce very different and sometimes better results compared to gradient based
methods, arguably due to the intrinsical behavior of the optimization
mechanism. However it remains an open problem to understand how in general
setting ES method compares with gradient methods. We expect future works
investigate further into broader spectrum of art forms beyond the minimalism
explored here.

Our dealing with evolutionary algorithms provides an insight into a different
paradigm that can be applied to computational creativity. Widely adopted
gradient-based methods are fine-tuned for specific domains, i.e., diff
rendering for edges, parameterized shapes, or data-driven techniques for
rendering better textures. Each of the applications requires tunes and tweaks
that are domain-specific and are hard to transfer. In contrast, ES is agnostic
to the domain, i.e., how the renderer works. We envision that ES-inspired
approaches could potentially unify various domains with significantly less
effort for adaption in the future.

##  7 Acknowledgement

We thank Toru Lin, Jerry Li, Yujin Tang, Yanghua Jin, Jesse Engel, Yifu Zhao
for their comments and suggestions. We specially thank Yanghua Jin for his
kind help with nvdiffrast. The experiments in this work performed on multi-GPU
Linux virtual machines provided by Google Cloud Platform.

## References

  * Alteredqualia [2008] Alteredqualia.  Evolution of mona lisa in javascript and canvas, 2008.  URL https://alteredqualia.com/visualization/evolve/.
  * Berg et al. [2019] Joachim Berg, Nils Gustav Andreas Berggren, Sivert Allergodt Borgeteien, Christian Ruben Alexander Jahren, Arqam Sajid, and Stefano Nichele.  Evolved art with transparent, overlapping, and geometric shapes.  In _Symposium of the Norwegian AI Society_ , pages 3–15. Springer, 2019.  URL https://arxiv.org/abs/1904.06110.
  * Bertoni [2002] Franco Bertoni.  _Minimalist architecture_.  Birkhauser, 2002.  URL https://www.worldcat.org/title/minimalist-architecture/oclc/469368801?loc=.
  * Beyer [2001] Hans-Georg Beyer.  _The theory of evolution strategies_.  Springer Science & Business Media, 2001.  URL https://www.springer.com/gp/book/9783540672975.
  * Beyer and Schwefel [2002] Hans-Georg Beyer and Hans-Paul Schwefel.  Evolution strategies–a comprehensive introduction.  _Natural computing_ , 1(1):3–52, 2002.
  * Cason [2016] Kenny Cason.  Genetic draw, 2016.  URL https://github.com/kennycason/genetic_draw.
  * Commons [2020] Wikimedia Commons.  File:040 okertalsperre.jpg — wikimedia commons the free media repository, 2020.  URL https://commons.wikimedia.org/w/index.php?title=File:040_Okertalsperre.jpg&oldid=496749636.  [Online; accessed 25-August-2021].
  * Dabrowski [2004] Magdalena Dabrowski.  Geometric abstraction, 2004.  URL https://www.metmuseum.org/toah/hd/geab/hd_geab.htm.
  * Fernando [2021] Chrisantha Fernando.  Royal academy summer exhibition 2021 submission, 2021.  URL https://www.chrisantha.co.uk/post/royal-academy-summer-exhibition-2021-submission.
  * Fernando et al. [2021] Chrisantha Fernando, SM Eslami, Jean-Baptiste Alayrac, Piotr Mirowski, Dylan Banarse, and Simon Osindero.  Generative art using neural visual grammars and dual encoders.  _arXiv preprint arXiv:2105.00162_ , 2021.  URL https://arxiv.org/abs/2105.00162.
  * Fogleman [2016] Michael Fogleman.  Primitive pictures, 2016.  URL https://github.com/fogleman/primitive.
  * Frans et al. [2021] Kevin Frans, LB Soros, and Olaf Witkowski.  Clipdraw: Exploring text-to-drawing synthesis through language-image encoders.  _arXiv preprint arXiv:2106.14843_ , 2021.  URL https://arxiv.org/abs/2106.14843.
  * Galatolo et al. [2021] Federico A Galatolo, Mario GCA Cimino, and Gigliola Vaglini.  Generating images from caption and vice versa via clip-guided generative latent space search.  _arXiv preprint arXiv:2102.01645_ , 2021.  URL https://arxiv.org/abs/2102.01645.
  * Ganin et al. [2018] Yaroslav Ganin, Tejas Kulkarni, Igor Babuschkin, SM Ali Eslami, and Oriol Vinyals.  Synthesizing programs for images using reinforced adversarial learning.  In _International Conference on Machine Learning_ , pages 1666–1675. PMLR, 2018.  URL http://proceedings.mlr.press/v80/ganin18a.html.
  * Gonsalves [2021] Robert A. Gonsalves.  Ganscapes: Using ai to create new impressionist paintings, 2021.  URL https://towardsdatascience.com/ganscapes-using-ai-to-create-new-impressionist-paintings-d6af1cf94c56.
  * Gregor et al. [2015] Karol Gregor, Ivo Danihelka, Alex Graves, Danilo Rezende, and Daan Wierstra.  Draw: A recurrent neural network for image generation.  In _International Conference on Machine Learning_ , pages 1462–1471. PMLR, 2015.  URL http://proceedings.mlr.press/v37/gregor15.html.
  * Ha [2017] David Ha.  Evolving stable strategies.  _blog.otoro.net_ , 2017.  URL http://blog.otoro.net/2017/11/12/evolving-stable-strategies/.
  * Ha and Eck [2017] David Ha and Douglas Eck.  A neural representation of sketch drawings.  _arXiv preprint arXiv:1704.03477_ , 2017.  URL https://arxiv.org/abs/1704.03477.
  * Harvey [2009] Inman Harvey.  The microbial genetic algorithm.  In _European conference on artificial life_ , pages 126–133. Springer, 2009.  URL http://users.sussex.ac.uk/~inmanh/MicrobialGA_ECAL2009.pdf.
  * Hochreiter and Schmidhuber [1997] Sepp Hochreiter and Jürgen Schmidhuber.  Long short-term memory.  _Neural computation_ , 9(8):1735–1780, 1997.  URL https://direct.mit.edu/neco/article/9/8/1735/6109/Long-Short-Term-Memory.
  * Huang and Canny [2019] Forrest Huang and John F Canny.  Sketchforme: Composing sketched scenes from text descriptions for interactive applications.  In _Proceedings of the 32nd Annual ACM Symposium on User Interface Software and Technology_ , pages 209–220, 2019.  URL https://dl.acm.org/doi/10.1145/3332165.3347878.
  * Huang et al. [2020] Forrest Huang, Eldon Schoop, David Ha, and John Canny.  Scones: towards conversational authoring of sketches.  In _Proceedings of the 25th International Conference on Intelligent User Interfaces_ , pages 313–323, 2020.  URL https://dl.acm.org/doi/10.1145/3377325.3377485.
  * Huang et al. [2019] Zhewei Huang, Wen Heng, and Shuchang Zhou.  Learning to paint with model-based deep reinforcement learning.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 8709–8718, 2019.  URL https://openaccess.thecvf.com/content_ICCV_2019/html/Huang_Learning_to_Paint_With_Model-Based_Deep_Reinforcement_Learning_ICCV_2019_paper.html.
  * Jia et al. [2021] Chao Jia, Yinfei Yang, Ye Xia, Yi-Ting Chen, Zarana Parekh, Hieu Pham, Quoc V Le, Yunhsuan Sung, Zhen Li, and Tom Duerig.  Scaling up visual and vision-language representation learning with noisy text supervision.  _arXiv preprint arXiv:2102.05918_ , 2021.  URL https://arxiv.org/abs/2102.05918.
  * Johansson [2008] Roger Johansson.  Genetic programming: Evolution of mona lisa, 2008.  URL https://rogerjohansson.blog/2008/12/07/genetic-programming-evolution-of-mona-lisa/.
  * Kato et al. [2020] Hiroharu Kato, Deniz Beker, Mihai Morariu, Takahiro Ando, Toru Matsuoka, Wadim Kehl, and Adrien Gaidon.  Differentiable rendering: A survey.  _arXiv preprint arXiv:2006.12057_ , 2020.  URL https://arxiv.org/abs/2006.12057.
  * Kingma and Ba [2014] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  _arXiv preprint arXiv:1412.6980_ , 2014.  URL https://arxiv.org/abs/1412.6980.
  * Kolmogorov [1965] Andrei Kolmogorov.  Three approaches to the quantitative definition of information.  _Problems of information transmission_ , 1(1):1–7, 1965.  URL https://www.tandfonline.com/doi/abs/10.1080/00207166808803030?journalCode=gcom20.
  * Kuiper [2021] Kathleen Kuiper.  Modernism, 2021.  URL https://www.britannica.com/art/Modernism-art.
  * Laine et al. [2020] Samuli Laine, Janne Hellsten, Tero Karras, Yeongho Seol, Jaakko Lehtinen, and Timo Aila.  Modular primitives for high-performance differentiable rendering.  _ACM Transactions on Graphics (TOG)_ , 39(6):1–14, 2020.  URL https://arxiv.org/abs/2011.03277.
  * Li et al. [2020] Tzu-Mao Li, Michal Lukáč, Michaël Gharbi, and Jonathan Ragan-Kelley.  Differentiable vector graphics rasterization for editing and learning.  _ACM Transactions on Graphics (TOG)_ , 39(6):1–15, 2020.  URL https://people.csail.mit.edu/tzumao/diffvg/.
  * Lindenmayer [1968] Aristid Lindenmayer.  Mathematical models for cellular interactions in development i. filaments with one-sided inputs.  _Journal of theoretical biology_ , 18(3):280–299, 1968.  URL https://www.sciencedirect.com/science/article/abs/pii/0022519368900799.
  * Liu et al. [2021] Songhua Liu, Tianwei Lin, Dongliang He, Fu Li, Ruifeng Deng, Xin Li, Errui Ding, and Hao Wang.  Paint transformer: Feed forward neural painting with stroke prediction.  _arXiv preprint arXiv:2108.03798_ , 2021.  URL https://arxiv.org/abs/2108.03798.
  * Lopes et al. [2019] Raphael Gontijo Lopes, David Ha, Douglas Eck, and Jonathon Shlens.  A learned representation for scalable vector graphics.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 7930–7939, 2019.  URL https://openaccess.thecvf.com/content_ICCV_2019/html/Lopes_A_Learned_Representation_for_Scalable_Vector_Graphics_ICCV_2019_paper.html.
  * Malkevitchn [2003] Joseph Malkevitchn.  Mathematics and art, 2003.  URL https://www.ams.org/publicoutreach/feature-column/fcarc-art1.
  * Mellor et al. [2019] John FJ Mellor, Eunbyung Park, Yaroslav Ganin, Igor Babuschkin, Tejas Kulkarni, Dan Rosenbaum, Andy Ballard, Theophane Weber, Oriol Vinyals, and SM Eslami.  Unsupervised doodling and painting with improved spiral.  _arXiv preprint arXiv:1910.01007_ , 2019.  URL https://arxiv.org/abs/1910.01007.
  * Modern [2018] Tate Modern.  Minimalism, 2018.  URL https://www.tate.org.uk/art/art-terms/m/minimalism.
  * Nakano [2019] Reiichiro Nakano.  Neural painters: A learned differentiable constraint for generating brushstroke paintings.  _arXiv preprint arXiv:1904.08410_ , 2019.  URL https://arxiv.org/abs/1904.08410.
  * Paauw and Van den Berg [2019] Misha Paauw and Daan Van den Berg.  Paintings, polygons and plant propagation.  In _International Conference on Computational Intelligence in Music, Sound, Art and Design (Part of EvoStar)_ , pages 84–97. Springer, 2019\.  URL https://link.springer.com/chapter/10.1007/978-3-030-16667-0_6.
  * Paul [2004] Stella Paul.  Abstract expressionism, 2004.  URL https://www.metmuseum.org/toah/hd/abex/hd_abex.htm.
  * Radford et al. [2021] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.  Learning transferable visual models from natural language supervision.  _arXiv preprint arXiv:2103.00020_ , 2021.  URL https://arxiv.org/abs/2103.00020.
  * Rewald [2014] Sabine Rewald.  Heilbrunn timeline of art history: Cubism.  _The Metropolitan Museum of Art_ , 2014.  URL https://www.metmuseum.org/toah/hd/cube/hd_cube.htm.
  * Rose [1965] Barbara Rose.  Abc art.  _Art in America_ , 53(5):57–69, 1965.  URL https://www.artnews.com/art-in-america/features/abc-art-barbara-rose-1234580665/.
  * Salimans et al. [2017] Tim Salimans, Jonathan Ho, Xi Chen, Szymon Sidor, and Ilya Sutskever.  Evolution strategies as a scalable alternative to reinforcement learning.  _arXiv preprint arXiv:1703.03864_ , 2017.  URL https://arxiv.org/abs/1703.03864.
  * @samburtonking [2021] @samburtonking.  Introduction to vqgan+clip, 2021.  URL https://docs.google.com/document/d/1Lu7XPRKlNhBQjcKr8k8qRzUzbBW7kzxb5Vu72GMRn2E/edit.
  * Schmidhuber [1997] Jürgen Schmidhuber.  Low-complexity art.  _Leonardo_ , 30(2):97–103, 1997.  URL https://www.idsia.ch/~juergen/locoart/locoart.html.
  * Sehnke et al. [2010] Frank Sehnke, Christian Osendorfer, Thomas Rückstieß, Alex Graves, Jan Peters, and Jürgen Schmidhuber.  Parameter-exploring policy gradients.  _Neural Networks_ , 23(4):551–559, 2010.  URL https://people.idsia.ch//~juergen/nn2010.pdf.
  * Shahrabi [2020] Shahriar Shahrabi.  Procedural paintings with genetic evolution algorithm, 2020.  URL https://github.com/IRCSS/Procedural-painting.
  * Studios" [2019] "Sizigi Studios".  Waifu labs, 2019.  URL https://waifulabs.com/.
  * Tate [2021] Tate.  Process art, 2021.  URL https://www.tate.org.uk/art/art-terms/p/process-art.
  * Toklu [2020] Nihat Engin Toklu.  Pgpelib.  https://github.com/nnaisense/pgpelib, 2020.  URL https://github.com/nnaisense/pgpelib.
  * Toklu et al. [2020] Nihat Engin Toklu, Paweł Liskowski, and Rupesh Kumar Srivastava.  Clipup: A simple and powerful optimizer for distribution-based policy evolution.  In _International Conference on Parallel Problem Solving from Nature_ , pages 515–527. Springer, 2020.  URL https://arxiv.org/abs/2008.02387.
  * Verostko [1994] Roman Verostko.  Algorithmic art, 1994.  URL http://www.verostko.com/algorithm.html.
  * Wang [2021a] Phil Wang.  Big sleep: A simple command line tool for text to image generation, using openai’s clip and a biggan, 2021a.  URL https://github.com/lucidrains/big-sleep.
  * Wang [2021b] Phil Wang.  Deep daze: A simple command line tool for text to image generation using openai’s clip and siren (implicit neural representation network), 2021b.  URL https://github.com/lucidrains/deep-daze.
  * White [2019] Tom White.  Shared visual abstractions.  _arXiv preprint arXiv:1912.04217_ , 2019.  URL https://arxiv.org/abs/1912.04217.
  * Zheng et al. [2019] Ningyuan Zheng, Yifan Jiang, and Dingjiang Huang.  Strokenet: A neural painting environment.  In _International Conference on Learning Representations_ , 2019.  URL https://openreview.net/forum?id=HJxwDiActX.