The *i*-th production of $\alpha$ is $A_i$, and its *i*-th attachment is $\eta_{i \to i+1}$.

For an *o*-Assembly Sequence $\alpha$, the notation $\lim \alpha = A_o$ is consistent with the case of
usual Assembly Sequences, i.e. $\omega$-Assembly Sequences.

These sequences correspond to the intuitive generalization of attachment sequences to
“times larger than infinity”. In particular, any tile attached through an ordinal attachment
sequence reaches the starting assembly of the sequence through a finite number of attachments.
Indeed, for a tile to have been added through an ordinal assembly sequence $\alpha$ at time $t$, the
neighbors to which it attaches must have been attached at some time $t' < t$. Since these
times are ordinals, such an decreasing sequence of times reaches 0 in a finite number of steps.
In particular, these ordinal assembly sequences produce the same productions as the usual
$\omega$-assembly sequences.

This remark formalizes thus:

► **Remark 64.** Let $S$ be an assembly system, $o$ a countable ordinal, $\alpha$ an $o$-Assembly Sequence, $k < o$ and $t@Z$ the $k$-th attachment of $\alpha$. There is a sequence $\alpha'$ of length $o'$ and a bijection $\iota : o \to o'$ such that:

- for all $t < o'$, the attachments $\alpha_t$ and $\alpha'_{\iota(t)}$ are the same,

- $\iota(k)$ is finite.

**Lemma 65.** Let $S$ be an assembly system, $o$ a countable ordinal, and $\alpha$ an $o$-Assembly Sequence. Then if $o$ is infinite, there is a $\omega$-Assembly Sequence $\alpha'$ such that $\lim \alpha' = \lim \alpha$.

**Proof.** The proof goes by transfinite induction.

If $o = \omega$, then $\alpha$ itself satisfies the conclusion of the lemma.

If $o = p + 1$ is a successor ordinal, by induction hypothesis, there is an $\omega$-assembly
sequence $\alpha''$ such that $\lim \alpha'' = A_p$, where $A_p$ is the $p$-th production of $\alpha$. Let $t@Z$ be the
$p$-th attachment of $\alpha$; there is an integer $k$ such that at time $k$, all the origin vertices of
the arcs in $Z$ are attached in $\alpha''$. The attachment sequence $\alpha' = (\alpha''_0, \dots, \alpha''_k, t@Z, \alpha''_{k+1}, \dots)$
satisfies the conclusion of the lemma.

If $o$ is a limit ordinal, there is a sequence $(\alpha'^i)_{i<o}$ of $\omega$-assembly sequences such that
for each $i$, $\lim \alpha'^i = A_i$. Let $A_j^i$ be the $j$-th production of $\alpha'^i$. Since $o$ is countable, it has
cofinality $\omega$; pick an increasing sequence $\epsilon : \omega \to o$ such that $\sup_{i<\omega} \epsilon(i) = o$. Let $\alpha'$ be the
enumeration of the set $\{A_j^{\epsilon(i)} | j \le i < \omega\}$ ordered lexicographically according to $(i, j)$. This
sequence $\alpha'$ is an attachment sequence, and it satisfies $\lim \alpha' = \bigcup_{i<o} (A_i^{\epsilon(i)}) = \lim \alpha$. $\quad$

## A.3 Holes and Fizziness

The Tree Pump lemma is about tree-like assemblies. Since trees are acyclic graphs, the *holes* of the assemblies play an important part in characterizing how these tree-like assemblies behave. Fizziness is the tendency of assembly sequences to create a lot of holes.

► **Definition 66 (Fizziness).** Let $A, A'$ be two free assemblies with $A$-support $\subset A'$-support, the fuzziness $fz(A, A')$ is the number of holes of $A'$ whose perimeter is not contained in $A$.

Let $\alpha = (A_0 \to A_1 \to \dots)$ be an Assembly Sequence in $\mathcal{H}^{\text{Free}}[S]$. The fuzziness of $\alpha$, noted $fz(\alpha)$ is the sequence $i \mapsto fz(A_i, A_{i+1})$.

A sequence $\alpha$ is more fizzy than $\beta$, written $\alpha >_{fz} \beta$ if $fz(\alpha) > fz(\beta)$ lexicographically.

► **Lemma 67 (Maximal Fizziness).** Let $S$ be a seeded assembly system. Assume $S$-seed has a finite number of non-null glues on its external edges. Then there is an $\omega$-Assembly Sequence $\alpha_{\max} \in \mathcal{H}^{\text{Free}}[S]$ with maximal fuzziness among $\omega$-Assembly Sequences.
The *i*-th production of $\alpha$ is $A_i$, and its *i*-th attachment is $\eta_{i \to i+1}$.

For an *o*-Assembly Sequence $\alpha$, the notation $\lim \alpha = A_o$ is consistent with the case of
usual Assembly Sequences, i.e. $\omega$-Assembly Sequences.

These sequences correspond to the intuitive generalization of attachment sequences to
“times larger than infinity”. In particular, any tile attached through an ordinal attachment
sequence reaches the starting assembly of the sequence through a finite number of attachments.
Indeed, for a tile to have been added through an ordinal assembly sequence $\alpha$ at time $t$, the
neighbors to which it attaches must have been attached at some time $t' < t$. Since these
times are ordinals, such an decreasing sequence of times reaches 0 in a finite number of steps.
In particular, these ordinal assembly sequences produce the same productions as the usual
$\omega$-assembly sequences.

This remark formalizes thus:

► **Remark 64.** Let $S$ be an assembly system, $o$ a countable ordinal, $\alpha$ an $o$-Assembly Sequence, $k < o$ and $t@Z$ the $k$-th attachment of $\alpha$. There is a sequence $\alpha'$ of length $o'$ and a bijection $\iota: o \to o'$ such that:

- for all $t < o'$, the attachments $\alpha_t$ and $\alpha'_{\iota(t)}$ are the same,

- $\iota(k)$ is finite.

► **Lemma 65.** Let $S$ be an assembly system, $o$ a countable ordinal, and $\alpha$ an $o$-Assembly Sequence. Then if $o$ is infinite, there is a $\omega$-Assembly Sequence $\alpha'$ such that $\lim \alpha' = \lim \alpha$.

**Proof.** The proof goes by transfinite induction.

If $o = \omega$, then $\alpha$ itself satisfies the conclusion of the lemma.

If $o = p + 1$ is a successor ordinal, by induction hypothesis, there is an $\omega$-assembly
sequence $\alpha''$ such that $\lim \alpha'' = A_p$, where $A_p$ is the $p$-th production of $\alpha$. Let $t@Z$ be the
$p$-th attachment of $\alpha$; there is an integer $k$ such that at time $k$, all the origin vertices of
the arcs in $Z$ are attached in $\alpha''$. The attachment sequence $\alpha' = (\alpha''_0, \dots, \alpha''_k, t@Z, \alpha''_{k+1}, \dots)$
satisfies the conclusion of the lemma.

If $o$ is a limit ordinal, there is a sequence $(\alpha'^i)_{i<o}$ of $\omega$-assembly sequences such that
for each $i$, $\lim \alpha'^i = A_i$. Let $A_j^i$ be the $j$-th production of $\alpha'^i$. Since $o$ is countable, it has
cofinality $\omega$; pick an increasing sequence $\epsilon: \omega \to o$ such that $\sup_{i<\omega} \epsilon(i) = o$. Let $\alpha'$ be the
enumeration of the set $\{A_j^{\epsilon(i)} | j \le i < \omega\}$ ordered lexicographically according to $(i, j)$. This
sequence $\alpha'$ is an attachment sequence, and it satisfies $\lim \alpha' = \bigcup_{i<o} (A_i^{\epsilon(i)}) = \lim \alpha$. $\quad$

## A.3 Holes and Fizziness

The Tree Pump lemma is about tree-like assemblies. Since trees are acyclic graphs, the *holes* of the assemblies play an important part in characterizing how these tree-like assemblies behave. Fizziness is the tendency of assembly sequences to create a lot of holes.

► **Definition 66 (Fizziness).** Let $A, A'$ be two free assemblies with $A$-support $\subset A'$-support, the fuzziness $fz(A, A')$ is the number of holes of $A'$ whose perimeter is not contained in $A$.

Let $\alpha = (A_0 \to A_1 \to \dots)$ be an Assembly Sequence in $\mathcal{H}^{\text{Free}}[S]$. The fuzziness of $\alpha$, noted $fz(\alpha)$ is the sequence $i \mapsto fz(A_i, A_{i+1})$.

A sequence $\alpha$ is more fizzy than $\beta$, written $\alpha >_{fz} \beta$ if $fz(\alpha) > fz(\beta)$ lexicographically.

► **Lemma 67 (Maximal Fizziness).** Let $S$ be a seeded assembly system. Assume $S$ seed has a finite number of non-null glues on its external edges. Then there is an $\omega$-Assembly Sequence $\alpha_{\max} \in \mathcal{H}^{\text{Free}}[S]$ with maximal fuzziness among $\omega$-Assembly Sequences.
**Proof.** Using Kőnig's Lemma.

<figure><img src="image_34.png" /></figure>

Note that this lemma does not hold for $o$-assembly sequences with $o > \omega$. Indeed, after time $\omega$, there might be an infinity of possible attachments, thwarting Kőnig's Lemma.

**Lemma 68 (Fizziness-Increase of Embeddings).** Let $S$ be a seeded TAS, and $\alpha \in \mathcal{H}^{\text{Free}}[S]$. Let $G = \text{support}(\lim \alpha)$, $G'$ an assembly support, and $e: G \to G'$ an embedding.

Then:

1. $e(\alpha)$ is a free assembly sequence,

2. $e(\alpha) \geq_{\text{fz}} \alpha$

3. $e(\alpha) >_{\text{fz}} \alpha$ if $e$ is not injective on $\text{dom}(\lim \alpha)$.

**Proof.** Let $\alpha = (t_i|_{z_i})_i$ be an assembly sequence in $\mathcal{H}^{\text{Free}}[S]$.

For Item 1, the attachments of $e(\alpha)$ are valid by definition of $k$. They are stable since for each attachment, the edges adjacent to $z_i$ which make this attachment stable are preserved by $e$.

For Item 2, for each assembly $A$ of $\alpha$ which is mapped injectively, each hole of $\alpha$ is mapped by $e$ to a hole of $e(A)$ with the same labels on its border.

For Item 3, if $e$ is not injective on $\text{dom}(\lim \alpha)$, then there are $i < k$ such that $e(z_i) = e(z_k)$. Since $\alpha$ is a valid assembly sequence, $z_i \neq z_k$, there are two different paths from the seed to $e(z_i)$. From these two paths, it is possible to construct a hole in $\text{dom}(e(\alpha))$ which is not a hole in $\text{dom}(\alpha)$. Hence, from the first attachment where this hole appears, the assemblies of $e(\alpha)$ are more fizzy than the corresponding ones in $\alpha$. ◀

**Lemma 69 (Maximal Sequences are Flat).** Let $S$ be a seeded TAS, $S$ an assembly of $S$ and $X \subset \mathcal{H}^{\text{Free}}[S, S]$ be a set of Free, Ordinal Assembly Sequences such that for any $\alpha \in X$, there is a $\alpha' \in X$ such that $\alpha' \geq_{\text{fz}}$ squash($\alpha$).

Assume $\alpha$ is a free assembly sequence with maximal fizziness within $X$. Then $\alpha$ embeds into $\mathbb{Z}^2$.

**Proof.** If $\alpha$ does not embed into $\mathbb{Z}^2$ by squash, then squash($\alpha$) is fizzier than $\alpha$ and thus $\alpha$ cannot be maximal in $X$. ◀

<figure><img src="image_35.png" /></figure>

## A.4 The Window Movie Lemma

The Window Movie Lemma [17] applies to free assembly sequences as well as to $\mathbb{Z}^2$ assembly sequences. In order to account for holes, movies need to record not only the glues appearing on either side of the window, but also the paths between edges of the window through either side.

**Definition 70 (Window, Frame, Movie).** Let $\alpha = (A_0, A_1, \dots)$ be an assembly sequence of $\mathcal{H}^{\text{Free}}[S]$. A window $W$ of $\alpha$ is a cut of $(\lim \alpha) \cdot \text{support separating it into two connected components Near}(W)$ and $\text{Far}(W)$.

The width $w(W)$ of a window is the maximum perspective difference between two vertices along the window.

Let $\text{Arcs}(W)$ be the set of arcs (either normal or external) of $(\lim \alpha) \cdot \text{support through } W$.

The frame $f$ on $W$ at time $k$ records for each arc $a = (v, d) \in \text{Arcs}(W)$:

- $f(a) \cdot \text{present:}$ whether $v \in A_k$; support, and if so,

- $f(a) \cdot \text{glue} = A_k \cdot \text{tile}(v)(d)$, as well as

- $f(a) \cdot \text{successor the arc of Arcs}(W)$ which is reachable from a through an exterior path which does not cross $W$, if there is one.
**Proof.** Using Kőnig's Lemma.

<figure><img src="image_33.png" /></figure>

Note that this lemma does not hold for $o$-assembly sequences with $o > \omega$. Indeed, after time $\omega$, there might be an infinity of possible attachments, thwarting Kőnig's Lemma.

**Lemma 68 (Fizziness-Increase of Embeddings).** Let $S$ be a seeded TAS, and $\alpha \in \mathcal{H}^{\text{Free}}[S]$. Let $G = \text{support}(\lim \alpha)$, $G'$ an assembly support, and $e: G \to G'$ an embedding.

Then:

1. $e(\alpha)$ is a free assembly sequence,

2. $e(\alpha) \geq_{\text{fz}} \alpha$

3. $e(\alpha) >_{\text{fz}} \alpha$ if $e$ is not injective on $\text{dom}(\lim \alpha)$.

**Proof.** Let $\alpha = (t_i|z_i)_i$ be an assembly sequence in $\mathcal{H}^{\text{Free}}[S]$.

For Item 1, the attachments of $e(\alpha)$ are valid by definition of $k$. They are stable since for each attachment, the edges adjacent to $z_i$ which make this attachment stable are preserved by $e$.

For Item 2, for each assembly $A$ of $\alpha$ which is mapped injectively, each hole of $\alpha$ is mapped by $e$ to a hole of $e(A)$ with the same labels on its border.

For Item 3, if $e$ is not injective on $\text{dom}(\lim \alpha)$, then there are $i < k$ such that $e(z_i) = e(z_k)$. Since $\alpha$ is a valid assembly sequence, $z_i \neq z_k$, there are two different paths from the seed to $e(z_i)$. From these two paths, it is possible to construct a hole in $\text{dom}(e(\alpha))$ which is not a hole in $\text{dom}(\alpha)$. Hence, from the first attachment where this hole appears, the assemblies of $e(\alpha)$ are more fizzy than the corresponding ones in $\alpha$. ◀

**Lemma 69 (Maximal Sequences are Flat).** Let $S$ be a seeded TAS, $S$ an assembly of $S$ and $X \subset \mathcal{H}^{\text{Free}}[S, S]$ be a set of Free, Ordinal Assembly Sequences such that for any $\alpha \in X$, there is a $\alpha' \in X$ such that $\alpha' \geq_{\text{fz}}$ squash($\alpha$).

Assume $\alpha$ is a free assembly sequence with maximal fizziness within $X$. Then $\alpha$ embeds into $\mathbb{Z}^2$.

**Proof.** If $\alpha$ does not embed into $\mathbb{Z}^2$ by squash, then squash($\alpha$) is fizzier than $\alpha$ and thus $\alpha$ cannot be maximal in $X$. ◀

<figure><img src="image_34.png" /></figure>

## A.4 The Window Movie Lemma

The Window Movie Lemma [17] applies to free assembly sequences as well as to $\mathbb{Z}^2$ assembly sequences. In order to account for holes, movies need to record not only the glues appearing on either side of the window, but also the paths between edges of the window through either side.

**Definition 70 (Window, Frame, Movie).** Let $\alpha = (A_0, A_1, \dots)$ be an assembly sequence of $\mathcal{H}^{\text{Free}}[S]$. A window $W$ of $\alpha$ is a cut of $(\lim \alpha) \cdot \text{support separating it into two connected components Near}(W)$ and $\text{Far}(W)$.

The width $w(W)$ of a window is the maximum perspective difference between two vertices along the window.

Let $\text{Arcs}(W)$ be the set of arcs (either normal or external) of $(\lim \alpha) \cdot \text{support through } W$.

The frame $f$ on $W$ at time $k$ records for each arc $a = (v, d) \in \text{Arcs}(W)$:

- $f(a) \cdot \text{present:}$ whether $v \in A_k$; support, and if so,

- $f(a) \cdot \text{glue} = A_k \cdot \text{tile}(v)(d)$, as well as

- $f(a) \cdot \text{successor the arc of Arcs}(W)$ which is reachable from a through an exterior path which does not cross $W$, if there is one.
<figure><img src="image_36.png" /><figcaption>■ **Figure 21** An assembly sequence $\alpha'$ of some temperature 1 aTAM, which yields a terminal production $F = \lim \alpha'$. Below a sequence $\alpha >_{fz} \alpha'$ which yields a smaller production $P$ which is bounded in direction $\vec{d}$ (vertically). The assembly sequence $\alpha$ is fizzier since by step 50, it just closed its seventh hole while $\alpha'$ is lagging at only 5 holes. Because $\alpha$ skips any unprofitable attachment, $\lim \alpha$ is very much not terminal: its seed (tile number 0) as well as every tile attached at a time of the form $16 + 7k$ has an attachable yet unfilled position. It can thus be extended into an ordinal assembly sequence $\alpha''$ which does reach $\lim \alpha$, but in time $3\omega$. The attachment times of the form $i.j$ in $\alpha''$ should be read as $i\omega + j$. At time $\omega$, $\alpha''$ has assembled $\lim \alpha$; at time $2\omega$, it has added the upwards path, and at time $3\omega$, it has added the last tile to each hole of $\lim \alpha$.</figcaption></figure>
<figure><img src="image_35.png" /><figcaption>■ **Figure 21** An assembly sequence $\alpha'$ of some temperature 1 aTAM, which yields a terminal production $F = \lim \alpha'$. Below a sequence $\alpha >_{fz} \alpha'$ which yields a smaller production $P$ which is bounded in direction $\vec{d}$ (vertically). The assembly sequence $\alpha$ is fizzier since by step 50, it just closed its seventh hole while $\alpha'$ is lagging at only 5 holes. Because $\alpha$ skips any unprofitable attachment, $\lim \alpha$ is very much not terminal: its seed (tile number 0) as well as every tile attached at a time of the form $16 + 7k$ has an attachable yet unfilled position. It can thus be extended into an ordinal assembly sequence $\alpha''$ which does reach $\lim \alpha$, but in time $3\omega$. The attachment times of the form $i.j$ in $\alpha''$ should be read as $i\omega + j$. At time $\omega$, $\alpha''$ has assembled $\lim \alpha$; at time $2\omega$, it has added the upwards path, and at time $3\omega$, it has added the last tile to each hole of $\lim \alpha$.</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Figure 22 Example assembly sequence α and window W. On the main picture, the labels of the tiles of lim α are the order in which they attach. The small pictures are the successive assemblies of α, their associated frames and the obtained movie. The movie is made up of the sequence of the unique frames, in order of apparition.</figcaption></figure>

The movie associated with W is the ordered set MOVIE(W) of distinct frames appearing on W.

Figure 22 gives an example free assembly sequence α with a window W. The arrows between the edges on either side of W represent the relation “successor”.

**▶ Lemma 71 (Window Movie Lemma).** Let $\alpha \in \mathcal{H}^{\text{Free}}[S, \sigma_A]$ and $\beta \in \mathcal{H}^{\text{Free}}[S, \sigma_B]$ two assembly sequences. Assume there are two windows A in $\alpha$ and B in $\beta$ and a translation $\tau$ such that $\tau(A) = B$, MOVIE(A) is the same as MOVIE(B) up to translation by $\tau$, and lastly $\tau$ can be extended to a translation from Far(A) to Far(B) which maps $\sigma_B \cap \text{Far}(B)$ to $\sigma_A \cap \text{Far}(A)$.

Let $\alpha^\dagger = \alpha|_{\text{Far}(A)} \in \mathcal{H}^{\text{Free}}[S, \lim \alpha \cap \text{Near}(A)]$ be the sequence of attachments of $\alpha$ within Far(A), and likewise $\beta^\dagger = \alpha|_{\text{Far}(B)} \in \mathcal{H}^{\text{Free}}[S, \lim \beta \cap \text{Near}(B)]$ be the sequence of attachments of $\beta$ within Far(B).

Let $\alpha'$ be the candidate assembly sequence consisting of $\alpha$ where for every $k$, the attachment $\alpha_k^\dagger$ is replaced with $\beta_k^\dagger$.

Then there is a window $A'$ on $\alpha'$ and two translations $\tau_N : \text{Near}(A') \to \text{Near}(A)$ and $\tau_F : \text{Far}(A') \to \text{Far}(B)$ such that:

$$ \begin{cases} \forall z \in \text{Near}(A'), & (\lim \alpha') \cdot \text{tile}(z) = (\lim \alpha) \cdot \text{tile}(\tau_N(z)) \\ \forall z \in \text{Far}(A'), & (\lim \alpha') \cdot \text{tile}(z) = (\lim \beta) \cdot \text{tile}(\tau_F(z)) \end{cases} \qquad (1) $$

Moreover, if $\beta^\dagger >_{\text{fz}} \alpha^\dagger$, then $\alpha' >_{\text{fz}} \alpha$.
<figure><img src="image_36.png" /><figcaption>Figure 22 Example assembly sequence $\alpha$ and window $W$. On the main picture, the labels of the tiles of $\lim \alpha$ are the order in which they attach. The small pictures are the successive assemblies of $\alpha$, their associated frames and the obtained movie. The movie is made up of the sequence of the unique frames, in order of apparition.</figcaption></figure>

The movie associated with $W$ is the ordered set $MOVIE(W)$ of distinct frames appearing on $W$.

Figure 22 gives an example free assembly sequence $\alpha$ with a window $W$. The arrows between the edges on either side of $W$ represent the relation “successor”.

**▶ Lemma 71 (Window Movie Lemma).** Let $\alpha \in \mathcal{H}^{\text{Free}}[S, \sigma_A]$ and $\beta \in \mathcal{H}^{\text{Free}}[S, \sigma_B]$ two assembly sequences. Assume there are two windows $A$ in $\alpha$ and $B$ in $\beta$ and a translation $\tau$ such that $\tau(A) = B$, $MOVIE(A)$ is the same as $MOVIE(B)$ up to translation by $\tau$, and lastly $\tau$ can be extended to a translation from $\text{Far}(A)$ to $\text{Far}(B)$ which maps $\sigma_B \cap \text{Far}(B)$ to $\sigma_A \cap \text{Far}(A)$.

Let $\alpha^\dagger = \alpha|_{\text{Far}(A)} \in \mathcal{H}^{\text{Free}}[S, \lim \alpha \cap \text{Near}(A)]$ be the sequence of attachments of $\alpha$ within $\text{Far}(A)$, and likewise $\beta^\dagger = \alpha|_{\text{Far}(B)} \in \mathcal{H}^{\text{Free}}[S, \lim \beta \cap \text{Near}(B)]$ be the sequence of attachments of $\beta$ within $\text{Far}(B)$.

Let $\alpha'$ be the candidate assembly sequence consisting of $\alpha$ where for every $k$, the attachment $\alpha_k^\dagger$ is replaced with $\beta_k^\dagger$.

Then there is a window $A'$ on $\alpha'$ and two translations $\tau_N : \text{Near}(A') \to \text{Near}(A)$ and $\tau_F : \text{Far}(A') \to \text{Far}(B)$ such that:

$$ \begin{cases} \forall z \in \text{Near}(A'), & (\lim \alpha') \cdot \text{tile}(z) = (\lim \alpha) \cdot \text{tile}(\tau_N(z)) \\ \forall z \in \text{Far}(A'), & (\lim \alpha') \cdot \text{tile}(z) = (\lim \beta) \cdot \text{tile}(\tau_F(z)) \end{cases} \qquad (1) $$

Moreover, if $\beta^\dagger >_{\text{fz}} \alpha^\dagger$, then $\alpha' >_{\text{fz}} \alpha$.
<figure><img src="image_38.png" /><figcaption>Figure 23 The Window Movie Lemma: consider two assembly sequences $\iota$ (the ibex) and $\beta$ (the rabbit), with their productions depicted in the top row. The sequence $\iota$, has two windows $W_t$ and $W_h$. In $W_t$, Far($W_t$) is the head of the ibex, Near($W_t$) its body. In $W_h$, Far($W_h$) is the tip of the horn. Each of them is associated with a translation, $\tau_t$ and $\tau_h$ respectively, which sends the window to another window with the same movie, and satisfying the hypothesis of Lemma 71. Applying Lemma 71 in each case gives two new assembly sequences represented on the bottom row. Applying it with $W_t$ yields a fearsome chimera, while its application to $W_h$ yields an ibex with a horn so long it needs to bend in the third dimension to avoid piercing its spine, i.e. avoiding the conflict that would arise were our attachment sequences not free.</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>Figure 23 The Window Movie Lemma: consider two assembly sequences $\iota$ (the ibex) and $\beta$ (the rabbit), with their productions depicted in the top row. The sequence $\iota$, has two windows $W_t$ and $W_h$. In $W_t$, Far($W_t$) is the head of the ibex, Near($W_t$) its body. In $W_h$, Far($W_h$) is the tip of the horn. Each of them is associated with a translation, $\tau_t$ and $\tau_h$ respectively, which sends the window to another window with the same movie, and satisfying the hypothesis of Lemma 71. Applying Lemma 71 in each case gives two new assembly sequences represented on the bottom row. Applying it with $W_t$ yields a fearsome chimera, while its application to $W_h$ yields an ibex with a horn so long it needs to bend in the third dimension to avoid piercing its spine, i.e. avoiding the conflict that would arise were our attachment sequences not free.</figcaption></figure>
**Proof.** Define $\alpha'$ as $\alpha$ where for each $k$, the $k$-th attachment $t@Z$ within $\text{Far}(A)$ has been replaced by the $k$-th attachment within $\text{Far}(B)$ of $\beta$. When doing this, for an attachment $t@Z$ within $\text{Far}(B)$, any arc $a = (v, d)$ in $Z$ with $v \in \text{Near}(B)$ is replaced by the arc $\tau^{-1}(a)$.

The translations $\tau_N$ and $\tau_F$ can be defined inductively on the attachments of $\alpha'$, such that Equation (1) holds.

The proof that $\alpha'$ is a valid free assembly sequence is the same as in the $\mathbb{Z}^2$ case. Note that since $B$ is a cut of $(\lim \beta) \cdot$ support, any attachment of $\beta$ in $\text{Far}(B)$ can be replayed in $\alpha'$ in $\text{Far}(A')$ without conflicts since they do not create any adjacency outside $\text{Far}(\beta)$.

Observe that every attachment in $\text{Far}(B)$ which affects holes in $\beta$ affects the same number of holes in $\alpha'$: holes which are wholly within $\text{Far}(B)$ have had all of their tiles attached in $\alpha'$, and holes which go through $B$ become holes through $A$ because the part within $\text{Near}(B)$ has the same connections in $\text{Near}(A)$ at that frame in the movie. ◀

The fact that the grafting process of Lemma 71 is increasing in the fuzziness of the far part of the assembly sequences implies that when the original assembly sequences have maximal fuzziness, so does the chimera sequence $\alpha'$.

Lemma 71 will be most useful in this paper in the case where the cuts $A$ and $B$ are on the same branch of the assembly. In this case, by iterating Lemma 71, it is possible to get a production with a periodic subassembly.

**Corollary 72.** Let $\alpha = (A_i)_{i<o}$ be an assembly sequence with two windows $A$ and $B$ satisfying the hypothesis of Lemma 71 and such that $\text{Far}(B) \subset \text{Far}(A)$. Let $\tau$ be the translation between $A$ and $B$. Then, there is an assembly sequence $\alpha^\tau$ such that $F = \lim \alpha^\tau \cap \text{Far}(A)$ verifies $\tau(F) \subset F$, and within $\text{Near}(B)$, $\alpha^\tau$ does the same attachments in the same order as $\alpha$.

**Proof.** Let $\tau$ be the translation sending $A$ to $B$.

Show by induction that for any $k$, there is a sequence $\alpha^k$ such that the movie on the windows $A$ and $B$ within $\alpha^k$ are the same as in $\alpha$, and for each $z \in \text{Near}(B) \cap \text{Far}(A)$ and $j \le k$, $(\lim \alpha^k)(\tau^j(z)) = (\lim \alpha)(z)$, and the attachments done by $\alpha^k$ and $\alpha^j$ within $\bigcup_{i \le j} \tau^i(\text{Near}(B) \cap \text{Far}(A))$ are the same.

For $k=0$, it suffices to pick $\alpha^0 = \alpha$. Assume that $\alpha^k$ satisfies the induction hypothesis. Then Lemma 71 applies, and the assembly sequence it yields, $\alpha^{k+1}$, satisfies the induction hypothesis at rank $k+1$.

For a pair $(j, k) \in \mathbb{N}$, if for all $i \le j$, $\alpha_i^k \in \bigcup_{l \le k} \tau^k(\text{Near}(B) \cap \text{Far}(A))$, then the $j$ first attachments are unchanging after rank $k$: for all $m \ge k$ and $i \le j$, $\alpha_i^m = \alpha_i^k$. Let $\alpha^{\vec{p}}$ be the sequence of such unchanging attachments. Let $m$ be the supremum of the $j \in \mathbb{N}$ such that the $j$ first attachments are unchanging after some rank $k$. Define $\alpha^{\vec{p}}$ as the sequence of length $m$ with $\alpha_j^{\vec{p}} = \alpha_j^k$, where $k$ is such that the $j$ first attachments are unchanging.

It is easy to check that indeed, $\lim \alpha^{\vec{p}} \cap \text{Far}(A)$ has period $\vec{p}$: any attachment in $\alpha^{\vec{p}}$ within $\text{Far}(A)$ gets picked up by subsequent applications of Lemma 71 which translate it by $\vec{p}$. ◀

In this situation with a branch and two cuts with the same movie, it is also possible to cut assembly sequences short, so that they can do their business in $\text{Near}(A)$ without needing to mess with $\text{Far}(B)$.

**Corollary 73.** Let $\alpha$ be an assembly sequence with two windows $A$ and $B$ satisfying the hypothesis of Lemma 71 and such that $\text{Far}(B) \subset \text{Far}(A)$. Then there is an assembly sequence $\alpha'$ which does the same attachments as $\alpha$ within $\text{Near}(A)$, but has no attachment in $\text{Far}(B)$.

**Proof.** Let $j$ be the last time that $\alpha$ does an attachment in $\text{Near}(A)$ next to $A$, i.e. the date of the last frame in the movie of $A$ where a tile is attached on the near side of the window. By Remark 64, up to a reordering of $\alpha$, $j$ is finite.
**Proof.** Define $\alpha'$ as $\alpha$ where for each $k$, the $k$-th attachment $t@Z$ within Far($A$) has been replaced by the $k$-th attachment within Far($B$) of $\beta$. When doing this, for an attachment $t@Z$ within Far($B$), any arc $a = (v, d)$ in $Z$ with $v \in \text{Near}(B)$ is replaced by the arc $\tau^{-1}(a)$.

The translations $\tau_N$ and $\tau_F$ can be defined inductively on the attachments of $\alpha'$, such that Equation (1) holds.

The proof that $\alpha'$ is a valid free assembly sequence is the same as in the $\mathbb{Z}^2$ case. Note that since $B$ is a cut of $(\lim \beta) \cdot$ support, any attachment of $\beta$ in Far($B$) can be replayed in $\alpha'$ in Far($A'$) without conflicts since they do not create any adjacency outside Far($\beta$).

Observe that every attachment in Far($B$) which affects holes in $\beta$ affects the same number of holes in $\alpha'$: holes which are wholly within Far($B$) have had all of their tiles attached in $\alpha'$, and holes which go through $B$ become holes through $A$ because the part within Near($B$) has the same connections in Near($A$) at that frame in the movie. ◀

The fact that the grafting process of Lemma 71 is increasing in the fuzziness of the far part of the assembly sequences implies that when the original assembly sequences have maximal fuzziness, so does the chimera sequence $\alpha'$.

Lemma 71 will be most useful in this paper in the case where the cuts $A$ and $B$ are on the same branch of the assembly. In this case, by iterating Lemma 71, it is possible to get a production with a periodic subassembly.

**Corollary 72.** Let $\alpha = (A_i)_{i<o}$ be an assembly sequence with two windows $A$ and $B$ satisfying the hypothesis of Lemma 71 and such that Far($B$) $\subset$ Far($A$). Let $\tau$ be the translation between $A$ and $B$. Then, there is an assembly sequence $\alpha^\tau$ such that $F = \lim \alpha^\tau \cap \text{Far}(A)$ verifies $\tau(F) \subset F$, and within Near($B$), $\alpha^\tau$ does the same attachments in the same order as $\alpha$.

**Proof.** Let $\tau$ be the translation sending $A$ to $B$.

Show by induction that for any $k$, there is a sequence $\alpha^k$ such that the movie on the windows $A$ and $B$ within $\alpha^k$ are the same as in $\alpha$, and for each $z \in \text{Near}(B) \cap \text{Far}(A)$ and $j \le k$, $(\lim \alpha^k)(\tau^j(z)) = (\lim \alpha)(z)$, and the attachments done by $\alpha^k$ and $\alpha^j$ within $\bigcup_{i \le j} \tau^i(\text{Near}(B) \cap \text{Far}(A))$ are the same.

For $k=0$, it suffices to pick $\alpha^0 = \alpha$. Assume that $\alpha^k$ satisfies the induction hypothesis. Then Lemma 71 applies, and the assembly sequence it yields, $\alpha^{k+1}$, satisfies the induction hypothesis at rank $k+1$.

For a pair $(j, k) \in \mathbb{N}$, if for all $i \le j$, $\alpha_i^k \in \bigcup_{l \le k} \tau^k(\text{Near}(B) \cap \text{Far}(A))$, then the $j$ first attachments are unchanging after rank $k$: for all $m \ge k$ and $i \le j$, $\alpha_i^m = \alpha_i^k$. Let $\alpha^{\vec{p}}$ be the sequence of such unchanging attachments. Let $m$ be the supremum of the $j \in \mathbb{N}$ such that the $j$ first attachments are unchanging after some rank $k$. Define $\alpha^{\vec{p}}$ as the sequence of length $m$ with $\alpha_j^{\vec{p}} = \alpha_j^k$, where $k$ is such that the $j$ first attachments are unchanging.

It is easy to check that indeed, $\lim_{\vec{p}} \alpha^{\vec{p}} \cap \text{Far}(A)$ has period $\vec{p}$: any attachment in $\alpha^{\vec{p}}$ within Far($A$) gets picked up by subsequent applications of Lemma 71 which translate it by $\vec{p}$. ◀

In this situation with a branch and two cuts with the same movie, it is also possible to cut assembly sequences short, so that they can do their business in Near($A$) without needing to mess with Far($B$).

**Corollary 73.** Let $\alpha$ be an assembly sequence with two windows $A$ and $B$ satisfying the hypothesis of Lemma 71 and such that Far($B$) $\subset$ Far($A$). Then there is an assembly sequence $\alpha'$ which does the same attachments as $\alpha$ within Near($A$), but has no attachment in Far($B$).

**Proof.** Let $j$ be the last time that $\alpha$ does an attachment in Near($A$) next to $A$, i.e. the date of the last frame in the movie of $A$ where a tile is attached on the near side of the window. By Remark 64, up to a reordering of $\alpha$, $j$ is finite.
Let $\alpha^0$ be the prefix of length $j$ of $\alpha$. For each $k$, if $\alpha^k$ has any attachment in $\text{Far}(B)$, it is possible to use Lemma 71 between $B$ and $A$ to obtain a sequence $\alpha^{k+1}$ with fewer attachments within $\text{Far}(B)$. Hence, there is a finite $k$ such that $\alpha^k$ has no attachments in $\text{Far}(B)$, and $\lim \alpha^k \cap \text{Near}(A) = \lim \alpha^0 \cap \text{Near}(A)$. Since by time $j$, the movie on $A$ is over, the rest of the attachments of $\alpha$ which take place in $\text{Near}(A)$ can be replayed after $\alpha^k$. ◀

## A.5 The Tree Pump Lemma

After all these considerations about the fantastic properties of Ordinal Free Assembly Sequences, it is now time to come back to Earth, or rather $\mathbb{Z}^2$. The object is now to prove Lemma 8, for which an investigation of the properties systems whose $\mathbb{Z}^2$-assembly sequences do not circle large squares is in order.

The statement of Lemma 8 is given again, recall that it deals with the $\mathbb{Z}^2$-productions of the aTAM system $S$, hence in its statement, $A_{\square}[S]$, $C_m[S]$, $B_{F(n,m),\vec{d}}[S]$ and $P_{\vec{d}}[S]$ are sets of $\mathbb{Z}^2$-assemblies.

**Lemma 8 (Tree Pump).** For any aTAM system $S$ with $S$-seed finite and connected, define the following sets of assemblies:

- for any integer $m$, $C_m[S]$ is the set of assemblies of $S$ which encircle an $m \times m$ square;

- for any real $k$ and vector $\vec{d}$, $B_{k,\vec{d}}[S]$ is the set of assemblies of $S$ which do not cover any position $\vec{p}$ such that $\vec{p} \cdot \vec{d} > k + |S$-seed|

- for any vector $\vec{d}$, $P_{\vec{d}}[S]$ is the set of ultimately periodic assemblies of $S$ such that there is a vector $\vec{p}$ with $|\vec{p} \cdot \vec{d}| > 0$ and a non-empty sub-assembly $a \subseteq A$ such that $a + \vec{p} \subseteq a$.

Then, there is a function $F: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that for aTAM system $S$ with $n$ tiles and a 1-tile seed, integer $m$ and unit vector $\vec{d}$ of $\mathbb{R}^2$,

$$A_{\square}[S] \cap (C_m[S] \cup B_{F(n,m),\vec{d}}[S] \cup P_{\vec{d}}[S]) \neq \emptyset.$$

When considering the statement of Lemma 8, the holes of the $\mathbb{Z}^2$-productions of $S$ might as well be filled in. Hence, the definition of their fill-in, illustrated on Figure 24.

**Definition 74 (Fill-in).** For a subgraph $D \subset \mathbb{Z}^2$, define the fill-in $D^\bullet$ as the subgraph of $\mathbb{Z}^2$ induced by the positions $p$ such that there is no infinite path from $p$ in $\mathbb{Z}^2 \setminus D$.

A $\mathbb{Z}^2$ assembly $A$ which does not circle any square larger than $m \times m$ looks like a tree, as expressed by the notion of *Connected Treewidth* [3]. This tree is obtained by grouping the positions of $A$-support in connected sets of size at most $2m$, known as *bags*, organized in a tree in such a way that:

- any arc of $A$-support is between two positions which appear in the same bag, and

- for any position $z \in A$-support, the set of bags which contain $z$ forms a subtree.

This decomposition is represented on Figure 24.

**Lemma 75.** Let $S$ an aTAM system, $m$ an integer and $A$ a $\mathbb{Z}^2$ assembly of $S$ such that $A \notin C_m[S]$.

Then $A$-support has *Connected Tree Width* $2m$.

**Proof.** Indeed, it has treewidth $m$: otherwise it would contain an $m \times m$ grid as a minor; that minor would have to be realized in $\mathbb{Z}^2$ as a subgraph which would encircle some $m \times m$ square. Moreover, since $P^\bullet$ does not encircle any empty position, it does not have any geodesic cycle of length more than 4. ◀
Let $\alpha^0$ be the prefix of length $j$ of $\alpha$. For each $k$, if $\alpha^k$ has any attachment in $\mathrm{Far}(B)$, it is possible to use Lemma 71 between $B$ and $A$ to obtain a sequence $\alpha^{k+1}$ with fewer attachments within $\mathrm{Far}(B)$. Hence, there is a finite $k$ such that $\alpha^k$ has no attachments in $\mathrm{Far}(B)$, and $\lim \alpha^k \cap \mathrm{Near}(A) = \lim \alpha^0 \cap \mathrm{Near}(A)$. Since by time $j$, the movie on $A$ is over, the rest of the attachments of $\alpha$ which take place in $\mathrm{Near}(A)$ can be replayed after $\alpha^k$. ◀

## A.5 The Tree Pump Lemma

After all these considerations about the fantastic properties of Ordinal Free Assembly Sequences, it is now time to come back to Earth, or rather $\mathbb{Z}^2$. The object is now to prove Lemma 8, for which an investigation of the properties systems whose $\mathbb{Z}^2$-assembly sequences do not circle large squares is in order.

The statement of Lemma 8 is given again, recall that it deals with the $\mathbb{Z}^2$-productions of the aTAM system $S$, hence in its statement, $A_{\square}[S]$, $C_m[S]$, $B_{F(n,m),\vec{d}}[S]$ and $P_{\vec{d}}[S]$ are sets of $\mathbb{Z}^2$-assemblies.

**Lemma 8 (Tree Pump).** For any aTAM system $S$ with $S$-seed finite and connected, define the following sets of assemblies:

- for any integer $m$, $C_m[S]$ is the set of assemblies of $S$ which encircle an $m \times m$ square;

- for any real $k$ and vector $\vec{d}$, $B_{k,\vec{d}}[S]$ is the set of assemblies of $S$ which do not cover any position $\vec{p}$ such that $\vec{p} \cdot \vec{d} > k + |S$-seed|

- for any vector $\vec{d}$, $P_{\vec{d}}[S]$ is the set of ultimately periodic assemblies of $S$ such that there is a vector $\vec{p}$ with $|\vec{p} \cdot \vec{d}| > 0$ and a non-empty sub-assembly $a \subseteq A$ such that $a + \vec{p} \subseteq a$.

Then, there is a function $F: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that for aTAM system $S$ with $n$ tiles and a 1-tile seed, integer $m$ and unit vector $\vec{d}$ of $\mathbb{R}^2$,

$$A_{\square}[S] \cap (C_m[S] \cup B_{F(n,m),\vec{d}}[S] \cup P_{\vec{d}}[S]) \neq \emptyset.$$

When considering the statement of Lemma 8, the holes of the $\mathbb{Z}^2$-productions of $S$ might as well be filled in. Hence, the definition of their fill-in, illustrated on Figure 24.

**Definition 74 (Fill-in).** For a subgraph $D \subset \mathbb{Z}^2$, define the fill-in $D^\bullet$ as the subgraph of $\mathbb{Z}^2$ induced by the positions $p$ such that there is no infinite path from $p$ in $\mathbb{Z}^2 \setminus D$.

A $\mathbb{Z}^2$ assembly $A$ which does not circle any square larger than $m \times m$ looks like a tree, as expressed by the notion of *Connected Treewidth* [3]. This tree is obtained by grouping the positions of $A$-support in connected sets of size at most $2m$, known as *bags*, organized in a tree in such a way that:

- any arc of $A$-support is between two positions which appear in the same bag, and

- for any position $z \in A$-support, the set of bags which contain $z$ forms a subtree.

This decomposition is represented on Figure 24.

**Lemma 75.** Let $S$ an aTAM system, $m$ an integer and $A$ a $\mathbb{Z}^2$ assembly of $S$ such that $A \notin C_m[S]$.

Then $A$-support has *Connected Tree Width* $2m$.

**Proof.** Indeed, it has treewidth $m$: otherwise it would contain an $m \times m$ grid as a minor; that minor would have to be realized in $\mathbb{Z}^2$ as a subgraph which would encircle some $m \times m$ square. Moreover, since $P^\bullet$ does not encircle any empty position, it does not have any geodesic cycle of length more than 4. ◀
<figure><img src="image_39.png" /><figcaption>■ **Figure 24** The domain *D* of a production *P*, its fill-in *D*• and an associated connected tree decomposition of *TREE* − *DEC[D*•*]. The blue part of each bag *b* of *TREE* − *DEC[D*•*] is its intersection *W*<sub>*b*</sub> with its parent, which disconnects the vertices appearing in its subtree from the rest of *D*•. The choice of the root of *TREE* − *DEC[D*•*] is arbitrary and does not affect the sets *W*<sub>*b*</sub>, up to renaming.</figcaption></figure>
<figure><img src="image_38.png" /><figcaption>■ **Figure 24** The domain *D* of a production *P*, its fill-in *D*• and an associated connected tree decomposition of *TREE - DEC[D*•*]. The blue part of each bag *b* of *TREE - DEC[D*•*] is its intersection *W*<sub>*b*</sub> with its parent, which disconnects the vertices appearing in its subtree from the rest of *D*•. The choice of the root of *TREE - DEC[D*•*] is arbitrary and does not affect the sets *W*<sub>*b*</sub>, up to renaming.</figcaption></figure>
The point of using Connected Treewidth rather than the usual treewidth is that thanks to the connectivity of the bags of $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DE}\mathcal{C}[P^\bullet]$, the distances in $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DE}\mathcal{C}[P^\bullet]$ reflect those in $P^\bullet$:

- there is a function $r$ such that for any vertex $v \in P^\bullet$, the subtree of the bags of $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DE}\mathcal{C}[P^\bullet]$ containing $v$ has a size at most $r(m)$,

- there are two increasing functions $d_m, D_m$ such that for any vertices $u, v$ at distance $\delta$ in $P^\bullet$, any bags $B_u \ni u$ and $B_v \ni v$ of $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DE}\mathcal{C}[P^\bullet]$ are at distance $\Delta$ with $d_m(\delta) \le \Delta \le D_m(\delta)$.

As a consequence, if two tiles are far enough in an assembly $A$ and the path between them does not go through the seed, there must be two windows cutting that path which satisfy the hypothesis of Corollary 72.

**Lemma 76.** Let $S$ be an aTAM system with $n$ tiles and $m$ an integer.

Let $\alpha = (A_i)_{i \le o} \in \mathcal{H}^{\text{Free}}[S, A_0]$ be such that for all $i \le o$, squash$(A_i) \notin C_m[S]$.

There is a constant $F(n, m)$ such that if $z, z' \in$ squash($\lim \alpha \cdot$ support $\setminus A_0$, support are such that the distance between $z'$ and $z$ is greater than $F(n, m)$, then there are two windows $A$ and $B$ with $z \in \text{Near}(A)$ and $z' \in \text{Far}(B)$ which satisfy the hypothesis of Lemma 71.

**Proof.** let $W(n, m)$ be the number of movies with $n$ glues on a window of width $2m$. Note that $W(n, m)$ counts the number of arrangements of the edges within the window, the perspective difference between the connected components of the window, as well as the events on the frames of the movie. Let $F(n, m) = d_m^{-1}(W(n, m))$.

Let $P = \lim \alpha$-support. By Lemma 75, pick a tree decomposition $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DE}\mathcal{C}[P^\bullet]$ with connected bags of size at most $2m$. Let $\delta$ be the distance between $z$ and $z'$. If $\delta \le F(n, m)$, by the pigeonhole principle, there must be two windows $A$ and $B$ between $z$ and $z'$ with the same movie. ◀

In the case where the assembly embeds into $\mathbb{Z}^2$, one actually controls the direction of the vector between the two windows.

**Lemma 77.** Let $S$ be an aTAM system with $n$ tiles, $m$ an integer and $\vec{d}$ a unit vector.

Let $\alpha = (A_i)_{i \le o} \in \mathcal{H}^{\text{Free}}[S, A_0]$ be such that for all $i \le o$, $A_i$ embeds into $\mathbb{Z}^2$.

There is a constant $F'(n, m)$ such that if $z, z' \in \lim \alpha$-support $\setminus A_0$, support are such that the distance between $(z' - z) \cdot \vec{d} > F'(n, m)$, then there are two windows $A$ and $B$ with $z \in \text{Near}(A)$ and $z' \in \text{Far}(B)$ which satisfy the hypothesis of Lemma 71, and such that the vector $\vec{p}$ of the translation between $A$ and $B$ satisfies $\vec{p} \cdot \vec{d} > 1$.

**Proof.** Pick $F'(n, m) = 2F(n, m)+1$. If $(z'-z) \cdot \vec{d} > F'(n, m)$, then in $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DE}\mathcal{C}[\lim \alpha \cdot \text{support}^\bullet]$ there are two bags, one $b$ containing $z$, the other $b' \supseteq z'$ such that on the path between $b_z$ and $b_{z'}$, there are $2F(n, m)+1$ bags $(b_i)_i$ such that for all $x \in b_i, y \in b_j$, $(x-y) \cdot \vec{d} \ge i-j$. Thus, there are $i, j$ such $j > i+1$ and two windows, $A$ between $b_i$ and $b_{i+1}$, and $B$ between $b_j$ and $b_{j+1}$ which have the same movie. Because the windows $A$ and $B$ are separated by at least two elements of $(b_i)$, the translation vector $\vec{p}$ between them satisfies $\vec{p} \cdot \vec{d} > 1$. ◀

Thus, there is a simple argument to prove Lemma 8: pick a sequence of maximal fuzziness which produces a terminal assembly of $S$ which is neither in $B_{F'(n,m), \vec{d}}[S]$ nor in $C_m[S]$. Lemma 77 yields two windows with the same movie; by applying Corollary 72 between them, a production in $P_{\vec{d}}[S]$ appears. Alas, the sequence on which this argument is founded may simply fail to exist: the $\omega$-sequences of maximal fuzziness may fail to reach a terminal production. One could extend them in order to reach a terminal production, but there may not be a sequence of maximal fuzziness among these (ordinal) extensions.
The point of using Connected Treewidth rather than the usual treewidth is that thanks to the connectivity of the bags of $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DEC}[P^\bullet]$, the distances in $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DEC}[P^\bullet]$ reflect those in $P^\bullet$:

- there is a function $r$ such that for any vertex $v \in P^\bullet$, the subtree of the bags of $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DEC}[P^\bullet]$ containing $v$ has a size at most $r(m)$,

- there are two increasing functions $d_m, D_m$ such that for any vertices $u, v$ at distance $\delta$ in $P^\bullet$, any bags $B_u \ni u$ and $B_v \ni v$ of $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DEC}[P^\bullet]$ are at distance $\Delta$ with $d_m(\delta) \le \Delta \le D_m(\delta)$.

As a consequence, if two tiles are far enough in an assembly $A$ and the path between them does not go through the seed, there must be two windows cutting that path which satisfy the hypothesis of Corollary 72.

**Lemma 76.** Let $S$ be an $aTAM$ system with $n$ tiles and $m$ an integer.

Let $\alpha = (A_i)_{i \le o} \in \mathcal{H}^{\text{Free}}[S, A_0]$ be such that for all $i \le o$, squash$(A_i) \notin C_m[S]$.

There is a constant $F(n, m)$ such that if $z, z' \in$ squash$(\lim \alpha) \cdot \text{support } \setminus A_0 \cdot \text{support } A$ such that the distance between $z'$ and $z$ is greater than $F(n, m)$, then there are two windows $A$ and $B$ with $z \in \text{Near}(A)$ and $z' \in \text{Far}(B)$ which satisfy the hypothesis of Lemma 71.

**Proof.** let $W(n, m)$ be the number of movies with $n$ glues on a window of width $2m$. Note that $W(n, m)$ counts the number of arrangements of the edges within the window, the perspective difference between the connected components of the window, as well as the events on the frames of the movie. Let $F(n, m) = d_m^{-1}(W(n, m))$.

Let $P = \lim \alpha$-support. By Lemma 75, pick a tree decomposition $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DEC}[P^\bullet]$ with connected bags of size at most $2m$. Let $\delta$ be the distance between $z$ and $z'$. If $\delta \le F(n, m)$, by the pigeonhole principle, there must be two windows $A$ and $B$ between $z$ and $z'$ with the same movie. ◀

In the case where the assembly embeds into $\mathbb{Z}^2$, one actually controls the direction of the vector between the two windows.

**Lemma 77.** Let $S$ be an $aTAM$ system with $n$ tiles, $m$ an integer and $\vec{d}$ a unit vector.

Let $\alpha = (A_i)_{i \le o} \in \mathcal{H}^{\text{Free}}[S, A_0]$ be such that for all $i \le o$, $A_i$ embeds into $\mathbb{Z}^2$.

There is a constant $F'(n, m)$ such that if $z, z' \in \lim \alpha$-support $\setminus A_0$-support $A$ such that the distance between $(z' - z) \cdot \vec{d} > F(n, m)$, then there are two windows $A$ and $B$ with $z \in \text{Near}(A)$ and $z' \in \text{Far}(B)$ which satisfy the hypothesis of Lemma 71, and such that the vector $\vec{p}$ of the translation between $A$ and $B$ satisfies $\vec{p} \cdot \vec{d} > 1$.

**Proof.** Pick $F'(n, m) = 2F(n, m)+1$. If $(z'-z) \cdot \vec{d} > F'(n, m)$, then in $\mathcal{TR}\mathcal{E}\mathcal{E} - \mathcal{DEC}[\lim \alpha \cdot \text{support}^\bullet]$ there are two bags, one $b$ containing $z$, the other $b' \supseteq z'$ such that on the path between $b_z$ and $b_{z'}$, there are $2F(n, m) + 1$ bags $(b_i)_i$ such that for all $x \in b_i, y \in b_j, (x-y) \cdot \vec{d} \ge i-j$. Thus, there are $i, j$ such $j > i+1$ and two windows, $A$ between $b_i$ and $b_{i+1}$, and $B$ between $b_j$ and $b_{j+1}$ which have the same movie. Because the windows $A$ and $B$ are separated by at least two elements of $(b_i)$, the translation vector $\vec{p}$ between them satisfies $\vec{p} \cdot \vec{d} > 1$. ◂

Thus, there is a simple argument to prove Lemma 8: pick a sequence of maximal fuzziness which produces a terminal assembly of $S$ which is neither in $B_{F'(n,m), \vec{d}}[S]$ nor in $C_m[S]$. Lemma 77 yields two windows with the same movie; by applying Corollary 72 between them, a production in $P_{\vec{d}}[S]$ appears. Alas, the sequence on which this argument is founded may simply fail to exist: the $\omega$-sequences of maximal fuzziness may fail to reach a terminal production. One could extend them in order to reach a terminal production, but there may not be a sequence of maximal fuzziness among these (ordinal) extensions.
Thus, it is necessary to prevent the pesky tendency ordinal sequences have to try and buy time by spacing out their attachment so as to generate an infinite family of sequences with increasing fuzziness. Straight sequences are the answer: they do not thave the wiggle room to misbehave.

**Definition 78.** Let $\mathcal{S}$ be an aTAM system, and $\prec$ an arbitrary total order on its sequences. A free, ordinal assembly sequence $\alpha = (A_i)_{i<\alpha} \in \mathcal{H}^{\text{Free}}[\mathcal{S}]$ is $\prec$-straight if for every pair of window $N, F$ such that $N$ and $F$ have the same movie and $A_0$-support $\subset \text{Near}(N)$, $\alpha$ is the smallest sequence for $\prec$ obtained by applying Corollary 72 to $N$ and $F$ in $\alpha$.

The order $\prec$ is henceforth fixed and left implicit.

**Lemma 79.** Let $\mathcal{S}$ be an aTAM system and $w$ an integer.

Let $X \subset \mathcal{H}^{\text{Free}}[\mathcal{S}]$ be a set of assembly sequences, and $d$ an integer. If for any sequence $\alpha \in X$ and any position $z \in \lim \alpha$-support at distance more than $d$ from $\mathcal{S}$-seed, there are two windows $A$ and $B$ such that $\mathcal{S}$-seed-support $\subset \text{Near}(A)$ and $z \in \text{Far}(B)$, then there is a finite number of straight sequences in $X$.

**Proof.** In this case, each straight sequence in $X$ is determined by what it does in a radius $d$ around $\mathcal{S}$-seed. ◀

**Lemma 80.** Let $\mathcal{S}$ be an aTAM system and $\alpha$ a straight assembly sequence of $\mathcal{S}$. There is a straight assembly sequence $\alpha'$ which extends $\alpha$ such that $\lim \alpha' \in \mathcal{A}_{\square}[\mathcal{S}]$.

**Proof.** Let $P = \lim \alpha$. If $P \notin \mathcal{A}_{\square}[\mathcal{S}]$, then there is an attachment $t@Z$ which is possible in $P$.

Let $\alpha'$ be $\alpha$ with $t@Z$ appended at its end, and $z$ the position of that last attachment in $\lim \alpha'$-support. Assume that $Z$ is such that the distance between $\mathcal{S}$-seed-support and $z$ is minimal.

If there are no pairs of windows $(N, F)$ in $\alpha'$ such that $\text{Far}(F) \subset \text{Far}(N)$, $\mathcal{MOVIE}(N) = \mathcal{MOVIE}(F)$, $\mathcal{S}$-seed-support $\subset \text{Near}(N)$ and $z \in \text{Far}(N)$, then $\alpha'$ is straight.

If there is a pair of windows $(N, F)$ in $\alpha'$ such that $\text{Far}(F) \subset \text{Far}(N)$, $\mathcal{MOVIE}(N) = \mathcal{MOVIE}(F)$, $\mathcal{S}$-seed-support $\subset \text{Near}(N)$ and $z \in \text{Far}(F)$, then by Corollary 73, there is a place closer to $\mathcal{S}$-seed than $z$ where an attachment was possible.

Lastly, if there is a pair of windows $(N, F)$ in $\alpha'$ such that $\text{Far}(F) \subset \text{Far}(N)$, $\mathcal{MOVIE}(N) = \mathcal{MOVIE}(F)$, $\mathcal{S}$-seed-support $\subset \text{Near}(N)$ and $z \in \text{Far}(N) \cap \text{Near}(F)$, then the sequence $\alpha''$ obtained by applying 72 in $\alpha'$ is straight and has $\alpha$ as a prefix, since the new attachments (the repetitions of $t@Z$) are done last in each repetition of the movie in $\text{Far}(N) \cap \text{Near}(F)$.

This process yields a straight assembly sequence $\alpha'$ which strictly extends $\alpha$. It can be repeated until $\lim \alpha' \in \mathcal{A}_{\square}[\mathcal{S}]$. ◀

**Lemma 81.** Let $\mathcal{S}$ be an aTAM system and $\alpha$ a straight assembly sequence of $\mathcal{S}$. Then squash($\alpha$) is a prefix of a straight sequence $\alpha'$. Moreover, the connected treewidth of $\lim \alpha'$-support is no greater than that of $(\lim \text{squash}(\alpha))$-support.

**Proof.** If $\alpha$ embeds cleanly in $\mathbb{Z}^2$, there is nothing to prove since squash($\alpha$) = $\alpha$.

Otherwise, squash($\alpha$) stops because one of its attachment $t@z$ creates a hole which does not exist in $\alpha$. Because $\alpha$ is straight, there cannot be a pair of windows with the same movie separating $\mathcal{S}$-seed and $z$. Applying Corollary 72 in squash($\alpha$) on each pair of windows with the same movie which are the closest to $\mathcal{S}$-seed yields a straight sequence $\alpha'$ extending squash($\alpha$).

Fix a tree decomposition of $(\lim \text{squash}(\alpha))$-support. There is a tree decomposition of $\lim \alpha'$-support where each bag which lies in the far part of a pair of windows with the same
Thus, it is necessary to prevent the pesky tendency ordinal sequences have to try and buy time by spacing out their attachment so as to generate an infinite family of sequences with increasing fuzziness. Straight sequences are the answer: they do not thave the wiggle room to misbehave.

**Definition 78.** Let $\mathcal{S}$ be an aTAM system, and $\prec$ an arbitrary total order on its sequences. A free, ordinal assembly sequence $\alpha = (A_i)_{i<\alpha} \in \mathcal{H}^{\text{Free}}[\mathcal{S}]$ is $\prec$-straight if for every pair of window $N, F$ such that $N$ and $F$ have the same movie and $A_0$-support $\subset \text{Near}(N)$, $\alpha$ is the smallest sequence for $\prec$ obtained by applying Corollary 72 to $N$ and $F$ in $\alpha$.

The order $\prec$ is henceforth fixed and left implicit.

**Lemma 79.** Let $\mathcal{S}$ be an aTAM system and $w$ an integer.

Let $X \subset \mathcal{H}^{\text{Free}}[\mathcal{S}]$ be a set of assembly sequences, and $d$ an integer. If for any sequence $\alpha \in X$ and any position $z \in \lim \alpha$-support at distance more than $d$ from $\mathcal{S}$-seed, there are two windows $A$ and $B$ such that $\mathcal{S}$-seed-support $\subset \text{Near}(A)$ and $z \in \text{Far}(B)$, then there is a finite number of straight sequences in $X$.

**Proof.** In this case, each straight sequence in $X$ is determined by what it does in a radius $d$ around $\mathcal{S}$-seed. ◀

**Lemma 80.** Let $\mathcal{S}$ be an aTAM system and $\alpha$ a straight assembly sequence of $\mathcal{S}$. There is a straight assembly sequence $\alpha'$ which extends $\alpha$ such that $\lim \alpha' \in \mathcal{A}_{\square}[\mathcal{S}]$.

**Proof.** Let $P = \lim \alpha$. If $P \notin \mathcal{A}_{\square}[\mathcal{S}]$, then there is an attachment $t@Z$ which is possible in $P$.

Let $\alpha'$ be $\alpha$ with $t@Z$ appended at its end, and $z$ the position of that last attachment in $\lim \alpha'$-support. Assume that $Z$ is such that the distance between $\mathcal{S}$-seed-support and $z$ is minimal.

If there are no pairs of windows $(N, F)$ in $\alpha'$ such that $\text{Far}(F) \subset \text{Far}(N)$, $\mathcal{MOVIE}(N) = \mathcal{MOVIE}(F)$, $\mathcal{S}$-seed-support $\subset \text{Near}(N)$ and $z \in \text{Far}(N)$, then $\alpha'$ is straight.

If there is a pair of windows $(N, F)$ in $\alpha'$ such that $\text{Far}(F) \subset \text{Far}(N)$, $\mathcal{MOVIE}(N) = \mathcal{MOVIE}(F)$, $\mathcal{S}$-seed-support $\subset \text{Near}(N)$ and $z \in \text{Far}(F)$, then by Corollary 73, there is a place closer to $\mathcal{S}$-seed than $z$ where an attachment was possible.

Lastly, if there is a pair of windows $(N, F)$ in $\alpha'$ such that $\text{Far}(F) \subset \text{Far}(N)$, $\mathcal{MOVIE}(N) = \mathcal{MOVIE}(F)$, $\mathcal{S}$-seed-support $\subset \text{Near}(N)$ and $z \in \text{Far}(N) \cap \text{Near}(F)$, then the sequence $\alpha''$ obtained by applying 72 in $\alpha'$ is straight and has $\alpha$ as a prefix, since the new attachments (the repetitions of $t@Z$) are done last in each repetition of the movie in $\text{Far}(N) \cap \text{Near}(F)$.

This process yields a straight assembly sequence $\alpha'$ which strictly extends $\alpha$. It can be repeated until $\lim \alpha' \in \mathcal{A}_{\square}[\mathcal{S}]$. ◀

**Lemma 81.** Let $\mathcal{S}$ be an aTAM system and $\alpha$ a straight assembly sequence of $\mathcal{S}$. Then squash($\alpha$) is a prefix of a straight sequence $\alpha'$. Moreover, the connected treewidth of $\lim \alpha'$-support is no greater than that of $(\lim \text{squash}(\alpha))$-support.

**Proof.** If $\alpha$ embeds cleanly in $\mathbb{Z}^2$, there is nothing to prove since squash($\alpha$) = $\alpha$.

Otherwise, squash($\alpha$) stops because one of its attachment $t@z$ creates a hole which does not exist in $\alpha$. Because $\alpha$ is straight, there cannot be a pair of windows with the same movie separating $\mathcal{S}$-seed and $z$. Applying Corollary 72 in squash($\alpha$) on each pair of windows with the same movie which are the closest to $\mathcal{S}$-seed yields a straight sequence $\alpha'$ extending squash($\alpha$).

Fix a tree decomposition of $(\lim \text{squash}(\alpha))$-support. There is a tree decomposition of $\lim \alpha'$-support where each bag which lies in the far part of a pair of windows with the same
movie is a copy of the corresponding bag close to the seed. This tree decomposition has the
same width as the decomposition of $(\lim \text{squash}(\alpha)) \cdot \text{support.}$ ▶

At last, all the ingredients are there to prove Lemma 8.

**Proof of Lemma 8.** Define a sequence $(\alpha_i)$ of straight assembly sequences as follows:

- Fix $\alpha_0$ to be the assembly sequence with zero attachments: $\alpha_0 = (S \cdot \text{seed})$;

- for $i$ even, $\alpha_{i+1}$ is a straight extension of $\alpha_i$ which reaches a terminal production;

- for $i$ odd, $\alpha_{i+1}$ is obtained from $\alpha_i$ by Lemma 81.

For every $i$, either $\alpha_{i+1} = \alpha_i$ or $\alpha_{i+1} >_{\text{fz}} \alpha_i$: at the even steps, if $\alpha_{i+1} \neq \alpha_i$, then it is a proper extension and is thus more fizzy; at the odd steps, if $\alpha_i$ does not embed into $\mathbb{Z}^2$, then squash $(\alpha_i)$ is more fizzy, and so is its extension $\alpha_{i+1}$.

If $\mathcal{A}_{\square}[S] \cap C_m[S] = \emptyset$, then for each $i$ odd, the Connected Treewidth of squash $(\alpha_i)$ is at most $2m$, hence the Connected Treewidth of $\alpha_{i+1}$ is at most $2m$. But then, by Lemma 79, $\alpha_i$ can only take a finite number of different values for $i$ even.

Thus, the sequence $(\alpha_i)$ is eventually stationnary, and its fixpoint $\beta$ is a straight sequence which embeds into $\mathbb{Z}^2$ and reaches a terminal production. If $\mathcal{A}_{\square}[S] \cap B_{F(n,m),\vec{d}} \neq \emptyset$, this terminal production must reach at least $F(n,m)$ in direction $\vec{d}$. Thus $\lim \beta$-support has a branch with two windows $A, B$ such that $\mathcal{M}\mathcal{O}\mathcal{V}\mathcal{I}\mathcal{E}(A) = \mathcal{M}\mathcal{O}\mathcal{V}\mathcal{I}\mathcal{E}(B)$ and the vector sending $A$ to $B$ verifies $\vec{p} \cdot \vec{d} > 1$ (by Lemma 77). Because $\beta$ is straight, that branch is periodic, thus $\lim \beta \in \mathcal{A}_{\square}[S] \cap P_{\vec{d}}$. ▶

References

1. Kimberly Barth, David Furcy, Scott M. Summers, and Paul Totzke. Scaled tree fractals do not strictly self-assemble. In *Oscar H. Ibarra, Lila Kari, and Steffen Kopecki, editors, *Unconventional Computation and Natural Computation - 13th International Conference, UCNC 2014, London, ON, Canada, July 14-18, 2014, Proceedings*, volume 8553 of *Lecture Notes in Computer Science*, pages 27–39. Springer. URL: https://doi.org/10.1007/978-3-319-08123-6_3, doi:10.1007/978-3-319-08123-6_3.

2. Matthew Cook, Yunhui Fu, and Robert Schweller. Temperature 1 self-assembly: Deterministic assembly in 3d and probabilistic assembly in 2d. In *Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 570–589. Society for Industrial and Applied Mathematics. URL: https://epubs.siam.org/doi/10.1137/1.9781611973082.45, doi:10.1137/1.9781611973082.45.

3. Reinhard Diestel and Malte Müller. Connected tree-width. *Comb.*, 38(2):381–398, 2018. URL: https://doi.org/10.1007/s00493-016-3516-5, doi:10.1007/S00493-016-3516-5.

4. Bruno Durand, Andrei E. Romashchenko, and Alexander Shen. Fixed-point tile sets and their applications. *J. Comput. Syst. Sci.*, 78(3):731–764, 2012. URL: https://doi.org/10.1016/j.jcss.2011.11.001, doi:10.1016/J.JCSS.2011.11.001.

5. Jérôme Durand-Lose, Jacob Hendricks, Matthew J. Patitz, Ian Perkins, and Michael Sharp. Self-assembly of 3-d structures using 2-d folding tiles. *Nat. Comput.*, 19(2):337–355, 2020. URL: https://doi.org/10.1007/s11047-019-09751-9, doi:10.1007/S11047-019-09751-9.

6. David Furcy and Scott M. Summers. Scaled pier fractals do not strictly self-assemble. *16(2):317–338*, doi:10.1007/s11047-015-9528-z.

7. Chaim Goodman-Strauss. Matching rules and substitution tilings. *Annals of Mathematics*, pages 181–223, 1998.

8. Daniel Hader, Matthew Patitz, and Scott Summers. Fractal dimension of assemblies in the abstract tile assembly model. *Natural Computing*, pages 1–16, 04 2023. doi:10.1007/s11047-023-09942-5.
movie is a copy of the corresponding bag close to the seed. This tree decomposition has the
same width as the decomposition of $(\lim \text{squash}(\alpha)) \cdot \text{support.}$ ▶

At last, all the ingredients are there to prove Lemma 8.

**Proof of Lemma 8.** Define a sequence $(\alpha_i)$ of straight assembly sequences as follows:

- Fix $\alpha_0$ to be the assembly sequence with zero attachments: $\alpha_0 = (S \cdot \text{seed})$;

- for *i* even, $\alpha_{i+1}$ is a straight extension of $\alpha_i$ which reaches a terminal production;

- for *i* odd, $\alpha_{i+1}$ is obtained from $\alpha_i$ by Lemma 81.

For every *i*, either $\alpha_{i+1} = \alpha_i$ or $\alpha_{i+1} >_{\text{fz}} \alpha_i$: at the even steps, if $\alpha_{i+1} \neq \alpha_i$, then it is a proper extension and is thus more fizzy; at the odd steps, if $\alpha_i$ does not embed into $\mathbb{Z}^2$, then squash $(\alpha_i)$ is more fizzy, and so is its extension $\alpha_{i+1}$.

If $\mathcal{A}_{\square}[S] \cap C_m[S] = \emptyset$, then for each $i$ odd, the Connected Treewidth of squash $(\alpha_i)$ is at most $2m$, hence the Connected Treewidth of $\alpha_{i+1}$ is at most $2m$. But then, by Lemma 79, $\alpha_i$ can only take a finite number of different values for $i$ even.

Thus, the sequence $(\alpha_i)$ is eventually stationnary, and its fixpoint $\beta$ is a straight sequence which embeds into $\mathbb{Z}^2$ and reaches a terminal production. If $\mathcal{A}_{\square}[S] \cap B_{F(n,m),\vec{d} \neq \emptyset}$, this terminal production must reach at least $F(n,m)$ in direction $\vec{d}$. Thus $\lim \beta$-support has a branch with two windows $A, B$ such that $\mathcal{M}\mathcal{O}\mathcal{V}\mathcal{I}\mathcal{E}(A) = \mathcal{M}\mathcal{O}\mathcal{V}\mathcal{I}\mathcal{E}(B)$ and the vector sending $A$ to $B$ verifies $\vec{p} \cdot \vec{d} > 1$ (by Lemma 77). Because $\beta$ is straight, that branch is periodic, thus $\lim \beta \in \mathcal{A}_{\square}[S] \cap P_{\vec{d}}$. ◀

References

1. Kimberly Barth, David Furcy, Scott M. Summers, and Paul Totzke. Scaled tree fractals do not strictly self-assemble. In *Oscar H. Ibarra, Lila Kari, and Steffen Kopecki, editors, Unconventional Computation and Natural Computation - 13th International Conference, UCNC 2014, London, ON, Canada, July 14-18, 2014, Proceedings*, volume 8553 of *Lecture Notes in Computer Science*, pages 27–39. Springer. URL: https://doi.org/10.1007/978-3-319-08123-6_3, doi:10.1007/978-3-319-08123-6_3.

2. Matthew Cook, Yunhui Fu, and Robert Schweller. Temperature 1 self-assembly: Deterministic assembly in 3d and probabilistic assembly in 2d. In *Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 570–589. Society for Industrial and Applied Mathematics. URL: https://epubs.siam.org/doi/10.1137/1.9781611973082.45, doi:10.1137/1.9781611973082.45.

3. Reinhard Diestel and Malte Müller. Connected tree-width. *Comb.*, 38(2):381–398, 2018. URL: https://doi.org/10.1007/s00493-016-3516-5, doi:10.1007/S00493-016-3516-5.

4. Bruno Durand, Andrei E. Romashchenko, and Alexander Shen. Fixed-point tile sets and their applications. *J. Comput. Syst. Sci.*, 78(3):731–764, 2012. URL: https://doi.org/10.1016/j.jcss.2011.11.001, doi:10.1016/J.JCSS.2011.11.001.

5. Jérôme Durand-Lose, Jacob Hendricks, Matthew J. Patitz, Ian Perkins, and Michael Sharp. Self-assembly of 3-d structures using 2-d folding tiles. *Nat. Comput.*, 19(2):337–355, 2020. URL: https://doi.org/10.1007/s11047-019-09751-9, doi:10.1007/S11047-019-09751-9.

6. David Furcy and Scott M. Summers. Scaled pier fractals do not strictly self-assemble. *16(2):317–338*, doi:10.1007/s11047-015-9528-z.

7. Chaim Goodman-Strauss. Matching rules and substitution tilings. *Annals of Mathematics*, pages 181–223, 1998.

8. Daniel Hader, Matthew Patitz, and Scott Summers. Fractal dimension of assemblies in the abstract tile assembly model. *Natural Computing*, pages 1–16, 04 2023. doi:10.1007/s11047-023-09942-5.
9 Daniel Hader, Matthew J. Patitz, and Scott M. Summers. Fractal dimension of assemblies in the abstract tile assembly model. In Irina Kostitsyna and Pekka Orponen, editors, *Unconventional Computation and Natural Computation*, Lecture Notes in Computer Science, pages 116–130. Springer International Publishing. doi:10.1007/978-3-030-87993-8_8.

10 Jacob Hendricks, Meagan Olsen, Matthew J. Patitz, Trent A. Rogers, and Hadley Thomas. Hierarchical self-assembly of fractals with signal-passing tiles. 17(1):47–65. URL: https://doi.org/10.1007/s11047-017-9663-9, doi:10.1007/S11047-017-9663-9.

11 Jacob Hendricks, Joseph Opseth, Matthew J. Patitz, and Scott M. Summers. Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals. 19(2):357–374. doi:10.1007/s11047-019-09777-z.

12 Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers, and Scott M. Summers. The power of duples (in self-assembly): It's not so hip to be square. 743:148–166. URL: https://doi.org/10.1016/j.tcs.2015.12.008, doi:10.1016/J.TCS.2015.12.008.

13 Steven M. Kautz and Brad Shutters. Self-assembling rulers for approximating generalized sierpinski carpets. 67(2):207–233. URL: https://doi.org/10.1007/s00453-012-9691-x, doi:10.1007/S00453-012-9691-X.

14 Paul Lafargue. *Le droit à la paresse*. La découverte, la découverte edition.

15 James I. Lathrop, Jack H. Lutz, and Scott M. Summers. Strict self-assembly of discrete sierpinski triangles. *Theor. Comput. Sci.*, 410(4-5):384–405, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.062, doi:10.1016/J.TCS.2008.09.062.

16 Jack H. Lutz and Brad Shutters. Approximate self-assembly of the sierpinski triangle. 51(3):372–400. URL: https://doi.org/10.1007/s00224-011-9345-4, doi:10.1007/S00224-011-9345-4.

17 Pierre-Etienne Meunier, Matthew J. Patitz, Scott M. Summers, Guillaume Theyssier, Andrew Winslow, and Damien Woods. Intrinsic universality in tile self-assembly requires cooperation. In *Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms*, SODA '14, pages 752–771. Society for Industrial and Applied Mathematics.

18 Shahar Mozes. Tilings, substitution systems and dynamical systems generated by them. *Journal d'analyse mathématique*, 53(1):139–186, 1989.

19 Matthew J. Patitz. An introduction to tile-based self-assembly and a survey of recent results. *Nat. Comput.*, 13(2):195–224, 2014. URL: https://doi.org/10.1007/s11047-013-9379-4, doi:10.1007/S11047-013-9379-4.

20 Matthew J. Patitz and Scott M. Summers. Self-assembly of discrete self-similar fractals. 9(1):135–172. URL: https://doi.org/10.1007/s11047-009-9147-7, doi:10.1007/S11047-009-9147-7.

21 Erik Winfree, Furong Liu, Lisa A. Wenzler, and Nadrian C. Seeman. Design and self-assembly of two-dimensional DNA crystals. 394(6693):539–544. Publisher: Nature Publishing Group UK London. URL: https://www.nature.com/articles/28998.
9 Daniel Hader, Matthew J. Patitz, and Scott M. Summers. Fractal dimension of assemblies in the abstract tile assembly model. In Irina Kostitsyna and Pekka Orponen, editors, *Unconventional Computation and Natural Computation*, Lecture Notes in Computer Science, pages 116–130. Springer International Publishing. doi:10.1007/978-3-030-87993-8_8.

10 Jacob Hendricks, Meagan Olsen, Matthew J. Patitz, Trent A. Rogers, and Hadley Thomas. Hierarchical self-assembly of fractals with signal-passing tiles. 17(1):47–65. URL: https://doi.org/10.1007/s11047-017-9663-9, doi:10.1007/S11047-017-9663-9.

11 Jacob Hendricks, Joseph Opseth, Matthew J. Patitz, and Scott M. Summers. Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals. 19(2):357–374. doi:10.1007/s11047-019-09777-z.

12 Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers, and Scott M. Summers. The power of duples (in self-assembly): It's not so hip to be square. 743:148–166. URL: https://doi.org/10.1016/j.tcs.2015.12.008, doi:10.1016/J.TCS.2015.12.008.

13 Steven M. Kautz and Brad Shutters. Self-assembling rulers for approximating generalized sierpinski carpets. 67(2):207–233. URL: https://doi.org/10.1007/s00453-012-9691-x, doi:10.1007/S00453-012-9691-X.

14 Paul Lafargue. *Le droit à la paresse*. La découverte, la découverte edition.

15 James I. Lathrop, Jack H. Lutz, and Scott M. Summers. Strict self-assembly of discrete sierpinski triangles. *Theor. Comput. Sci.*, 410(4-5):384–405, 2009. URL: https://doi.org/10.1016/j.tcs.2008.09.062, doi:10.1016/J.TCS.2008.09.062.

16 Jack H. Lutz and Brad Shutters. Approximate self-assembly of the sierpinski triangle. 51(3):372–400. URL: https://doi.org/10.1007/s00224-011-9345-4, doi:10.1007/S00224-011-9345-4.

17 Pierre-Etienne Meunier, Matthew J. Patitz, Scott M. Summers, Guillaume Theyssier, Andrew Winslow, and Damien Woods. Intrinsic universality in tile self-assembly requires cooperation. In *Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms*, SODA '14, pages 752–771. Society for Industrial and Applied Mathematics.

18 Shahar Mozes. Tilings, substitution systems and dynamical systems generated by them. *Journal d'analyse mathématique*, 53(1):139–186, 1989.

19 Matthew J. Patitz. An introduction to tile-based self-assembly and a survey of recent results. *Nat. Comput.*, 13(2):195–224, 2014. URL: https://doi.org/10.1007/s11047-013-9379-4, doi:10.1007/S11047-013-9379-4.

20 Matthew J. Patitz and Scott M. Summers. Self-assembly of discrete self-similar fractals. 9(1):135–172. URL: https://doi.org/10.1007/s11047-009-9147-7, doi:10.1007/S11047-009-9147-7.

21 Erik Winfree, Furong Liu, Lisa A. Wenzler, and Nadrian C. Seeman. Design and self-assembly of two-dimensional DNA crystals. 394(6693):539–544. Publisher: Nature Publishing Group UK London. URL: https://www.nature.com/articles/28998.