Assembly

Informal Description

An assembly generally refers to the structure created from tiles binding together.

Definition

An assembly in \(n\)-dimensional space (most commonly taken to be \(n=2\) or \(n=3\)) is a partial function \({\alpha}:{\mathbb{Z}^n} \dashrightarrow {T}\).

Tau-stable Assembly

An assembly is \(\tau\)-stable if it cannot be broken up into smaller assemblies without breaking bonds of total strength at least \(\tau\) for some \(\tau \in \mathbb{N}\). For example, in models such as the Abstract Tile Assembly Model (aTAM) and the Two-Handed Assembly Model (2HAM) all assemblies are \(\tau\) stable where \(\tau\) is the Temperature of the system, but in models such as the Kinetic Tile Assembly Model (kTAM) there can exist an assembly that is not \(\tau\)-stable.

Seed Assembly

In a seeded tile assembly model, a seed assembly refers to the initial structure from which growth begins.

Terminal Assembly

Let \(\mathcal{T} = (T, \sigma, \tau)\) be a TAS. An assembly \(\alpha \in \mathcal{A}[\mathcal{T}]\) is terminal, and we write \(\alpha \in \mathcal{A}_{\Box}[\mathcal{T}]\), if no tile can be \(\tau\)-stably added to it. It is clear that \(\mathcal{A}_{\Box}[\mathcal{T}] \subset \mathcal{A}[\mathcal{T}]\), where \(\mathcal{A}[\mathcal{T}]\) is the set of all assemblies that can arise from \(\mathcal{T}\). Note that similar definitions of terminal assembly hold for other tile assembly models such as the 2HAM and derivatives of the 2HAM and aTAM.

Assembly Sequence

An assembly sequence in a TAS \(\mathcal{T}\) is a (finite or infinite) sequence \(\vec{\alpha} = (\alpha_0,\alpha_1,\ldots)\) of assemblies in which each \(\alpha_{i+1}\) is obtained from \(\alpha_i\) by the addition of a single tile. The result \(\res{\vec{\alpha}}\) of such an assembly sequence is its unique limiting assembly. (This is the last assembly in the sequence if the sequence is finite.)