IU Results in Diffusion-Restricted and Directed aTAMs

An \(n\)-dimensional TAS is diffusion-restricted if and only if the paths along which attaching tiles diffuse are confined to the \(n\)-plane containing its assembly.

Let \(\mathcal{T}\) be a diffusion-restricted 2D TAS. We call \(\mathcal{T}\) "planar" because diffusion-paths are embedded in the plane. Suppose over the course of its growth \(\mathcal{T}\)'s assembly comes to enclose a tileless region. From the time of enclosure onward it will be impossible for a tile to attach within the enclosed region, as the confinement of its path of diffusion to the plane would force it to move from the outside of the assembly through a wall of existing tiles, which is impossible.

Contrast this to the regular 2D TAS wherein tiles are permitted to attach within an enclosed region by diffusing into it from the third dimension along an axis orthogonal to the plane.

The 3D diffusion-restricted TAS, or "spatial TAS", prohbits attachment within enclosed volumes, while the regular 3D TAS permits it via the fourth dimension.

Without added constraint the general 1D aTAM is diffusion-restricted. This is because the 1D aTAM cannot enclose a nonzero length, as doing so would require there to be tiles on either side of the enclosed space, and one side could only have been reached from the other by a bridge of tiles filling the gap between.

It has been shown that the 1D diffusion-restricted aTAM and the planar aTAM are not intrinsically universal. It has been shown that the spatial aTAM is intrinsically universal.