3 edited body

I am wondering if it is hopeless to obtain any firm results on the following model of a "random polycube shape." First, a polycube in $\mathbb{R}^3$ is a connected face-to-face gluing of unit cubes. (This is a term prominent in Computer Graphics.) By a random polycube shape I mean the following. Start with an $n \times n \times n$ polycube, forming a cube of side length $n$. For example, for $n=3$, we start with $3^3=27$ unit cubes.

Now iterate the following process: (1) Identify a random exposed cube face. (2) Adjoin a new cube there. (3) Remove a randomly selected cube on the boundary of the shape (i.e., a cube with at least one exposed face), but only if the resulting polycube remains connected (in the face-to-face dual). So the shape grows by one cube and shrinks by one cube, therefore always maintaining $n^3$ cubes, and always maintaining connectivity.

The (distracting!) animation below (a snapshot every 100 iterations over 10000 iterations) may not animate in your browser (Also, it, at points, wanders "off-screen"—double apology!):

I am interested in even gross parameters: What is the mean diameter $d$ (longest cube-to-cube path in the dual) of the shape? How does the genus $g$ grow as a function of $n$? Presumably both $d \rightarrow \infty$ and $g \rightarrow \infty$ as $n \rightarrow \infty$, but it might be difficult to determine the rates of growth. Pointers to relevant related literature would be appreciated. Thanks!

The (distracting!) animation below (a snapshot every 100 iterations over 10000 iterations) may not animate in your browser (Also, it, at points, wanders "off-screen"—double apology!):

2 URL correction

I am wondering if it is hopeless to obtain any firm results on the following model of a "random polycube shape." First, a polycube in $\mathbb{R}^3$ is a connected face-to-face gluing of unit cubes. (This is a term prominent in Computer Graphics.) By a random polycube shape I mean the following. Start with an $n \times n \times n$ polycube, forming a cube of side length $n$. For example, for $n=3$, we start with $3^3=27$ unit cubes.

Now iterate the following process: (1) Identify a random exposed cube face. (2) Adjoin a new cube there. (3) Remove a randomly selected cube on the boundary of the shape (i.e., a cube with at least one exposed face), but only if the resulting polycube remains connected (in the face-to-face dual). So the shape grows by one cube and shrinks by one cube, therefore always maintaining $n^3$ cubes, and always maintaining connectivity.

The (distracting!) animation below (a snapshot every 100 iterations over 10000 iterations) may not animate in your browser (sorry!): Also, it, at points, wanders "off-screen"—double apology!):

I am interested in even gross parameters: What is the mean diameter $d$ (longest cube-to-cube path in the dual) of the shape? How does the genus $g$ grow as a function of $n$? Presumably both $d \rightarrow \infty$ and $g \rightarrow \infty$ as $n \rightarrow \infty$, but it might be difficult to determine the rates of growth. Pointers to relevant related literature would be appreciated. Thanks!

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# Random polycube shapes

I am wondering if it is hopeless to obtain any firm results on the following model of a "random polycube shape." First, a polycube in $\mathbb{R}^3$ is a connected face-to-face gluing of unit cubes. (This is a term prominent in Computer Graphics.) By a random polycube shape I mean the following. Start with an $n \times n \times n$ polycube, forming a cube of side length $n$. For example, for $n=3$, we start with $3^3=27$ unit cubes.

Now iterate the following process: (1) Identify a random exposed cube face. (2) Adjoin a new cube there. (3) Remove a randomly selected cube on the boundary of the shape (i.e., a cube with at least one exposed face), but only if the resulting polycube remains connected (in the face-to-face dual). So the shape grows by one cube and shrinks by one cube, therefore always maintaining $n^3$ cubes, and always maintaining connectivity.

The animation below (a snapshot every 100 iterations over 10000 iterations) may not animate in your browser (sorry!):

I am interested in even gross parameters: What is the mean diameter $d$ (longest cube-to-cube path in the dual) of the shape? How does the genus $g$ grow as a function of $n$? Presumably both $d \rightarrow \infty$ and $g \rightarrow \infty$ as $n \rightarrow \infty$, but it might be difficult to determine the rates of growth. Pointers to relevant related literature would be appreciated. Thanks!