Single marble stacking operation that fills out a 3-dimensional space?

Question

My question is:

Is there a way to stack marbles by using only a single one-marble stacking operation such that an infinite 3-dimensional stack is constructed?

For example:

In 1-dimension one can start with a stack such as

-o-o-o-o

and by adding a marble -o infinitely, with a single operation of attaching it from the right, one can construct an infinite 1-dimensional stack.

In 2-dimensions, one can start with a stack such as

 o-o-o-o-o-o
/| | | | | |
 o-o-o-o-o-o-o-o-o-o-o-o-o
/| | | | | | | | | | | | |/
 o-o-o-o-o-o-o-o-o-o-o-o-o
/| | | | | | | | | | | | |/
 o-o-o-o-o-o-o-o-o-o-o-o-o
/| | | | | | | | | | | | |

where the last marble on the right in each row is connected to the first marble on the left one row higher. With this, the single operation of adding a marble with two connections:

-o
 |

can be used infinitely to construct an infinite homogeneous 2-dimensional stack of marbles. (Note that there are no edge effects as we pass from one row to the next. The rows themselves are an artifact of the visualization. After adding one marble to the stack above, we can rearrange our viewpoint by pushing all marbles one position to the left and arrive at the original viewpoint, but the stack is now one marble longer.)

Can something similar be done for a three-dimensional stack as well? If yes, how does the stack and the stacking operation look like?

Answer 1

A more usual way to stack the marbles in 2D is $$\begin{align} &6\ \ 9\\&3\ \ 5\ \ 8\\&1\ \ 2\ \ 4\ \ 7\end {align}$$, putting a new one on the end and going up diagonally to the left. The same idea works in any number of dimensions. In 3D you would place $1,2,3$ as shown, then place $4$ in front of $1$, and continue.

Answer 2

I'm going to offer a slightly different mathematical footing for this question that gives it both a positive (locally) and a negative (globally) answer.

One useful model for thinking about structures like this is the notion of a Cayley graph for (a presentation of) a group; the nodes of the graph are the elements of the group, and the links between nodes represent the generators of the presentation. For instance, the 'classic' presentation of the group $\mathbb{Z}$ is literally just $\langle a\rangle$; the group has a single generator, and there are no relations between any of the elements. The elements are all of the form $a^n, n\in\mathbb{Z}$, and we associate the element $a^n$ with the number $n$. The Cayley graph for this presentation is just the (doubly infinite) line of nodes. The standard presentation of $\mathbb{Z}^2$ is $\langle a,b\mid ab=ba\rangle$; two generators that commute with each other, with the group elements of the form $a^mb^n$ for $m,n\in\mathbb{Z}$. The Cayley graph for this presentation is the infinite two-dimensional graph, with each node $a^mb^n$ connected to the nodes $a^{m\pm1}b^n$ and $a^mb^{n\pm1}$; the relation $ab=ba$ corresponds to the fact that starting from the node $a^mb^n$ and taking an $a$-step to $a^{m+1}b^n$ and then a $b$-step to $a^{m+1}b^{n+1}$ — that is, going along the bottom and right edges of a square — brings you to the same place as taking a $b$-step and then an $a$-step. From this perspective, the squares in the Cayley graph represent the commutativity of the generators.

Cayley graphs automatically give the sort of 'uniformity' that you're looking for, because the neighborhood around each node of the graph is identical to the neighborhood around any other node.

Now, we can think of your construction as a Cayley graph for a different presentation of the group $\mathbb{Z}$. The fact that the group is $\mathbb{Z}$ corresponds to your notion of adding elements 'one by one' — there's a single chain through the entire structure that can be followed. But when you connect e.g. the seventh element in the string back to the first, you're introducing a second generator $b$ (which represents the 'y connection'), and a relation of the form $b=a^7$ (seven steps 'around' the ring is equivalent to one step 'up'). This means that the Cayley graph for $\mathbb{Z}$ with the presentation $\langle a,b\mid b=a^7\rangle$ is a cylinder, with a chain of elements spiraling up the cylinder. The idea that this is a 'two-dimensional' grid comes from the fact that there are two generators; and in fact, it's 'locally indistinguishable' from $\mathbb{Z}\times\mathbb{Z}$ in any patch that doesn't go so far as to actually show the 'wrap' represented by the $b=a^7$ relation (i.e., any strip that's less than seven elements wide). Note that we could add a relation of the form $ab=ba$ to make the squares explicit, but we actually get this 'for free'; it's already implied by the relation $b=a^7$. By taking larger and larger values of $7$, you can make this local indistinguishability happen over a larger and larger patch.

From this perspective, what you're looking for is a presentation of $\mathbb{Z}$ that's locally indistinguishable from $\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$; every node in the Cayley graph will have two links each along three different axes, that form commuting cubes. This corresponds to the presentation $\langle a,b,c\mid ab=ba, ac=ca, bc=cb\rangle$ for $\mathbb{Z}^3$. Now, similarly to how we wrapped the cylinder by adding a relation $b=a^n$, we can consider adding two distinct relations; one for $b=a^m$ will mean that every $m$ steps in the 'x' direction ($a$), we'll take one 'y' step ($b$). Likewise, a relation $c=b^n$ will mean that every $n$ steps in the 'y' direction ($b$), we'll take one 'z' step ($c$). Since $b^n=(a^m)^n=a^{(mn)}$, every $mn$ 'x' steps will also correspond to that one 'z' step. Essentially, this corresponds to filling in the rows of an $m\times n$ grid, one by one; the end of one row is linked to the start of the next exactly analagous to the 2d case, but then after we've done $n$ rows' worth of this, we wrap again, back to the start of the grid, one level higher (the 'z' step). So a presentation of the form $\langle a,b,c\mid b=a^m, c=b^n\rangle$ gives a 3d analog of your cylindrical spiral; as long as your patch is smaller than $m\times n$, you won't have any links that let you distinguish it from a 'real' $\mathbb{Z}^3$ grid (that is, these relations are minimal in that sense; a single $b$ step can't be represented with fewer than $m$ $a$ steps, and a single $c$ step can't be represented with fewer than $n$ $b$ steps, or with any combination of $a$ and $b$ steps smaller than that).

So given this, you can take $m$ and $n$ as large as you want, and get presentations of $\mathbb{Z}$ that look locally like $\mathbb{Z}^3$ for larger and larger ranges of locality; and of course, there's nothing that restricts us to three dimensions here, so you can get a similar 'locally $\mathbb{Z}^d$' presentation for $\mathbb{Z}$ for any dimension $d$. But these presentations always have to be local; there will always have to be some loop in your structure. This is because the groups $\mathbb{Z}$ and $\mathbb{Z}^2$ (and $\mathbb{Z}^3$, etc.) are all distinct from each other (which is a very nontrivial statement!), and two distinct groups will always have distinct Cayley graphs, whatever presentation you give them.