# Zoom into the classical Laver table fractals

At this link, you may zoom into the fractal representing the classical Laver tables.

For all $n$, let $L_{n}:\{0,\ldots,2^{n-1}\}\rightarrow\{0,\ldots,2^{n-1}\}$ be the mapping that reverses the digits in the binary expansion of a natural number. Let $L_{n}^{\sharp}:\{1,\ldots,2^{n}\}\rightarrow\{1,\ldots,2^{n}\}$ be the mapping where
$L_{n}^{\sharp}(x)=L_{n}(x-1)+1$. Let $\#_{n}$ be the operation on $\{1,\ldots,2^{n}\}$ defined by $x\#_{n}y=L_{n}^{\sharp}(L_{n}^{\sharp}(x)*_{n}L_{n}^{\sharp}(y))$ where $*_{n}$ is the classical Laver table operation on $\{1,\ldots,2^{n}\}$.

Let $C_{n}=\{(\frac{x}{2^{n}},\frac{x\#_{n}y}{2^{n}})\mid x,y\in\{1,\ldots,2^{n}\}\}$. Then $C_{n}$ is a subset of $[0,1]\times[0,1]$ and the sets $C_{n}$ converge in the Hausdorff metric to a compact subset $C\subseteq[0,1]\times[0,1]$. The link that I gave gives images of $C$ that you may zoom in to.

Since $A_{48}$ is still the largest classical Laver table ever computed, we are only able to zoom into $C$ with $2^{48}\times 2^{48}$ resolution (which is about 281 trillion by 281 trillion so we can see microscopic detail).

As I kind of expected, these images of the classical Laver tables are quite tame compared to the wildness of the final matrix which one obtains from the generalized Laver tables $(A^{\leq 2^{n}})^{+}$; the generalized Laver tables give more fractal-like images while the classical Laver tables give more geometric images. I conjecture that the set $C$ has Hausdorff dimension $1$ though I do not have a proof. The simplicity of these images of the classical Laver tables gives some hope for computing the classical Laver tables past even $A_{96}$.

Some regions in the set $C$ may look to be simply smooth vertical or diagonal lines, but if there exists a rank-into-rank cardinal, then every single neighborhood in $C$ has fractal features if you zoom in far enough (I suspect that you will need to zoom in for a very very long time before you see any fractal features and I also suspect that you will need to zoom into the right location to see the fractal behavior).