As the question title asks for, how do others "visualize" Witt vectors? I just think of them as algebraic creatures. Bonus points for pictures.



How to visualize a Witt vector?
You can view the spectrum of the ring Witt vectors, in the sense of scheme theory, pretty reasonably.


If $R$ is $p$-torsion free, then $\mathrm{Spec}(W\_n(R))$ is $n+1$ copies (or $n$ if you use the traditional indexing) of $\mathrm{Spec}(R)$ glued together in a suitable way along their fibers over $p$. But there are two qualifications. First, there is some "Frobenius twisting" involved, which is impossible to visualize because the Frobenius morphism is impossible to visualize, as far as I know. Second, each component is not simply glued transversally to the previous ones, but there is some higher order gluing.


A simple example is $\mathrm{Spec}(W\_n(\mathbf{Z}))$. It consists of $n+1$ copies of $\mathrm{Spec}(\mathbf{Z})$, numbered $0$ to $n$, where the $k$-th copy is glued to the $(k-1)$-st copy modulo $p^k$. So copy $1$ is glued to copy $0$ transversally. Copy $2$ is glued to copy $1$ tangentially, but only to order $1$, and so on.


There is a little picture of $W\_1$ on page 5 of my paper [The basic geometry of Witt vectors, I. The affine case](https://arxiv.org/abs/0801.1691).


