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

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    $\begingroup$ Do you mean an element? the ring of Witt vectors? it's spectrum? And over a field, I guess... $\endgroup$ – Xarles Jul 15 '18 at 7:38
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    $\begingroup$ Here is a picture on Wikipedia: en.wikipedia.org/wiki/P-adic_number#/media/… $\endgroup$ – Jason Starr Jul 15 '18 at 9:12
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    $\begingroup$ Do you mean the $p$-typical Witt vectors of a perfect ring in characteristic $p$, or something more general? The $p$-typical Witt vectors of $\mathbb F_q$ are the unique unramified extension of $\mathbb Z_p$ with residue field $\mathbb F_q$, which I think is pretty explicit (but bigger examples become more annoying to compute). $\endgroup$ – R. van Dobben de Bruyn Jul 15 '18 at 10:10
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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.

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