  For a prime number $p$ and any natural number $n$ we introduce, by giving an
explicit recursive formula, the $p$-Jones-Wenzl projector
${}^p\operatorname{JW}_n$, an element of the Temperley-Lieb algebra $TL_n(2)$
with coefficients in ${\mathbb F}_p$. We prove that these projectors give the
indecomposable objects in the $\tilde{A}_1$-Hecke category over ${\mathbb
F}_p$, or equivalently, they give the projector in
$\mathrm{End}_{\mathrm{SL}_2(\overline{{\mathbb F}_p})}(({\mathbb
F}_p^2)^{\otimes n})$ to the top tilting module. The way in which we find these
projectors is by categorifying the fractal appearing in the expression of the
$p$-canonical basis in terms of the Kazhdan-Lusztig basis for $\tilde{A}_1$.
