  If $f:\mathbb{R}^3 \to \mathbb{R}^3$ is a uniformly quasiregular mapping with
Julia set $J(f)$ a genus $g$ Cantor set, for $g\geq 1$, then for any linearizer
$L$ at any repelling periodic point of $f$, the fast escaping set $A(L)$
consists of a spiders' web structure containing embedded genus $g$ tori on any
sufficiently large scale. In other words, $A(L)$ contains a spiders' web of
doughnuts. This type of structure is specific to higher dimensions, and cannot
happen for the fast escaping set of a transcendental entire function in the
plane. We also show that if $f:\mathbb{R}^n \to \mathbb{R}^n$ is uqr, for
$n\geq 2$ and $J(f)$ is a Cantor set, then every periodic point is in $J(f)$
and is repelling.
