  We show that the knot quandle of the $3$-, $4$-, or $5$-twist-spun trefoil is
isomorphic to a quandle related to the $16$-, $24$-, or $600$-cell
respectively. We further show that the cardinality of the knot quandle of the
$m$-twist-spun trefoil is finite if and only if $1 \leq m \leq 5$. This
phenomenon is attributable to the fact that the regular tessellation $\{ 3, m
\}$, in the sense of the Schl\"{a}fli symbol, consists of infinite triangles if
$m$ is greater than or equal to 6.
