On the knot quandle of the twist-spun trefoil

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 $1\le m\le 5$. 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.