  In this paper we present novel $ADE$ correspondences by combining an earlier
induction theorem of ours with one of Arnold's observations concerning
Trinities, and the McKay correspondence. We first extend Arnold's indirect link
between the Trinity of symmetries of the Platonic solids $(A_3, B_3, H_3)$ and
the Trinity of exceptional 4D root systems $(D_4, F_4, H_4)$ to an explicit
Clifford algebraic construction linking the two ADE sets of root systems
$(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$ and $(I_2(n), I_2(n)\times I_2(n),
D_4, F_4, H_4)$. The latter are connected through the McKay correspondence with
the ADE Lie algebras $(A_n, D_n, E_6, E_7, E_8)$. We show that there are also
novel indirect as well as direct connections between these ADE root systems and
the new ADE set of root systems $(I_2(n), A_1\times I_2(n), A_3, B_3, H_3)$,
resulting in a web of three-way ADE correspondences between three ADE sets of
root systems.
