The rotational symmetry group of the $n$-simplex (not permitting reflections) is always the alternating group $A_{n+1}$. When $n=4$, this coincides with the rotational symmetry group of the icosahedron and dodecahedron, which is also $A_5$. (I believe this is the only "exceptional isomorphism" between symmetry groups of regular polytopes in any dimension, i.e. one besides the isomorphism between dual polytopes.)
Obviously one can prove this fact by showing that each group happens to be the alternating group on $5$ elements, but I am curious whether there is a geometric approach to exhibiting the isomorphism - some natural bijection between rotations of the $4$-simplex and rotations of the icosahedron which shows they exhibit the same multiplicative structure, without determining what that structure is.