This is a follow-up paper to the report [R.~T.~Živaljević and D.~R.~Ži\-va\-lje\-vić, {ı Icosahedron and a paper dragon}, The Teaching of Mathematics {\bf 28}, 2 (2025), 118--124] on an animated mathematical experiment (simulation) involving the icosahedron. The basic idea of the experiment was to create the simplest possible combinatorial geometric environment, for studying the mathematics behind the morphogenesis of icosahedral shapes in nature. Our objective is to present, in the form accessible to students, teachers, and non-specialists, some of the not so well-known facts about the geometry and combinatorics of the icosahedron, related to this mathematical simulation, emphasizing the unity of mathematics and the importance of the multidisciplinary approach in mathematical education.