Leonardo polyhedron

Leonardo polyhedron is a polyhedron with a Platonic solid's rotational symmetry and has genus ${\displaystyle g\geq 2}$. Here, a polyhedron is the unbounded 2-manifold embedded in three-dimensional Euclidean space. The polyhedron is named after Leonardo da Vinci, who illustrated geometrical shapes in Luca Pacioli's De divina proportione in three phases: drawing Platonic solids and Archimedean solids; replacing the edges of those solids by struts, forming a convex polygon, and this results in the first polyhedron with many genera; and placing each hole with the skeleton of a pyramid.^[1]

Alicia Boole Stott discovered the first regular Leonardo polyhedron (its property has transitivity by the set consisting of vertex, edge, and face of a polyhedron). Similar to Leonardo's work, she began the construction with a four-dimensional polytope, projecting to a Schlegel diagram, and replacing its edges with quadrilateral-shaped struts.^[2] Coxeter later discovered the regular skew polyhedron.^[3] Felix Klein discovered the three genera.^[4] Together with Robert Fricke, they found the five genera of Leonardo polyhedra.^[5] Some colleagues further discovered the locally regular and the genus up to 14.^[6]

• Gevay, G.; Schulte, E.; Wills, J. M. (2014). "The regular Grünbaum polyhedron of genus 5". Advances in Geometry. 14 (3): 465–482. arXiv:1212.6588. doi:10.1515/advgeom-2013-0033.