Leonardo polyhedron

Rhombicuboctahedron, one of the Leonardo da Vinci's geometrical shapes illustration in 1509 Divina proportione

A Leonardo polyhedron is a polyhedron with a Platonic solid's rotational symmetry and genus $g\ge 2$. 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.Template:Sfnp

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.Template:Sfnmp Coxeter later discovered the regular skew polyhedron.Template:Sfnp Felix Klein discovered the three genera.Template:Sfnmp Together with Robert Fricke, they found the five genera of Leonardo polyhedra.Template:Sfnp Some colleagues further discovered the locally regular and the genus up to 14.Template:Sfnmp

References

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Further reading

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