# Kepler and the Rhombic Dodecahedron: Investigation Links

Author(s):
Roberto Cardil (matematicasVisuales)

 Using derivatives, we can solve an isoperimetric problem that interested 18th-century mathematicians such as Colin Maclaurin (1698–1708): We want to close a hexagonal prism as bees do, using three congruent rhombi. What is the shape of the three such rhombi with minimum total surface area?

 A cuboctahedron is an Archimedean solid. It can be visualized as made by cutting off the corners of a cube. We study some of its properties. This beautiful polyhedron is used as a decoration in art and jewelry.

 The trapezo-rhombic dodecahedron is a polyhedron that also tessellates space. It has twelve faces, but six of them are trapezoids

 TO INVESTIGATE    More about Leonardo da Vinci’s drawings of polyhedra for Luca Pacioli's book De divina proportione, with video animations. Leonardo da Vinci: Drawing of a cuboctahedron made for Luca Pacioli's De divina proportione. Leonardo da Vinci: Drawing of an augmented rhombicuboctahedron made for Luca Pacioli's De divina proportione.

 TO INVESTIGATE     Angles of the rhombic faces of a rhombic dodecahedron. We need some basic trigonometry to calculate angles of the faces of the rhombic dodecahedron. The obtuse angle is called a Maraldi Angle. It was named after Giacomo Maraldi (1665–1729), the first mathematician to study and publish about the angles of the rhombi at the bottom of bee cells.

 TO INVESTIGATE     Density of the optimal sphere packing. Using what we now know about the volume of the rhombic dodecahedron and its space-tessellation property, we can also calculate the density of the optimal sphere packing, using the rhombic dodecahedron as a unit cell.

Space holder.

Roberto Cardil (matematicasVisuales), "Kepler and the Rhombic Dodecahedron: Investigation Links," Convergence (March 2022)