Kepler and the Rhombic Dodecahedron: Investigation Links
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TO INVESTIGATE Kepler: The Volume of a Wine Barrel. To learn more about Kepler’s contribution to the development of calculus, read this article published in Convergence in 2011. |
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TO INVESTIGATE A property of minima in honeycombs. 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? |
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TO INVESTIGATE Properties of the cuboctahedron. 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. |
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TO INVESTIGATE The trapezo-rhombic dodecahedron. The trapezo-rhombic dodecahedron is a polyhedron that also tessellates space. It has twelve faces, but six of them are trapezoids |
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More about Leonardo da Vinci’s drawings of polyhedra for Luca Pacioli's book De divina proportione, with video animations. |
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Leonardo da Vinci: Drawing of a cuboctahedron made for Luca Pacioli's De divina proportione.
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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. |
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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.