Make Classic 3D Shapes with Balls and Sticks
For thousands of years geometers have explored classic polyhedra. From Plato to Kepler, from da Vinci to Escher, the classics have been a source of inspiration.
Nobel prizes in physics and chemistry were given to people exploring novel geometries of carbon that would be familiar to these forebears. Architect Buckminster
Fuller designed geodesic domes based on classic polyhedra and the name Buckminsterfullerene (Bucky Ball) was given to 1996 Nobel Prize winning discovery of a C60
molecule which many of us recognize as a soccer ball. Geometers call it a truncated icosahedron. Archimedes wrote about this geometric shape and Leonardo da Vinci
depicted it in De Divina Proportione. Current artist Ai Weiwei was inspired by da Vinci to make a sculpture of it out of wood. The headmaster's office in
Hogwarts castle features classic polyhedra, benefitting from their timeless aesthetic. Viruses employ classic polyhedra for their protein shells because of their
simple design and their structural integrity.
Plato wrote about these five regular, convex polyhedra around 360 B.C. He associated each with a classical element,
earth with the cube, water with the icosahedron, air with the octahedron, fire with the tetrahedron and the fifth element, the heavens, with the dodecahedron.
Euclid's The Elements, concerned with the elements of geometry, and perhaps the greatest textbook of all times, concludes with a detailed examination
of these five solids.
An Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons. The Archimedean
solids take their name from Archimedes of Syracuse (287 BC – c. 212 BC). While working at the library of Alexandria, Archimedes made many copies of Euclid's
The Elements. Rather than inventing a printing press, he came up with the 13 solids that could have been the next chapter. During the Renaissance, artists
and mathematicians revived the classics and rediscovered all of these forms.
A Kepler–Poinsot polyhedron is any of four regular star polyhedra. They have regular pentagrammic faces or vertex figures and are obtained by stellating the regular convex
dodecahedron and icosahedron. Regular star polyhedra first appear in Renaissance art. The small and great stellated dodecahedra were first recognized as regular by Johannes
Kepler in 1619. In 1809, Louis Poinsot rediscovered Kepler's figures and also discovered two more regular stars, the great icosahedron and great dodecahedron. Both M. C. Escher's
Gravitation and his Contrast (Order and Chaos) feature a small stellated dodecahedron.
The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865. Each is a
dual polyhedron to an Archimedean solid. Unlike Platonic and
Archimedean Solids, their faces are not regular polygons. However, the vertex figures of Catalan solids are regular, and they have constant dihedral angles. Naturally occurring crystal
formations of garnet can take a rhombic dodecahedral form. Tetrahexahedra are observed in copper and fluorite systems.
A figure that is composed of several polyhedra sharing a common center is considered a compound polyhedron. M.C Escher was a twentieth century artist who incorporated many of these
classic figures in his work. His works Waterfall and Stars are notable examples. Escher's brother was a crystallographer. Like Catalan whose father was a jeweler, Escher had family
who shared his interest in these classic polyhedra.
Honeycomb geometry is a space filling tessellation or 3D tiling. Polyhedra stack in a way where there are no gaps and vertices meet neighboring vertices. Linus Pauling discussed
cubic and hexagonal closest packed structures in his epic tome "The Nature of the Chemical Bond and the Structure of Molecules and Crystals".
Stars and stellations
A polyhedron is stellated by extending the edges or face planes of a polyhedron until they meet again to form a new polyhedron or compound. The Archimedean solids and their
duals can also be stellated.