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.
Platonic Solids
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.
Archimedean 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.
Kepler–Poinsot polyhedra
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.
Catalan Solids
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.
Compound Polyhedra
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
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.