Today, guest scientist Andrew Boyd looks at some divine shapes.
The University of Houston presents
this series about the machines that make our civilization
run, and the people whose ingenuity created them.

For many of us, the word *mathematics* evokes
fear or, what may be worse, dread. Yet, throughout history, mathematics
has often inspired awe and been the focus of mystical or religious devotion.

One alluring mathematical artifact is the collection of regular solids --
three-dimensional shapes with identical sides and corners. The cube is a
regular solid. It has six identical sides, all squares. It also has eight
identical corners.

The simplest regular solid is the tetrahedron, made of four identical triangles.
It looks a lot like a pyramid, but has a triangle rather than a square for its
base. Altogether there are only five regular solids. The remaining three are
the octahedron, the dodecahedron, and the icosahedron.

The fact that there are only five regular solids can be traced to Euclid, who
devotes much of the final chapter of his work the Elements to various facts about
the regular solids. However, it's Plato whose name is most closely attached to
these solids. Plato associated one solid with each of the four basic elements --
fire, earth, air, and water. He reserved the fifth for the heavens beyond the stars
and planets. Plato's theory of how the universe worked was so intimately connected
with the regular solids that they're often referred to as the *Platonic solids.*

Two thousand years later, Johannes Kepler also fell under the spell of these ethereal
forms. Best known for his three laws of planetary motion, Kepler's earliest published
work, the *Mysterium Cosmographicum,* focused on how the regular solids could be
used to explain the orbits of the planets. He alternately nested spheres and solids
inside of one another so they all fit tightly together. This nesting was such that the
relative sizes of the spheres were close to the relative sizes of the planets' orbits.
(At the time, only six planets were known.)

Unlike Kepler's laws of planetary motion, which were profound insights, his nested solids
idea added nothing to our knowledge of planetary orbits. It's often pointed to as an
example of misguided pseudo-science. Yet both ideas stemmed from Kepler's devout
conviction that God expressed himself in the beauty of nature. The planets and stars
were long considered the Creator's most perfect expressions of his being. By connecting
the planets' orbits with the regular solids, Kepler was elevating these mathematical forms
to the level of the divine.

Today, we may not treat the regular solids with the same esteem as Euclid or Plato or Kepler,
but their allure remains as strong as ever. A quick search on the Web yields hundreds of
links to the regular solids, almost all with pictures, some with fancy three-dimensional
renderings. Many of these Web pages are set up by individuals who simply find beauty in
these shapes, and perhaps puzzle over why nature only gave us five. And I have to admit,
as I stare at the stick models in my office that I made many years ago, I can't help but
still smile at the elegance of their symmetry.

I'm Andy Boyd, at the University of Houston,
where we're interested in the way inventive minds
work.

(Theme music)

Dr. Andrew Boyd is Chief Scientist and Senior Vice President at PROS, a provider of
pricing and revenue optimization solutions. Dr. Boyd received his A.B. with Honors at
Oberlin College with majors in Mathematics and Economics in 1981, and his Ph.D. in
Operations Research from MIT in 1987. Prior to joining PROS, he enjoyed a successful
ten year career as a university professor. *His latest book is: The Future of Pricing:
How Airline Ticket Pricing Has Inspired a Revolution.* (New York: Palgrave MacMillan, 2007).
For more on the Platonic Solids, see:
http://math.ucr.edu/home/baez/platonic.html

or http://en.wikipedia.org/wiki/Platonic_solid

For more on Kepler, see: http://en.wikipedia.org/wiki/Johannes_Kepler

The five Platonic solids above are displayed in the following sequence, top to bottom: The
hexahedron or cube (6 sides), tetrahedron (4 sides), octahe-dron (8 sides), dodecahedron,
and icosahedron. Below, see Kepler's model of the solar system with the planets riding in
nested Platonic solids. (Public domain images courtesy of Wikipedia).

The Engines of Our Ingenuity is
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Lienhard.