regular polygons revealed a surprising connection between geometry and
an interesting class of prime numbers. Indeed, this is one of the persist-
ently surprising and appealing aspects of mathematics—there are unex-
pected connections between not only areas of mathematics, but also
between mathematics and other areas.
However, mathematics sometimes causes people to leap to unsup-
ported conclusions. In trying to fit the orbits of the planets into a coher-
ent pattern, Johannes Kepler was struck by the coincidence that, at the
time, there were six planets and five regular solids. More planets would
be discovered, but Greek mathematicians had proved that there were
only five regular solids: the four-sided tetrahedron, the six-sided cube,
the eight-sided octahedron, the twelve-sided dodecahedron, and the
twenty-sided icosahedron. Based on inadequate data, Kepler constructed
this model.
“The Earth’s orbit is the measure of all things; circumscribe around it a
dodecahedron, and the circle containing this will be Mars. Circumscribe
around Mars a tetrahedron, and the circle containing this will be Jupiter.
Circumscribe around Jupiter a cube, and the circle containing this will
be Saturn. Now inscribe within the Earth an icosahedron, and the circle
contained within it will be Venus. Inscribe within Venus an octahedron,
and the circle contained within it will be Mercury. You now have the rea-
son for the number of planets.”^19
An exquisitely beautiful scheme—but dead wrong. The lure of pattern
is so strong that just as we think we see a face on Mars when it is merely
a land formation seen in light that accentuates features that appear to be
human, we sometimes see mathematical patterns based on inadequate
data or information. It is to Kepler’s eternal credit that he did something
that must have been extremely difficult: when Tycho Brahe supplied him
with better data to which he could not get the model to fit, he abandoned
the model. In so doing, he formulated Kepler’s laws of planetary motion,
which led to Newton’s discovery of the theory of universal gravitation.
Pythagoreans Redux
The fundamental tenet of the Pythagoreans was that the universe was
constructed of whole numbers and the ratios of whole numbers. The dis-
covery that the square root of 2 was incommensurable destroyed this
worldview—at the time of the Pythagoreans. However, in an intriguing
twist, the Pythagoreans may have been right after all! Quantum mechan-
ics, so far the most accurate depiction of the universe that we have, is es-
sentially a modern version of the view espoused by the Pythagoreans. As
76 How Math Explains the World