Today, let's think about fives. The University of
Houston's College of Engineering presents this
series about the machines that make our
civilization run, and the people whose ingenuity
We bipedal people are made
up of two similar left/right halves, and we do love
symmetry. We balance our designs and our
architecture into complex symmetries and partial
symmetries. We like to play with three-dimensional
symmetry. A cube, for example, has nine different
planes of symmetry and, if you want some fun, you
might try figuring out what they are.
The concept of symmetry can be confusing. Hand a
person a triangle, a square, and a pentagon, and
ask which is least symmetrical. Three out of four
people pick the pentagon, even though its five axes
of symmetry are the largest of the set. The
pentagon, with all that symmetry, doesn't strike us
as symmetrical at all.
But its symmetry does play on our subconscious.
What fun it is when children learn to draw
five-pointed stars with five pencil strokes -- so
easy, yet so subtle! And a small pentagon sits in
its center. Leonardo da Vinci's spread-eagled human
inscribed in a pentagon has become a major icon
just because it's so unexpected. Add our head to
our arms and legs and you and I become pentagons.
Nature offers so much pentagonal symmetry: the
armor of pineapples, cross sections of apples --
starfish, flowers, sand dollars. But we also force
five-fold symmetry on many things. The Greeks spoke
of five Platonic solids: cubes, octahedrons,
tetrahedrons, icosahedrons, and dodecahedrons. The
first four (formed from triangles or squares) stood
for earth, air, fire, and water. But the fifth, the
dodecahedron (made from pentagons) was the symbol
for pure celestial matter. That's where the movie
The Fifth Element gets its title.
Hindu philosophers added celestial ether to their
four earthly elements, and they also got the
mystical number, five.
Ask musicians how they feel a five-beat measure.
They'll tell you, "Twos and threes, Ya-ta
Ta-ta-ta, or Ya-ta-ta Ta-ta."
We have trouble hearing a full five-beat sequence
because the number five seems to lack symmetry in
time as well as in space.
Now Istvan Hargittai offers a book of essays,
Fivefold Symmetry. Twenty-nine authors
hold the idea up to the light and look at its
refractions. They find fives lurking everywhere in
both nature and iconography. Small wonder that,
when Mephistopheles tried to leave Faust, he
tripped upon, a "Thin goblin-foot upon the
threshold there." Faust had painted a pentagram on
the floor and the Devil had to struggle to get
Fivefold symmetry is shot through tile making,
architecture, basket weaving, fortifications, and
Christian, Islamic, and Egyptian iconography. Fives
seem to rise like mist out of our subconscious
sense of balance. They're almost never in our
mental forefront. Rather, the imagery of fives is
one of those odd driving forces within us -- and
one it might pay to be more aware of.
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds