a random shape and unequal sides and the next cell could then
be custom made to fit to that cell, without gaps. And so on, one
cell after another, each fit to the one before it. But this method
of constructing a honeycomb would require that the worker bees
work sequentially, one at a time, first making one cell, then fitting
the next cell to that, and so on. This procedure would be a waste
of time for the bees. Each insect would have to wait in line for the
guy in front to finish his cell. If you’ve ever seen bees building a
beehive, you know they don’t wait for each other. They work simultaneously. So the bees need to have a game plan in advance, knowing that all the cells will fit together automatically. Only equilateral
triangles, squares, and hexagons will do.
But why hexagons? Here lies another fascinating story. More
than two thousand years ago, in 36 bc, the Roman scholar Marcus Terentius Varro conjectured that the hexagonal grid is the
unique geometrical shape that divides a surface into equal cells
with the smallest total perimeter. And the smallest total perimeter, or smallest total length of sides, means the smallest amount
of wax needed by the bees to construct their honeycomb. For
every ounce of wax, a bee must consume about eight ounces of
honey. That’s a lot of work, requiring visits to thousands of flowers and much flapping of wings. The hexagon minimizes the
e=ort and expense of energy. But Varro had made only a conjecture. Astoundingly, Varro’s conjecture, known by mathematicians as the Honeycomb Conjecture, was proven only recently, in
1999, by the American mathematician Thomas Hales. The bees
knew it was true all along.
of a telescope and six-sided snowflakes on a cold winter day? The
answer must be partly psychological. I would claim that symmetry represents order, and we crave order in this strange universe we find ourselves in. The search for symmetry, and the
emotional pleasure derived when we find it, must help us make
sense of the world around us, just as we find satisfaction in the
repetition of the seasons and the reliability of friendships. Symmetry is also economy. Symmetry is simplicity. Symmetry is elegance.
And however we define the mysterious quality we call beauty,
we associate symmetry with it. Both Darwin and Freud have argued that our sense of beauty and the appeal of beauty originated
with the imperative for sexual reproduction and the association
of beauty with a vibrant mate. As Darwin wrote in The Descent
A sense of beauty has been declared to be peculiar to man. . . .
[But] when we behold a male bird elaborately displaying his
plumes and splendid colours before the female, whilst other
birds, not so decorated, make no such display, it is impossible to doubt that she admires the beauty of her male
partner. As women everywhere deck themselves with these
plumes, the beauty of such ornaments cannot be disputed.
Clearly, human-made art and architecture abound with symmetry. The Taj Mahal has a central dome and arch, two identical side domes, and four identical towers, symmetrically placed.
OF COURSE, NATURE OCCASIONALLY VIOLATES COMPLETE
SYMMETRY WITH IRREGULAR COASTLINES AND THE
AMORPHOUS SHAPES OF CLOUDS.
There’s more to the bee story. Bees relate to the question of
why flowers have so much symmetry. Bees need flowers for their
food and for making wax, and flowers need bees for pollination.
Experiments published in 2004 by researchers at the Freie Uni-versität in Berlin and the CNRS Université Paul-Sabatier in Toulouse show that bees are more attracted to flowers with more
symmetry. And why are bees attracted to flowers with more symmetry? The same researchers propose that symmetrical stimuli
from the flowers are more easily processed by the visual system
in the bee brain; that is, they require less neurological apparatus.
Again, the principle of economy at work.
BUT WHY ARE WE ATTRACTED to symmetry? Why do we human
beings delight in seeing perfectly round planets through the lens
Leading to the building is a rectangular pool with equally spaced
cypress trees on both sides and symmetrical gardens beyond. The
Octagon on Roosevelt Island in New York, designed by Alexan-
der Jackson Davis, is shaped like a you-know-what. The famous
drawing by Leonardo da Vinci called the Vitruvian Man depicts
a male figure with two identical sets of outstretched and equally
spaced arms and legs, one set inscribed within a circle and one
within a square. A widely reproduced image of Lakshmi shows
the Hindu goddess sitting in the center of a circular flower with
two identical arms raised upward and holding identical yellow
flowers, two more identical arms lowered and releasing flower
petals, and two identical elephants on each side of her pouring
water from identical jugs. However, if one looks at the image
closely, one can see that there is a slight departure from perfect