A GEOMETRIC NOTE BY RICHARD MOORE
A few years ago, Richard Moore, who is an amateur mathematician as well as a poet, sent me a reprint of the following article, which appeared in Hellas in 1990. I found it refreshing (despite the algebraic demonstrations which I couldn't follow at all - but that is precisely his point!), and of course especially of interest for its treatment of the hexagon!
Incidentally, a sort of geometric meditation had preceded the writing of "The Hexagon": I
had been assuaging the boredom of my last year in law school by making doodles that explore the
properties of this figure. In The Consciousness of Earth (ch. II) I had also written of "the small
Euclidean universe of feeling" from which modern science tends to abstract us.... which is also a
point made in Moore's article.- EC
HEXAGONS, CUBES AND PYTHAGOREANS
(first published in Hellas)
Everyone knows that purely geometric proofs are more cumbersome than algebraic ones
and that the methods of algebra are more efficient in every way. If any doubt remains about this,
just look into any up-to-date high school geometry book. Where's the geometry? It has all been
translated into algebra. As a small comment on the situation, let me present something that the
ancient Pythagoreans knew about but kept secret for two thousand years until I rediscovered it
last week. It raises some interesting questions, I think, about proofs and about what is interesting,
what is profound in mathematics.
The Pythagoreans have made us familiar with the triangular numbers :
and with the square numbers:
but they kept the hexagonal numbers (as they tried to keep the dodekahedron) to themselves:
Hexagonal numbers are "natural" because hexagons, like squares and equilateral triangles and like
no other regular figure, exactly fill plane space--as bees discovered in their honeycombs long
before man appeared on the scene. If you have a lot of pennies to play with on a tabletop, you will
come up with hexagonal numbers very quickly because six pennies fit snugly around a seventh,
twelve more pennies around these, eighteen more around these, etc.
But such hexagons have a connection with three dimensional space as well. The sum of the first n hexagonal numbers is n3 . That is,
and applying the familiar formula
which reduces to
We need now only show that
for all n - an easy exercise in mathematical induction. The problem is a nice illustration of the effectiveness of modern elementary mathematics. So now, on to the next problem, and...
"Hang on!" cries an ancient Greek, whom we have roused from the Shades by this
sacrilege. "What on earth are you doing with those wretched symbols? You juggle those ugly
little squiggles about according to a set of arbitrary, abstract rules until you get what you want,
and you say that it proves something? There is a beautiful relationship between a regular hexagon
and cube which your methods have missed entirely. Forget your numbers and symbols for a
moment and look at a revealing representation of the thing itself:
A cube may be positioned so that its outline is a hexagon. Let a cubical array of dots be positioned in this way. Then the outer layer of dots in the three visible faces forms a hexagonal number. Remove this outer layer, and a smaller cubical array (with one less dot on each edge) remains. The process may be carried out as many limes as there are dots on an edge after the first removal. By drawing this diagram and making the removals, we not only prove, but we also see that every cubical number is the sum of a corresponding series of hexagonal numbers.
"Is this not better than your arcane manipulation of symbols? Who but a student of
mathematics would even be able to follow your algebraic proof? My proof is not only profounder in its implications, but it is so clear, so simple, so sensuous, that an uneducated child (like the
uneducated slave in Plato's Meno) would not only understand it, but would delight in it. Would,
that is, if your textbooks and teaching methods had not already destroyed his capacity for delight.
"You are a poet. You ought to know that we judge anything profound and beautiful - mathematics included - in exactly the same way that we judge a metaphor in poetry: by its inclusiveness, justness, and sensuous impact. And if your educators had any sense of this, any understanding that mathematics can be a thing of beauty when our geometric methods are used to explain it, then there would be far less of the 'widespread mathematical illiteracy' among your young, about which those educators so loudly complain. Among us ancient Greeks, every educated man developed an appreciation of mathematics, and the tranquility of our souls benefitted enormously."