we have seen, the world according to quantum mechanics consists of a
collection of whole numbers of basic units—mass, energy, length, and
time are all measured in terms of quanta. The mathematics of the real
number system, the residence of the square root of 2, is an ideal construct
that has great utility and considerable intellectual interest. In the real
world, though, if a square can actually be constructed out of material ob-
jects, its diagonal (also constructed out of material objects) is either a lit-
tle too short to reach from corner to corner or extends a little beyond.
It makes one wonder if other ideas, known to previous civilizations but
long since discarded, are waiting quietly on the sidelines to make a come-
back in modern guise.
NOTES
1. h t t p : / / w w w. p e r s e u s. t u f t s. e d u / G r e e k S c i e n c e / T h u c .2 .47 -55 .html.- International Journal of Infectious Diseases, Papagrigorakis, Volume 11, 2006.
3. T. L. Heath, A History of Greek Mathematics I (New York: Oxford, 1931).
4. h t t p : / / w w w - g r o u p s. d c s. s t - n d. a c. u k / ~ h i s t o r y / H i s t T o p i c s / D o u b l i n g t h e c u b e
.html#s40. This gem of a Web site contains not only Archytas’s and Menaech-
mus’s solutions to duplicating the cube, but also Eratosthenes’ method of find-
ing roots. A certain comfort level with analytic geometry is required to stay up
with Archytas, but Menaechmus’s solutions are fairly straightforward and a
high-school graduate shouldn’t have much difficulty with them. Even if the
reader doesn’t intend to “do the math” necessary to follow the constructions, the
site is worth looking at simply to gain a greater appreciation of the sophistication
of the ancient Greeks. The fact that they could do all these things using only ge-
ometry (no analytic geometry, which greatly simplifies all things geometrical)
and not having access to pencil and paper, still causes me to shake my head in
disbelief—and we haven’t even gotten to Archimedes.
5. T. L. Heath, A History of Greek Mathematics I (New York: Oxford, 1931). - Ibid.
- Ib id.
8. A. K. Dewdney, Beyond Reason (New York: John Wiley & Sons, 2004), p. 135. This
construction is by no means Archimedes’ finest hour—but an off day for
Archimedes would make the career of many lesser mathematicians. He simply
uses the length of the unrolled circumference as the base of a right triangle, and
the radius of the circle as the height of that triangle. This results in a triangle
whose area is^1 ⁄ 2 (2r)rr^2 , and a standard construction will produce a
square with the same area as the triangle.
9. http:// www .jimloy .com/ geometry/ trisect .htm. This site probably sets the record
for most erroneous trisections of an angle—some extremely ingenious and only
subtly in error. I wish it had been in existence when I was a junior faculty mem-
ber at UCLA back in the late 1960s. Then as now, UCLA was the home of the
Pacific Journal of Mathematics. Back then it would receive numerous submis-
sions for trisection of the angle—and the editors, generally a polite group, would
respond not with a curt, “It’s impossible, don’t bother sending anything else,”
Impossible Constructions 77