## Thursday, August 4, 2011

### Cantor Ternary Set

The second issue of the Journal of Humanistic Mathematics has recently been posted -- with more new poemsThe first issue contained a poem by Philip Holmes about one of the most amazing collections of numbers in all of mathematics, the  Cantor Ternary SetThis set, discovered by Henry John Stephen Smith (1826-1883) but popularized by Georg Cantor (1845-1918) consists of all the real numbers whose base 3 or ternary representations involve only the digits 0 and 2. Like a fishing net, the Cantor Ternary Set is mostly holes. "Gaps" by Philip Holmes spreads it out before us -- and reflects on what else it may represent:

Gaps   by Philip Holmes

Take a line and take away
the middle third, and then
the middle thirds of two thirds
left behind, and middle thirds

of those four ninths remaining.
Go on and on: what's left at last
is utterly disjoint -- beginnings,
ends -- each point divided from

the next, but oh! so close,
infinitely numerous
as what you started with
and carefully have pried apart.

Will there be time to measure up
this dust of unremembering?

*  *

Take a line and take away the middle third,
and then the middle thirds of two thirds
left behind, and middle thirds of those four
ninths that still remain. Reiterate:

what's left at last is utterly disjoint --
beginnings, ends and more -- each point
divided from the next and yet uncountable
and numerous as what you had before.

Take a life and take the most part out,
for so it happens; only the best-rehearsed
of memories remain: a voice transformed
among the absences, a face, a hand.

You brought me here, but there was more:
dust that blows away, gaps that captivate.

Song lyrics ("Mandelbrot Set")  by Jonathan Coulton posted on 10 May 2010 also mention (but do not explain) the Cantor Ternary Set.