### Transcendental trivia?

Both e (2.7182...) and pi (3.1416...) are transcendental numbers of great significance in mathematics and the scientific description of nature. Instead of being neat and orderly (as we devoutly hope nature will be), the decimal expansions of these two numbers are patternless, some say ugly. Faint hope arises at the 710,150th digit of pi where a satisfying string of seven consecutive 3s appears (....353733333338...). More reassuring is the observation that (pi)^{4} + (pi)^{5} almost exactly equals e^{6} .

We are sure that great truths lie hidden in these two numbers (despite their unattractive decimals) when we find that a 5x5 magic square (first row: 17, 24, 1, 8, 15) can be transformed by the alchemy of pi into an unmagic but very strange square. To do this, replace the 17 by the 17th digit of pi (this is 2); 24 by the 24th digit (this is 4); and so on. The rows and columns of the new square add up to the same numbers: columns; 17, 19, 25, 24, 23; rows; 24, 23, 25, 29, 17. (Yes, the order given is correct.)

Gardner maintains that this astounding transformation is merely a coincidence, like all of the other peculiar relationships between 3 and pi. Millions and millions of relationships are possible and a few will certainly be remarkable, just as only a few of the many possible mathematical equations describe natural phenomena.

(Gardner, Martin, "Mathematical Games," *Scientific American*, 241:22, September 1979.)

**From Science Frontiers #9, Winter 1979. © 1979-2000 William R. Corliss**