Friday, September 9, 2011

From Number to Formula

You happen upon the number 1.6180339887. It looks vaguely familiar, but you can't quite place it. How can you find out whether this particular number is special in some way, perhaps as the output of a specific formula or the value of a familiar mathematical constant?

If you have the kind of phenomenal insight and prodigious memory that mathematician Karl Friedrich Gauss (1777-1855) had at his disposal to work out such problems, you might be able to figure it out on your own.

Otherwise, you could consult Plouffe’s Inverter (PI)—a kind of search engine for formulas. Enter your number (from 5 to 64 digits long) in the blank space provided, click "go," and await the result. In effect, you type in the answer to get back the question. The underlying database contains 215 million mathematical constants.

In the case of 1.6180339887, the database search produces pages of formulas and functions that could generate 1.6180339887 (rounded off). One intriguing possibility is the expression (1 + sqrt(5))/2, which represents the golden ratio.

The Inverter is largely the work of Simon Plouffe. It started out in 1995 as the Inverse Symbolic Calculator (ISC), a project of the Center for Experimental and Constructive Mathematics at Simon Fraser University. The current Inverter is the product of several iterations of that pioneering search engine.

Plouffe’s vast compendium makes it possible to identify all kinds of "special" numbers. But there's a catch.

Given a formula or expression such as "2 + 2", there's only one answer, 4. But, given the result 4, there are lots of different ways to get there besides "2 + 2". Thus, it can get very tricky to sift the "true" formula from a coincidental expression extracted from the Inverter database.

The hazard is greatest when only a small number of digits is used or when the number is truncated or rounded off.

In the original ISC, Plouffe addressed the problem by offering a "smart lookup" in addition to the "simple lookup" that you can start with. Higher levels of search, employing a variety of algorithmic strategems, narrowed down the possibilities further.

Plouffe’s Inverter is a useful research tool for mathematicians and a fascinating playground for number fanatics. Do you have a favorite number: pi, the golden ratio, e (Napier's number), gamma (Euler-Mascheroni constant), the square root of 2 (Pythagoras' constant), the natural logarithm of 10, or one of Feigenbaum's numbers? Now, you can find out what that number is made of.

Originally posted Nov. 11, 1996

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