Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/18/84 / ST 1.0; site saber.UUCP Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!decvax!decwrl!sun!idi!saber!skinner From: skinner@saber.UUCP (Robert Skinner) Newsgroups: net.math Subject: Re: Pascal's Inverse Triangle Message-ID: <1677@saber.UUCP> Date: Mon, 8-Jul-85 13:19:33 EDT Article-I.D.: saber.1677 Posted: Mon Jul 8 13:19:33 1985 Date-Received: Thu, 11-Jul-85 00:00:49 EDT References: <2216@utcsstat.UUCP> Organization: Saber Technology, San Jose, CA Lines: 74 > Now, consider this, which I shall refer to as Pascal's Inverse triangle... > > 1 > 1 1 > 1 1/2 1 > 1 2/3 2/3 1 > 1 3/5 3/4 3/5 1 > > this triangle also continues on for infinity. > > Each entry is the INVERSE of the sum of the two entries > directly above, for example > > 1/2 = 1/(1 + 1) > > and 2/3 = 1/(1/2 + 1) > > .... > > > 1 > X = ----------------- > 1 + 1 > ----------- > 1 + 1 > --------- > 1 + etc. > > and this is the same as > > > 1 > X = --------------- > 1 + X > > X * X + X - 1 = 0 > > and this equations one positive root is > > (sqrt(5)-1)/2 > > Has anyone come across this triangle before, and if so > have you anything of interest to share concerning it ? *** REPLACE THIS LINE WITH YOUR MESSAGE *** I've never seen the triangle you describe here, but the root you show is also the Golden Ratio. (Or the inverse: (sqrt(5)-1)/2=.618=1/1.618= 1/(sqrt(5)+1)/2). The way I first heard it stated, the ancient Greeks thought that the most pleasing ratio for a (Golden) rectangle were such that if a square were chopped off one end, it would yield another Golden rectangle. In other words, x/1=1/(x-1) ==> x=1.618. The Golden Ratio also pops up in Fibonacci numbers, the ratio of fn to fn-1 converges to 1.618, regardless of the values of f0 and f1. Yet another odd place: the five pointed star. This is hard for me to describe, but suppose you draw a five pointed star like you did in grade school. Assuming you draw a perfect star, the ratio of the length from the body to the tip of one point, to the length of the rest of the same line segment is .618=1/1.618. It's nice to know the Golden Ratio lives in other places as well. Anybody know of other places in math or geometry where the Golden Ratio occurs? ------------------------------------------------------------------------------ Name: Robert Skinner Mail: Saber Technology, 2381 Bering Drive, San Jose, California 95131 AT&T: (408) 945-0518, or 945-9600 (mesg. only) UUCP: ...{decvax,ucbvax}!decwrl!saber!skinner ...{amd,ihnp4,ittvax}!saber!skinner