Phi like Pi is what is called an irrational number in that it has no end. Pi, as you know, is the circumference of a circle divided by its diameter which is 3.14159.....ad infinitum.Phi is credited to the ancient Greek mathematician Euclid, the father of geometry. Euclid discovered the following: if you draw a straight horizontal line on a peace of paper and you want to find the exact center of this line it can be done with a simple compass. With the compass point at one end of the line, strike an arc both above and below the line. Then do the same thing with the point of the compass at the other end of the line. The result will be two sets of arcs that intersect each other and if we then draw a line connecting these two points of intersection you will have a straight line that crosses the original line, in a perpendicular manner, at the exact middle of the original line. What Euclid discovered was the ratio of these two lines ( the length of the longer line divided by the length of the shorter line ), no matter how far the arc intersections were from the the original line and no matter the length of the original line, was always the same; 1.618033....ad infinitum. This number took the name Phi. Euclid also determined that Phi is the only number that is equal to one plus its own reciprocal, in other words, 1/Phi is equal 0.618033.... and, therefore, Phi = 1 + 1/Phi. This discovery of Phi received a lot of attention. Phi became known as the golden or Divine ratio and rectangles with base and sides that met this ratio became known as golden rectangles which greatly influenced ancient Greek and Roman architecture. Well, Phi is definitely a cool number, but hang on, things get more curious as time goes on.
In the twelfth century, a mathematician by the name Leonardo Fabonacci with a curious sequence of numbers that henceforth bare his name. Every number in a Fabonacci sequence is the sum of the previous two numbers, i.e.,
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ......
What's cool about the Fabonacci sequence is that if you take any pair of Fabonacci numbers and divide the larger by the smaller you get a number very similar to Phi and the further out in the Fabonacci sequence you go this ratio gets closer and closer to Phi. In other words, these ratios of Fabonacci numbers converge on Phi. Well, now people began see Phi and Fabonacci numbers as something mystical; they saw Phi and Fabonacci ratios every where, the shape of the galaxies, between the earth and the moon, in plant life and animals and etc. In the renaissance age artist including Leonardo da Vincci, use the Phi ratios in their constructs of human figures in their art. In other words they used these ratios to determine things, such as, the length of the face relative to the width, the position of the eyes, the position and width of mouth and all of the various proportions of the body. Critics claim that these proportions are very pleasing to the eye.
Like I said at the beginning, I've learned more than I really wanted to know. But what is strange to me personally is that only learned about Phi for the first time when I was about fifty years old. I say strange because I am an engineer and I spent a fair amount of time studying science and mathematics and never once did I come across Phi. Oh well, I'm already tired of Phi.
There is, however, a number Ive always found interesting and you may as well. This number doesn't have a name that I am aware of. I can't even remember where or when I first came across this number but it was a long time ago. The number is 142857. What is interesting about this number are its multiples as you can see in the following table:
1142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142
142857 x 7 = 99999
142857 x 8 = 1142856
142857 x 9 = 1285713
As you can see, for multiples 2 through 6 all of the results contain the exact same digits as the original number; only the sequence starts at a different point in each case. When we multiply by 7 we obviously get the source of our mysterious number because 999999 divided by 7 is 142857. If we look close at the multiples of 8 and 9 we note the the results now have seven digits instead of six and if we sum the first and the last digit will have the number that we are missing from the sequence. You've got to admit this is a cool number.
Enough for numbers! Sense or nonsense? You be the judge.
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