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Below is a short exploration of the Golden Ratio!
A NEW AND INTERESTING NUMBER CALLED PHI
From some brief introductions to the topic of the Golden Rectangle, you may not realize the underlying, fundamental nature of the constant involved which is the irrational number called Phi (or the Golden Ratio, also called the Golden Mean, approximately 1.618033...)
Like Pi, Phi is an irrational number so it is an infinite, non-repeating decimal that we can only approximate (since we can't use a number that never ends we have to pick an ending somewhere!)
Compare Pi and Phi to 10 decimal places:
Pi = 3.1415926536...
Phi = 1.6180339887...
Lets compare two simple geometric ways to discover Pi and Phi.
1. Finding Pi From A Circle:
the ratio of the circumference of a circle to its diameter is Pi.
2. Finding Phi From A Line:
Divide a line into a short (S) and long (L) segments. If you divide it such that the ratio of the longer to shorter segment (L:S) is the same as the ratio of its original length (L+S) to the longer segment L i.e., L:S = (L+S):L, this ratio is Phi. A line that is divided in this special way is called a Golden Section, it would seem better to me to call it a Golden Segment but no one does.
Start with a line segment
Divide the segment into 2 segments at a special point
The total length of original line is S+L. The special point at which the original line segment is to be divided is the one and only point that will make the ratio of (S+L):L = L:S. The numerical value of this ratio, for any starting line segment, is the Golden Ratio = Golden Mean = Phi = 1.6180339887...Pretty neat isn't it?
You can extend the "Phi From A Line" to finding Phi in many geometrical objects. For example:
Phi From A Rectangle:
Check out this web page for a short but interesting set of examples of where this number appears, as if by magic.
One of my favorite curves is called a logarithmic spiral. The sea creature called the Nautilus grows its shell in this shape. It also appears in the spiral way sunflower seeds grow and in the shape stars form in some spiral galaxies. You can see a cross section of a Nautilus shell in the upper left-hand corner of the Algebra 2 Home page.
I forgot where I bought the Nautilus shell, it was about 10 or 15 years ago. I took a digital camera picture of the shell next to the math textbook we used for the course. Well, the Golden Ratio turns up in the underlying equations for these types of spirals! Pretty cool and pretty amazing! The Nautilus is a very cool animal too, a very unique beast.
Another nice site that provides some good descriptions of the Golden Ratio is a web site by Cynthia Lanius of Rice University. It has a cool page for building your own Golden Rectangle, and a page that shows you the underlying Algebra in a very straightforward way.
Finally, a site that has more than you probably want to know about Phi and related topics like the Fibonacci sequence.
An extra note:
Not everyone is convinced Golden Rectangles are "the best" looking rectangles. After all, that is a matter of individual taste isn't it? But, there may be something in the natural symmetry of geometric figures that involve the Golden Ratio that makes them more appealing to the eye. But as the saying goes "Your mileage may vary!" By that I mean something so subtle surely won't be experienced the same way by everyone. Symmetry is an important component of many things we find beautiful, so I wouldn't discount the possibility that a Golden Rectangle "looks the best" to some, or even most people. I sort of like squares myself and a nice trapezoid beats them both in my book! Perhaps for the people that really like them, seeing Golden Rectangles is like "seeing music." Why do our brains like music? It is sort of odd if you think about it isn't it? I mean why are sounds (notes) played a certain way pleasing and in another way not? Many sounds are annoying, what is different about music we like? Something about the symmetry of certain noises just "clicks" right in our brains. Maybe the same thing happens visually with geometric shapes that just seem to "click" the right way in our brains. Maybe someday we will encounter an alien civilization that doesn't understand music at all but really likes to watch displays of rectangles and triangles that involve the Golden Ratio! It will be fun trying to get them to enjoy music and they will surely have a challenge getting us to enjoy a visual "Golden Ratio" show. Or perhaps all we earthlings are missing is for the right "composer" to appear. The person that can find the hidden ratios that "cllick" with us visually the way music does. Maybe all we are seeing are disorganized "notes." Are you the "composer" we have been waiting for?
Mr. Cantlin (1/16/02)
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