You know, I really have bitten off more than I can chew here by taking a look at phi - hardly an uncharacteristic move on my part. Phi is a very deep well, literally a topic as endless as the number of digits to the right of the decimal in the number itself. There are heaps of resources out there on the web discussing practically every conceivable aspect of phi, and a spate of books on the topic, so I'll leave it to interested readers to make these investigations for themselves.
One of my favorite books about numbers and their archtypes, simply taking the numbers 1~10 in detail, is The Beginner's Guide to Constructing the Universe. It's amazing how we take for granted today things that were occult mysteries at one point in time. Have you ever considered what it might be like to live at a time when there was no intellection of numbers at all - no distinguishing between unity (1) and duality (2) or trinity (3)? It's like the comment I read a while back on Newton's 'discovery' of gravity - before the apple hit his head and he had the grand realization, the concept of 'gravity' was unknown. Which is another way of saying that it didn't exist. Curious to think about, and leads me to wonder what things are outside of our current conceptual framework, that later generations, after these discoveries have come to light, will wonder how we ever failed to notice?
Yesterday I left off with the observation that the value of phi, 1.6180339887..., can be expressed by the formula (√5 +1)/2, which with a little rearrangement becomes
5ˆ.5 x .5 + .5
As noted, there are a lot of 5's in that formula, hinting at the connection between the number 5 and phi. This association can be seen the most clearly in the relationship between the 5-sided figures, the pentagon and the pentagram (5-pointed star). Now we get into the satanic worship part of my site, heh-heh-heh...need to find some candles now....
I'm not one for New Age interpretations of geometric forms, nor am I interested in numerology. I can't explain all the mysteries inherent in the archtypal meanings of numbers and the various polygons, and am perfectly fine with letting things be mysterious. Life should have some small corner of unknown and unexplained in it it seems to me. Without something to discover, without some mystery, life would be a stale endeavor indeed.
So if we're going to look at the relationship between pentagons and pentagrams, we first need to draw them. Anyone can draw a rough pentagram, but drawing an accurate one, or a pentagon, is a little more involved than the square or triangle. That said, the good news is that the pentagon can be constructed with the standard geometers tools of straightedge and compass. No mathematics or digital angle finders needed.
So let's walk through that drawing process - one method at least. This method proceeds from within a circle.
First draw a line and bisect it with a perpendicular: Next draw a circle, placing the compass point at the intersection of our lines. This circle will circumscribe our pentagon: Now we take our horizontal line, already bisected in two, and bisect one half of that line, giving point 'A':Using point 'A' as the center, swing an arc from the intersection of our vertical line and the circle, labeled as 'B' in the diagram:This generates point 'C'. You might note how similar this process is to the one shown yesterday for establishing a Golden Mean point from a given line segment and the square - one of the reasons I chose to show this particular method for describing a pentagon.
Take your compass, set the point on point 'B' and adjust the end to the distance B~C. Swing an arc of this distance out to the circle as shown:Almost there. The distance we just swung with our compass, B~C, makes the point 'D' on the circle. This distance is the length of the side of the pentagon - point 'D' is on of the corners of that pentagon. Keeping your compass at the same setting, 'walk' the compass around the circle:From here, it's simply a matter of connecting the marks just walked around the circle: Next, part is easy too: drawing the pentagram, simply by connecting the marks once again:Now that we have the drawn the pentagon and pentagram within the circle, we can look at some of the Golden Ratio relationships inherent in this construction. First, taking the length of the pentagon side as 1.0, we can see that the 'apothem' of the pentagon is ø in measure:Another is found as follows:In fact, there's no point going much further with this examination, since these phi relationships are extremely numerous in this construction. In fact, I understand there are 288 distinct phi relationships to be found in this interplay between pentagon and pentagram(!)
One more look though, this time comparing areas of two triangles formed in this construction, also in a phi relationship:
One thing that might be of interest, is the fact that the mix of pentagon and pentagram forms the 'Golden Triangle'. This one has a base of 1.0, and two sides each measuring 1.6180339..., ø, in length. This form of triangle is termed 'isosceles' ('iso'= 'equal'; 'sceles' = from the Greek 'skelos', meaning 'leg'): The Golden Triangle, it turns out, is more than just a leading area for opium production. Thinking about that, I realize we have another 'Golden' polygon, one in fact we have actually more or less drawn when we looked at finding the Golden Mean from the square in yesterday's post:The short side is multiplied by phi to give the long side measure.
Some readers my be getting a little itchy by this point and wondering where this discussion is going. Am I lost in a math vortex, never to return? Is it time to bring out the straight-jacket?
Not quite yet I'm afraid. We will take a look at how we can make some practical use of the geometrical drawing techniques using compass and straightedge, and producing phi as a means of designing attractive pieces. First off, take a look at a web article detailing the geometric relationships found in a piece of furniture made by the Herter Brothers in the late 19th century. Next post, I'll take a look at one way of using the Golden Ratio as a design tool.