I'd like to clarify a few things in my posting about the golden ratio:
When I said that for simplicity we could let B be 1, that's justified because, whatever length B is, we can define that length as one unit of measure. Then we just have one variable, A, to solve for.
When I said "solving (by the quadratic equation)", I meant to say "solving (by the quadratic formula)".
The quadratic formula solves all quadratic equations. All quadratic equations have an exact solution (though it isn't always in real numbers), and that solution can always be gotten by the quadratic formula.
What I meant by 1.618... is a number with a decimal fraction that starts out like that, but continues on with more digits. It's an
"irrational number", and so the decimal fraction doesn't repeat and has infinitely many digits. Square rootes, cube roots, trig functions, etc., are typically irrational numbers.
The reason I said to take that number's reciprocal was: At the outset I said that the golden section is B/A, where B and A are related as I described. Since I said to choose the unit of measure so that B = 1, A = A/B. So then, since I had an answer for A, or A/B, and I wanted B/A, that's why I said to take the reciprocal.
I should say that the exact expression of what the quadratic formula gives is (where "sqr" means "square root of":
(1 + sqr(5))/2
The reciprocal of that expression evaluates to the golden mean, which is about .618... Again, that's a nonrepeating decimal fraction with infinitely many digits, an irrational number. They're called "irrational" because they can't be expressed as a ratio of whole numbers.
Since I mentioned the quadratic formula, maybe I should state it here:
The general quadratic equation is: Ax^2 + Bx + C = 0.
Where "sqr" means "square root of", and +/- means plus or minus, the quadratic formula says:
x = (-B +/- sqr(B^2-4AC))/2A
That formula is derived by "completing the square":
A quadratic expression can be gotten by squaring a binomial like Ax + B. You get (AX)^2 + 2AxB + B^2. Any quadratic expression can be made into something of that form by adding something to its constant term. So that something is added there, and of course to the other side of the equation too (One is starting with the equation Ax^2 + Bx + C = 0), and then the square root of each side is taken. The resulting linear equation is easily solved for x. In that way the quadratic formula is derived.
I say all this because I feel that I shouldn't leave unexplained statements. I'd write out the derivation of the quadratic formula by completing the square, but that would probably be adding too much explanation and lengthening this post too much.
I may have glossed over the psychology of the pentagram too dismissively. Notice that the star has a rather attention-getting appearance, especially inside a more prosaic figure like a circle or polygon. Someone could say that its explosive or radiant appearance suggests energy, strength, power, etc. Maybe the radiance reminded someone of the Lucifer account, and suggested the notion of people seeking strength in the wrong place, etc.
I don't know much about paganism, and I don't even know if pagans really used the pentagram. I didn't know that it was often represented with a circle instead of a pentagon.
Since this thread has been about how the pentagram got its bad connotation, that sounds more like psychology than parapsychology. Does sciforums have a psychology forum? That's more where this thread fits best.
Mike Ossipoff