Question:
I was just looking at some information about the golden
proportion. One formula I saw was (a+b)/a=a/b. I want to know what a
and b could equal to make this true.
Replies:
The golden proportion (golden ratio or golden rectangle) is 0.618.
You can derive this ratio from the formula. Substitute c=b/a then
1+(b/a)=1/(b/a) and
1+c=1/c or
c^2 + c -1=0 a quadratic equation that has as one of it solutions c=0.618
It is the ratio (close to 5/8) that is important not the specific values of a
and b.
If you construct a rectangle with the sides b/a=0.618 you can take a square from
the rectangle and it will leave a rectangle with the golden ratio for its sides.
You can do the same for the smaller rectangle and so on (see Parthenon in
Athens).
David S. Kupperman
This equation states a relation between two line segments "a" and "b". Let "a" be
the longer, and "b" be the shorter line segments. In words, the equation states
that if the ratio of the total line, "(a+b)" to the longer segment "a" (
symbolically (a+b) < a <1) is in the same ratio as the longer segment "a" is to
the shorter segment "b" (symbolically a > b > 1) then both ratios:
(a+b)/a = a/b = (1+ sqrt (5))/2 = phi = ~1.618034.
You are "free" to choose any "a" and any "b" you want, but notice that those
choices determine "a/b" uniquely provided "a" and "b" are not zero. Your free
choice also determines the other ratio too -- that is -- (a+b) / a.
Perhaps an easier way to grasp what is going on is to rearrange the formula:
b(a+b) = a^2
which can be rearranged further to give the equation: a^2 - ab - b^2 = 0. It now
becomes apparent that once you have selected "b" for example, and require
equality you have in effect fixed the value of "a" by the requirement of "... =
0 ".
A specific example: Let "b = 1" then the equation reads: a^2 - a -1 = 0. The
quadratic formula gives the positive root "a" = phi and the negative root: "a"
= -phi^-1. If you choose any other non-zero value of "b" the quadratic formula
gives the same two solutions.
There are many excellent books on the topic and web sites as well, e.g.:
http://en.wikipedia.org/wiki/Golden_ratio
Vince Calder
One formula would be b = a(Sqrt(5)-1)/2. So, if a = 1, b = 0.618034,
approximately.
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