GOLDEN RATIO SPECTACULAR

 

 

View full size image 

 

Library page

 

 

 

Here’s something I didn’t know about the golden mean, or should I say “golden ratio”?  It would seem that “ratio” is the accepted term these days.

 

My caller told me she had a tidbit of information that she got at a recent NEARA  conference:

 

The reciprocal of the golden ratio has the identical decimal fraction as the golden ratio!  Start with the golden ratio which is 1.618 approximately.  Divide 1 by that.  Get 0.618 approximately.  Same digits of 618.  Now this was just stunning news to me even though I suppose it’s on every table of reciprocals ever printed.

 

I immediately got my calculator to verify this and learn more using a little high school math.

 

Golden digits in the reciprocal

 

The golden ratio is 1.6180339887498948482045868343646 etc.

 

To get the reciprocal of a number, divide 1 by that number.

 

1 / 1.6180339887498948482045868343646 = 0.61803398874989484820458683436626

 

Thus, the reciprocal of the golden ratio is 0.61803398874989484820458683436626 and has the same decimal fraction as the golden ratio (it works!).  It is as identical as the calculator will let it be.  Here I am using the 30+ place calculator that comes with my Windows operating system:  Start / Programs / Accessories / Calculator.

 

Golden digits in the square

 

Next I set up an equation representing the golden ratio’s reciprocal.

 

Let x = the golden ratio

The reciprocal = 1/x

As seen above, the reciprocal also = x – 1

 

equation

1/x = x – 1

 

multiplying both sides of the equation by x I get

1 = x² – x

transposing to solve for x²

x² = x + 1

 

Now this was another revelation for me.  From this equation I can see that the square of the golden ratio is the golden ratio plus one and the square therefore has the same decimal fraction!

 

The square is 2.6180339887498948482045868343603.  Same digits of 618 etc.

 

Golden sum equals golden product

 

Now if I turn the equation around it will help me make my next point.

x + 1 = x²

 

multiplying both sides of the equation by x I get

x² + x = x³

 

Now I suppose you thought that x² times x = x³.  (It does.)  But in the case of the golden ratio, x² plus x = x³ is also true.  Fantastic!

 

Golden series

 

I noticed something here and that is when I reverse the values being added I get:

 

x + x² = x³

 

multiplying both sides of the equation by x I get

x² + x³ = x⁴

 

repeating

x³ + x⁴ = x⁵

 

with this information I can set up a series where each value in the series is one power greater than the previous, and each value equals the sum of the previous two values:

 

x, x², x³, x⁴, x⁵ . . . . x^n-1, xⁿ

 

This is reminiscent of the Fibonacci Series where each value in the series equals the sum of the previous two values:

 

for example, 1, 1, 2, 3, 5, 8, 13, 21 . . . . 6765, 10946, 17711, etc.

 

Finding the golden ratio fraction

 

By solving my first equation above for x, I can get the fraction which represents the golden ratio.  That fraction is (1 + √5)/2.  I hadn’t known that and had waited a long time for it.

 

Here’s the calculation, using a technique called “completing the square” (ninth year math).

 

Let x = golden ratio

1/x = x – 1

1 = x² – x

x² – x = 1

x² – x + 1/4 = 1 + 1/4

(x – 1/2)² = 5/4

(x – 1/2) =  plus-minus √(5/4)

x = (1/2) + (√5 / 2) = (1  + √5)/2 = 1.6180339887498948482045868343656

and x = (1/2) – (√5 / 2) = (1 – √5)/2 = minus 0.6180339887498948482045868343656

 

So these are the two values that will fit the equation.  I discarded the negative value because I can’t use it for drawings.

 

The Fibonacci Series gives fractional equivalents that approach the golden ratio (as the series progresses, the quotients of successive numbers become closer and closer to the golden ratio), but the exact fraction representing the golden ratio is (1 + √5)/2.

 

By the way, my decimal value for the golden ratio developed from this fraction varies slightly from that in my earlier post and earlier in this post, a value I derived from sine values.  The derivation may explain why the final two digits are different.  Also, end digits tend to get fuzzy with numerous calculator operations.

 

Finding the golden ratio in the altitude

 

Many people know that the golden ratio can be found in the relationship among the various sized segments of the pentagram, as follows:  medium segment (M) to short segment (S), long (L) to medium (where “long” equals a short plus a medium segment), longest (LL) to long, all these ratios equal 1 + √5 to 2, or 1.6180339887498948482045868343656 to one, in other words, the “golden ratio.”  (See my graphic at top.)

 

These segments make up triangles within the pentagram design.  What about an altitude in such a triangle?

 

Visualize an isosceles triangle found in a pentagram design (angles of 72, 72, 36) with sides of 1 + √5, 1 + √5, and 2.  Within that are two right triangles, each with hypotenuse equal 1 + √5 and side opposite angle of 18 degrees equal 1.  Use Pythagoras to figure the third side, which is the altitude (a) of the isosceles triangle. 

 

Pythagoras says:  a² + b² = c²   (in a right triangle the sum of the squares of the legs equals the hypotenuse squared)

a = √(c² – b²)  = √((1  + √5)² – 1²) = √((1 + √5 + √5 + 5) – 1) = √(5 + 2√5) = 3.0776835371752534025702905760369

 

I played a bit with my calculator and found that this value for the altitude is the product of the golden ratio and the square root of the-golden-ratio-plus-2, as follows:

 

3.0776835371752534025702905760369 = 1.6180339887498948482045868343656 x √3.6180339887498948482045868343656

 

So the 618 etc. digits and the golden ratio are in the altitude also.

 

The sides of the 18-90-72 right triangle are respectively, 1, 2x, x(√(x + 2)), where x is the golden ratio.  Kind of neat.

 

The number 3.618 etc. is also the product of √5 and the golden ratio.  Just found that by clicking around.

 

For what it is worth, 3.618 can be expressed at least three different ways:

3.618 = x √5

3.618 = x + 2

3.618 = x² + 1

where x is the golden ratio.

 

Very nice that the golden ratio is there in the altitude also.

 

Conclusion

 

What can I conclude but that I have no idea why the golden ratio (or its decimal fraction) show up here and there and in so many disparate places:  in the proportions of the pentagram, the circles associated with the pentagram and pentagon, the altitude for a triangle in a pentagram design, the Fibonacci Series, the golden ratio reciprocal, the golden ratio square, and as some say (and I haven’t learned about this yet) in spirals in nature.  I suspect there is something to learn about the essence of our Existence in all this, but what?  Is it spectacular?