- 2076
- - Math, the Golden Ratio. If you divide a straight line segment into
“a” and “b” , two segments and make the length of the segments such that a +
b/a =
a/b, then, their ratio is always = 1.618... If you make the ratio of the entire line
divided by the larger segment = the larger segment divided by the smaller
segment the ratio is always the Golden Ratio which is = 1.618...
-
-
-
------------------------------ 2076 -
Math, the Golden Ratio
-
----------------------------- a + b / a = a / b
-
- If you divide
a straight line segment into “a” and “b” , two segments and make the length of
the segments such that a + b/a = a/b then their ratio is always =
1.618... In other words if you make the
ratio of the entire line divided by the larger segment = the
larger segment divided by the smaller segment the ratio is always the Golden
Ratio which is 1.618...
-
- Actually, the
ratio is an irrational number where the decimal digits go on forever and never
repeat themselves. The Golden Ratio =
1.6180339887 ……….
-
- If you make a
square inside a rectangle from this line segment and then a square outside the
rectangle and continue to do this you will create a series of Golden
Rectangles. This pattern of an expanding
logarithmic spiral is found in the shape of sea shells and the shape of spiral
galaxies. It is also the ratio chosen by
artist in paintings and sculptures who
see it as the most aesthetic form. The
same applies to buildings and ancient pyramids.
The rectangles of books and credit cards all approximate this unique
mathematical ratio, 1.618...
-
- I will use
the 1.618 to designate the ratio but remember this is only a portion of the
number which is irrational and extends forever.
But, the math relationships are amazing:
-
-----------
1.618 / 1 = 1 / (1.618 - 1)
-
-----------
1.618 = 1 /
0.618
-
------------ If
you cross multiply and set the equation equal to zero you get the quadratic
equation:
-
------------
1.618^2 - 1.618 -
1 = 0
-
------------
Solving this using the quadratic equation solution:
-
------------
Solution = (-b+(b^2 - 4*a*c)^0.5) / 2*a
-
------------
Solution = (1+(5)^0.5) / 2
-
------------
Solution = (1 + square root of 5) over 2
- The Fibonacci
number series is where each successive number is the sum of the previous two
numbers:
--------------
1, 1, 2, 3, 5, 8, 13, 21, 34, ………………
--------------
If you take the ratios of these successive numbers it approaches the
Golden Ratio and equals it at infinity:
-
------------
1/1 = 1
------------
2/1 = 2
------------
3/2 = 1.5
------------
5/3 = 1.667
------------
8/5 = 1.600
------------
13/18 = 1.625
------------
21/13 = 1.615
------------
34/21 = 1.619
------------
55/34 = 1.618
------------ 89/55
= 1.618
------------
etc. = 1.618.…
-
- The ratios
oscillate but continually approach the Golden Ratio, 1.618.….
-
- Any power of
1.618 is equal to the sum of the two immediate powers:
-
---------------
1.618^0 = 1
---------------
1.618^1 = 1.618
---------------
1.618^2 = 2.618
---------------
1.618^3 = 4.236
---------------
1.618^4 = 6.854
---------------
etc………
-
- The Golden
Ratio can be written as this infinite series of ratios of “1“:
-
------------
1.618... = 1+ 1 / ( 1 +
1 / ( 1 + 1 / ( 1 + 1 / ( 1 + 1 / ( etc ……..
-
- The Golden
Ratio can also be written as this infinite series of ratios of square roots:
-
-----------
1.618... = square root (1 + square root ( 1+ square root
(1+ square root ( 1+ etc…………….
-
- 1.618... = 2 cosine (pi / 5), where pi is also an
irrational number that goes on forever.
-
- Roger
Penrose, a great mathematician, created tile shapes that could cover any flat
area. The two shapes are 4 sided
polygons called “darts” and “kites”. As
the tiled area gets larger and larger the ratio of “darts” to “kites”
approaches 1.618.…
-
- Paul Davies,
a physicists, discovered that Black Holes transform from negative to positive
specific heat when the ratio of the square of the mass to the square of the
angular momentum = 1.618.…
-
- All of these
and many, many more mathematical relationships come out of nature when you make
a/b = 1.618 and a+b / a = 1.618... It
all seems like a simple ratio but the result goes on forever and the result is
found in many places in nature and in art.
-
- Amazing!
-
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to
whomever is interested. ----
---
Some reviews are at:
--------------
http://jdetrick.blogspot.com -----
--
email feedback, corrections, request for copies or Index of all reviews
------------------------- Sunday, April 29, 2018
--------------------------------
-----------------------------------------------------------------------------------------
No comments:
Post a Comment