- 3125 - 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... 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...
----------------------- 3125 - MATH - the Golden Ratio
- The ratio is an irrational number where the decimal digits go on forever and never repeat themselves. The Golden Ratio = 1.6180339887 ……….
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- 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.
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- 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...
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- 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:
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----------- 1.618 / 1 = 1 / (1.618 - 1)
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----------- 1.618 = 1 / 0.618
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------------ If you cross multiply and set the equation equal to zero you get the quadratic equation:
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------------ 1.618^2 - 1.618 - 1 = 0
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------------ Solving this using the quadratic equation solution:
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------------ Solution = (-b+(b^2 - 4*a*c)^0.5) / 2*a
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------------ Solution = (1+(5)^0.5) / 2
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------------ Solution = (1 + square root of 5) over 2
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- The Fibonacci number series is where each successive number is the sum of the previous two numbers:
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-------------- 1, 1, 2, 3, 5, 8, 13, 21, 34, ………………
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-------------- If you take the ratios of these successive numbers it approaches the Golden Ratio and equals it at infinity:
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------------ 1/1 = 1
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------------ 2/1 = 2
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------------ 3/2 = 1.5
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------------ 5/3 = 1.667
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------------ 8/5 = 1.600
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------------ 13/18 = 1.625
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------------ 21/13 = 1.615
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------------ 34/21 = 1.619
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------------ 55/34 = 1.618
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------------ 89/55 = 1.618
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------------ etc. = 1.618.…
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- The ratios oscillate but continually approach the Golden Ratio, 1.618.….
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- Any power of 1.618 is equal to the sum of the two immediate powers:
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--------------- 1.618^0 = 1
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--------------- 1.618^1 = 1.618
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--------------- 1.618^2 = 2.618
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--------------- 1.618^3 = 4.236
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--------------- 1.618^4 = 6.854
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--------------- etc………
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- The Golden Ratio can be written as this infinite series of ratios of “1“:
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------------ 1.618... = 1+ 1 / ( 1 + 1 / ( 1 + 1 / ( 1 + 1 / ( 1 + 1 / ( etc ……..
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- The Golden Ratio can also be written as this infinite series of ratios of square roots:
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----------- 1.618... = square root (1 + square root ( 1+ square root (1+ square root ( 1+ etc…………….
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- 1.618... = 2 cosine (pi / 5), where pi is also an irrational number that goes on forever.
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- 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.…
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- 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.…
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- 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...
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- 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!
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- April 15, 2021 MATH - the Golden Ratio 853 3125
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--------------------- --- Thursday, April 15, 2021 ---------------------------
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