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Golden ratio Lidium

From: Liddz.

72.81 divided by 45 = 1.618. Also 40.45 divided by 25 = 1.618. Also. 9.708 divided by 6 = 1.618. The Golden ratio can also be obtained in Trigonometry through the formula Cosine (36) = 0.809016994374948 multiplied by 2 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (36) = 0.809016994374948 x 2 = 1.618033988749895.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (72) = 0.309016994374948 multiplied by 2 = 0.618033988749895 plus 1 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (72) = 0.309016994374948 x 2 = 0.618033988749895 + 1 = 1.618033988749895. The Golden ratio can also be obtained in Geometry through Cosines of both multiples of 36 and 72 and remember to multiply the result twice to get the Golden ratio. The Golden ratio in Trigonometry can also be obtained through the formula 1+SIN (18) = 1.309016994374947 x 2 = 2.618033988749895 – 1 = 1.618033988749895. Please remember that 2.618033988749895 is The Golden section or Golden ratio squared and also has many applications. The Golden ratio squared can also be obtained in Trigonometry through the formula Cosine (36) x 2 + 1 = 2.618033988749895. The Golden ratio squared can also be obtained in Trigonometry through the formula 2+SIN (18) x 2 = 2.618033988749895.

Numerically with a digital calculator the Golden ratio is expressed as 1.618. The Golden ratio is connected to the square root of 5 and can also be obtained from the Square root of 5 by adding 1 to the Square root of 5 and then dividing the Square root of 5 plus 1 into 2. An example of obtaining the Golden ratio is 2.236 plus 1 equals 3.236. 3.236 divide by 2 equals 1.618. The Golden ratio can also be approximated from a mathematical infinite progression called the Fibonacci series with the remainder being the result of adding together the previous 2 numbers in the sequence. The Fibonacci progression is as follows: 1plus1 equals 2, 2 plus 1 equals 3, 3 plus 2 equals 5, 5 plus 3 equals 8, 8 add 5 equals 13, 13 add 8 equals 21, 21 add 13 equals 34, 34 add 21 equals 55, 55 add 34 equals 89. 89 divide by 55 is: 1.618181818181818818181818181818. Pi approximated to 864 divided by 275 = 3.141818181818182 divided by 1.2 = 2.618181818181818. 2.618181818181818 subtract = 1.618181818181818. 3.141592653589793 subtract 1.5235 = 1.618092653589793. An approximation of Pi can be obtained from the Golden ratio also if the Golden ratio is applied to Square root and the result of applying the Golden ratio to square root is the ratio 1.27 and if 4 is divided by 1.272727272727273 the result is Pi approximated to 22 divided by 7 = 3.142857142857143. The square root of the Golden ratio approximated to: 1.619834710743802 is 1.272727272727273. Cosine (36) multiplied by 2 is the Golden ratio of 3.144605511029693and if 1.618033988749895 is applied to square root the result is 1.272019649514069. 4 divided by 1.272019649514069 is Golden Pi = 3.144605511029693. Cosine (36) = the Golden ratio of 1.618033988749895. If the Golden ratio of 1.618033988749895 is divided by Pi approximated to 22 divided by 7 = 3.142857142857143 then the result is 0.514828996420421 and if 0.514828996420421 is multiplied by 5.28 then the result is an approximation for the mathematical constant that is known as E = 2.718297101099824. The sequence can also work with the numbers above being doubled such as 2,4,6,10,16,26,42,68,110,178. 178 divide by 110 is 1.618181818181818818181818181818. The further the progression is extended the closer to view the Golden ratio becomes.

Also 72.81 divided 45 = 1.618. Also 40.45 divided by 25 = 1.618. 9.708 divided by 6 = 1.618.

The Golden ratio can also be approximated very closely if 121393 divided by 75025 = 1.618033988670443. If 121393 is added to 75025 the result is 196418. If 196418 is divided by 121393 the result is 1.618033988780243.

If 196418 is added to 121393 the result is 317811. 317811 divided by 196418 = 1.618033988738303. If 317811 is added to 196418 = 514229. 514229 divided by 317811 = 1.618033988754323. If 514229 is added to 317811 the result is 832040. 832040 divided by 514229 = 1.618033988748204.

Please note the repeating numbers 1.6180339887….. after 196418 has been divided by 121393. So the Golden ratio can be approximated to 1.6180339887 plus an infinite amount of decimal point numbers that never repeat. So the Golden ratio can be reduced to 1.6180339887 because after 1.6180339887 there is an infinite amount of decimal point numbers that are never repeated but 1.6180339887 is repeated after 196418 is divided by 121393.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (36) = 0.809016994374948 multiplied by 2 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (36) = 0.809016994374948 x 2 = 1.618033988749895.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (72) = 0.309016994374948 multiplied by 2 = 0.618033988749895 plus 1 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (72) = 0.309016994374948 x 2 = 0.618033988749895 + 1 = 1.618033988749895. The Golden ratio can also be obtained in Geometry through Cosines of both multiples of 36 and 72 and remember to multiply the result twice to get the Golden ratio. The Golden ratio in Trigonometry can also be obtained through the formula 1+SIN (18) = 1.309016994374947 x 2 = 2.618033988749895 – 1 = 1.618033988749895. Please remember that 2.618033988749895 is The Golden section or Golden ratio squared and also has many applications. The Golden ratio squared can also be obtained in Trigonometry through the formula Cosine (36) x 2 + 1 = 2.618033988749895. The Golden ratio squared can also be obtained in Trigonometry through the formula 2+SIN (18) x 2 = 2.618033988749895.

Many mathematicians have called the Golden ratio the most pleasing ratio in existence.

Constructing Golden logarithmic spirals that all share the same centre in a circle:

A circle can be constructed with opposing logarithmic spirals originating from the centre of the circle. To divide a circle into spirals construct a rectangle that is based upon the Golden ratio with the longer length of the Golden rectangle with equal length to the radius of the circle. Now construct a second Golden rectangle that has its longest edge the same as the shorter edge of the larger Golden rectangle. Both of the large and small Golden rectangles must have an edge that emanates from the centre of the circle. The diagonals of the 2 Golden rectangles must then be divided into the golden ratio the same way that 3 divisions of the longest width of a Pentagram is divided. One rectangle must have a diagonal that is 31.71 degrees and the other rectangle must have a diagonal of 58.28 degrees or if the choice is to draw the rectangles tilted then another combination of 2 angles that add up to 90 degrees can be used just like both 31.71 and 58.28 add up to 90. 31.71 degrees and 58.28 degrees must come from the centre of the circle and can expand to infinity by the Golden ratio. Any 1 of the 2 longer divisions of the diagonals that are located on the sides of the smaller middle section must then be used to extend the diagonals through the centre of the circle, allowing 2 new larger golden rectangles to be constructed. Please remember for the purpose of accuracy that when a smaller Golden rectangle shares it’s longest edge length with the shorter edge length of a larger Golden rectangle the full length of the whole rectangle will have a ratio of the square root of 5. The longer length of the shorter Golden rectangle is the shorter length of the Square root of 5 rectangle. Remember again that the longest length of the larger Golden rectangle is equal to the radius of the circle so that the spirals that are drawn from the squares that make the Golden rectangle can all originate from the centre of the circle. Now Golden spirals can be constructed from the largest Golden rectangles with the spirals attempting to emanate from the centre of the circle. The circumference of the circle can be divided into equal sections with the process of constructing Golden spirals that attempt to emanate from the centre of the circle being repeated to infinity if desired by expanding the radius of the circle to form new circles that share the same centre with the method of the Fibonacci progression.

The radius of the circles that all share the same centre must expand by the Golden ratio if the sizes of the spirals are to be increased. If opposing spirals are drawn from the centre of the circle and are on opposite sides and above each other then they can be interpreted as the spirals of a Galaxy. Please note that an unfortunate fact exists and that is the spiral or spirals that can be created from the squares that form a Golden rectangle cannot completely touch the centre of the circle even though the spirals do come close to touching the centre of the circle. The reason why using squares that form the Golden rectangle to create spirals that are desired to touch the centre of the circle cannot be achieved 100% is because the Squares that form the Golden rectangle are created in ratio to each other based upon the Fibonacci sequence and the Fibonacci sequence begins with 1 plus 1 then 2 plus 1 equaling 3, so to obtain the size of the next square the number that proceeded the current number must be added to the current number.

If the Geometer is determined to make the Golden spiral proceed to the centre of the circle that must have both 31.71 degrees and 58.28 degrees emanating from the centre of the circle to indicate the point for the spiral to emanate then the only solution is to divide 1 of the 2 smallest squares into a Golden rectangle. The Golden rectangle that allows the logarithmic spiral to reach the centre of the circle must have its longest edge length equal to 1 of the 2 smallest squares that make up the whole largest Golden rectangle. The Golden rectangle that has its longest edge length equal to 1 of the 2 smallest squares that make up the largest Golden rectangle must also be divided into squares if possible.

Constructing a Golden rectangle inside of any random circle:

Please remember that a Golden rectangle can be constructed inside any random circle when a double square-rectangle is constructed inside of that circle that has the diameter of the circle being equal to the diagonal of the rectangle that is made of 2 squares and is constructed inside of the circle. Please remember that if a Golden rectangle is placed on the 2 parallel edges of a square then the total height of the new rectangle is Square root of 5. This knowledge is also used to construct an Icosahedron inside of a Sphere.

• The Golden ratio:

The Golden section is a ratio that has a line with the small section being the same as the larger section is to the whole length of the line. The Golden ratio first appears when a circle is divided into 5 equal parts producing a Pentagon with the Pentagon’s diagonals being connected to the 5 vertices of the Pentagon inside the circle touching the circle’s circumference. Applications for the Golden ratio are many and the author of this book does NOT have enough space to list all of them here. Some of the applications for the Golden ratio include, constructing a Pentagon inside a circle.

“Using the Golden ratio to construct a Pentagon and also a Decagon”:

The Golden ratio must be used to construct a Pentagon because the edge of a Pentagon in relation to it’s diagonal is the Golden ratio. Here is another method for constructing a Pentagon inside of a circle .The Golden ratio can also be used to divide the circumference of a circle into 10 equal parts because one tenth of a circle’s circumference is the smaller section of the Golden ratio in relation to the radius of a circle. Remember that a Pentagon is half of a Decagon. The radius of a circle can be divided into the Golden ratio Geometrically with just the use of a compass and a straight edge and obviously a stylus by first dividing the radius of a circle into half by constructing a Vesica Pisces with the base being equal to the radius of the circle, the base of the Vesica Pisces and the radius of the circle must share the same base. Next connect a line between the 2 opposing apexes of the Vesica Pisces so that the radius of the circle can be divided into 2. Next construct a semi-circle or a circle with the radius of the large circle that will contain the Pentagon being the diameter of the smaller circle, so that a rectangle made of double squares can be constructed on the radius of the large circle. The radius of the large circle must be the long edge of the rectangle that is made from 2 squares, the short length of the rectangle that is made from double squares is equal to half of the radius of the large circle that contains the Pentagon. Next swing an arc with a compass with a measurement of the short part of the rectangle that is made from double squares onto the diagonal of the rectangle that is made from double squares, next swing another arc from the larger division of the diagonal of the rectangle that is made from double squares onto the radius of the large circle that contains the Pentagon. The radius of the circle that is to contain the Pentagon has been divided into the Golden ratio, take the larger part of the circle’s radius that has been divided into the Golden ratio and multiply it 2 times from the point on the circumference of the circle that has the radius of the circle touching it so that one fifth of the circle’s circumference can be obtained. Now that one fifth of the circle’s circumference has been obtained by dividing the radius of the circle into the golden section and multiplying the larger part twice, next multiply one fifth of the circle’s circumference until the Pentagon is completed and connect the diagonals of the Pentagon up to the 5 vertices of the circle’s circumference so that a 5 pointed star that is based upon the Golden ratio can be observed.

“Inscribing a Pentagon in a circle when only the measure for the radius or diameter of the circle is known or finding the measure for the Pentagon’s radius”:

A method for determining the measure for the edge of a Pentagon when only the measure for the radius or diameter of the Pentagon’s circle is known is to add the result of the circle’s radius squared to the result of the circle’s decagon squared and then apply the result of adding together both the results of the circle’s radius squared and the circle’s decagon squared to square root. Remember the edge of a decagon is obtained by dividing the circle’s radius into the Golden ratio of 1.6180339887498. The edge of a decagon can easily be obtained by also dividing the circle’s circumference into 10 equal units of measure.

Another solution for determining the edge length of a Pentagon when only the radius and diameter of the Pentagon’s circle is known is to take the measure of the diameter of the Pentagon being added to the edge of a decagon that has a radius equal in measure to the Pentagon’s radius must be squared and added to the measure for the Pentagon’s diameter squared and then the result must be applied to square root and the result will be the hypotenuse of a scalene triangle with its shortest length being equal to the diameter of the Pentagon and the second longest length of this scalene triangle is equal in measure to the measure for the Pentagon’s diameter being added to and combined with the edge of a decagon with a radius equal in measure to the Pentagon’s radius. The hypotenuse of the scalene triangle with its shortest length being equal in measure to the Pentagon’s diameter while the second longest length of the scalene triangle is equal in measure to the Pentagons diameter plus the edge of a decagon with a radius equal in measure to the Pentagon’s radius must be divided into the square root of 3 = 1.732050807568877 and the result of dividing the hypotenuse of this scalene triangle into the square root of 3 = 1.732050807568877 and the larger part of the division of the hypotenuse into the square root of 3 must be divided into the Golden ratio and the larger part of that new division from the hypotenuse being divided into the Golden ratio is the measure for the edge length of the Pentagon. The equation above can be simply written as:

1. Diameter of circle added to the length of the edge of the circle’s decagon.

2. Diameter of circle added to the length of the edge of a decagon squared and the results remembered.

3. Diameter of circle squared and the results remembered.

4. The result of the circles diameter squared added to the length of the circle’s diameter combined with the circle’s decagon edge length squared

5. The sum of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root.

6. The result of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root results in a measure that must be divided by the square root of 3 = 1.732050807568877.

7. The result of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root results in a measure that must be divided by the square root of 3 = 1.732050807568877 and then divided by the Golden ratio of 1.6180339887498 and the result is the measure for the edge of the circle’s Pentagon.

Another method for determining the measure for the edge of a Pentagon when only the measure for the radius or diameter of the Pentagon’s circle is known is to multiply the measure for the radius of the Pentagon by: Cosine (54) multiplied by 2 So if the radius of the Pentagon is 55 equal units of measure then multiply 55 by Cosine (54) multiplied by 2 and the result for the edge of the Pentagon with a radius of 55 equal units of measure is 64.656377752172052. Remember to reduce the results to 6 or 5 or 4 or 2 decimal places then the measure for the edge of the Pentagon can also be known in addition to the radius and diameter for the Pentagon’s circle already being known. The ratio 1.175570504584946 is the result of dividing the edge length of a Pentagon by the radius of a Pentagon and can be obtained in Trigonometry through the formula Cosine (54) = 0.587785252292473 multiplied by 2 = 1.175570504584946. The ratio 1.175570504584946 can be achieved through the short written formula in Trigonometry as Cosine (54) = 0.587785252292473 x 2 = 1.175570504584946. The ratio 1.175570504584946 reduced to 4 decimal places reads as 1.175.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter a solution is to divide the measure for the edge of the Pentagon by Cosine (54) and the result can be the diameter for the Pentagon and the radius of the Pentagon is obtained by dividing the diameter of the Pentagon by 2 equal halves. So if the edge length of the Pentagon has 34 equal units of the measure and the desire is to know the diameter and radius of a Pentagon with a edge length of 34 equal units of measure the solution can be written as 34 divided by Cosine (54) = 57.844254967938708 equal units is the measure of a diameter for a Pentagon with 34 equal units of measure and 28.922127483969354 equal units is the measure for the radius of a Pentagon with a edge length of 34 equal units of measure. Written shorter the formula for obtaining the radius of a Pentagon with an edge length of 34 equal units of measure is 34/Cosine (54) = diameter of Pentagon = 57.844254967938708. Diameter of Pentagon = 57.844254967938708 divided by 2 = radius of Pentagon: 28.922127483969354.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter a solution is to divide the measure for the edge of the Pentagon by 0.6180339887498 or multiply the measure for the edge of the Pentagon by 1.6180339887498 and the result is the length of the Pentagon’s diagonal. Multiply the Pentagon’s diagonal by the Square root of 3 = 1.732050807568877. Remember the result of the Pentagon’s diagonal being multiplied by the Square root of 3. Divide the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 by the Square root of 5 = 2.23606797749979. Remember the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979. The result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = must now be squared and the result must also be remembered. The result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 must now be divided into the Golden ratio of 1.6180339887498 and the result must also be squared and remembered. The results of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then squared must then be added to the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then being divided by the Golden ratio of 1.6180339887498 and then squared and then applied to Square root. The results of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then squared must then be added to the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then being divided by the Golden ratio of 1.6180339887498 and then squared and then applied to Square root and then divided by the Square root of 3 = 1.732050807568877 and the new result can now be the radius of the Pentagon. So a simple formula for the equation above can be written as:

1. Edge of Pentagon divided by 0.6180339887498 or Edge of Pentagon multiplied by 1.6180339887498 resulting in the measure for the Pentagon’s diagonal.

2. Pentagon’s diagonal multiplied by the Square root of 3 = 1.732050807568877.The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 must then be divided by the Square root of 5 = 2.23606797749979.

3. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 must now be squared and remembered.

4. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then divided into the Golden ratio of 1.6180339887498.

5. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then be divided by the Square root of 5 = 2.23606797749979 and then divided into the Golden ratio of 1.6180339887498 and then squared and remembered.

6. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then be divided by the Square root of 5 = 2.23606797749979 and then squared and remembered must now be added to the length of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then divided by the Square root of 5 = 2.23606797749979 and then being divided into the Golden ratio of 1.6180339887498 and squared.

Now that the length of the Pentagon’s diagonal has been multiplied by the Square root of 3 = 1.732050807568877 and the result of such has been divided by the Square root of 5 = 2.23606797749979 and then squared and remembered. The result of the length of the Pentagon’s diagonal multiplied by the Square root of 3 = 1.732050807568877 has been divided by the Square root of 5 = 2.23606797749979 and has be divided by the Golden ratio of 1.6180339887498 and squared and remembered. The sum of the result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then divided by the Golden ratio of 1.6180339887498 and then squared and remembered must now be added to the result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 squared and remembered and the result is the measure for the length of the Pentagon’s radius.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter of the Pentagon another solution is to:

1. Have half of the edge of the Pentagon squared and the result remembered.

2. Half of the edge of the Pentagon multiplied by the ratio 1.376381920471173 and the result also squared and remembered. The ratio 1.376381920471173 can be obtained on a digital calculator through the Trigonometric formula TAN (54). .

3. Add the result of half of the edge of the Pentagon squared to the result of half of the edge of the Pentagon multiplied by 1.376381920471173 and then apply the result of both combined to Square root and the final result is the measure for the radius of the Pentagon.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter of the Pentagon another solution is to:

1. Multiply half the edge length of the Pentagon by TAN (54) = 1.376381920471173

2. Divide the result of multiplying the edge length of the Pentagon by TAN (54) = 1.376381920471173 by the Golden ratio of 1.618033988749895.

3. The result of the edge length for the Pentagon being multiplied by TAN (54) = 1.376381920471173 and then divided into the Golden ratio of 1.618033988749895 must now be multiplied by 2 resulting in the measure for the radius of the Pentagon.

“How to obtain the height of a Pentagon”:

The height of a Pentagon can be obtained by:

1. Multiplying half the edge of the Pentagon by the ratio 3.077683537175253 and the result will be the height of the Pentagon. The ratio 3.077683537175253 can be obtained digitally on a calculator through the Trigonometric formula TAN (72).

2. The height of a Pentagon can also be obtained if half of the edge of the Pentagon is multiplied by the ratio 1.376381920471173 and the result then added to the radius of the Pentagon from the centre of the circle. The ratio 1.376381920471173 can be obtained from a calculator through the Trigonometric formula TAN (54).

3. If the radius of a circle is divided in half and then divided by 0.6180339887498 or multiplied by the Golden ratio of 1.6180339887498 and then the result then added to the radius of the circle from the centre of the circle the total length is equal to the height of a Pentagon that can be created to fit in the circle.

“How to obtain the radius of a Pentagon with Trigonometry if the height of the Pentagon is already known”:

So the height of the Pentagon has already been determined by multiplying half the edge of the Pentagon by TAN (72) = 3.077683537175253 and the desire is to know the radius of the Pentagon and a simple solution is to multiply half the edge of the Pentagon by TAN (54) = 1.376381920471173 and subtract the result of multiplying the edge of the Pentagon by TAN (54) = 1.376381920471173 from the height of the Pentagon and the result is the radius of the Pentagon.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter another solution is to divide the measure for the edge of the Pentagon by 1.170 or 1.175 or 1.175570504584997 or 1.175851692873989 or 1.175656984785615 or 1.1756124220922 or 1.175454545454545 or 1.175636363636364 or 1.175505023709698 or 1.175570504584965 and the result will be an approximation for the radius of the Pentagon and the result must be reduced to 6 or 5 or 4 or 2 decimal places for better accuracy. Remember that if the radius of a circle is divided into the Golden ratio the result is the larger part of the division being equal to the measure for the edge of a decagon that has a radius equal in measure to the radius of the circle.

Geometrically the radius of a Pentagon can become visible if all the 5 edges of the Pentagon are divided in half with lines that pass through the centre of the Pentagon’s circle that also touch the centre edges of the Pentagon.

It should become evident from reading above that any line can be divided into the Golden ratio when a double square is constructed over that line and then swing an arc with a compass measurement equal to half of the double square on to the diagonal of the double square and then swinging an arc with compass measurement of the larger division of the diagonal of the rectangle that is made from double the squares. If the desire is to find the shorter part of the Golden ratio when you have a line and your desire is to make the line longer the use of a square that is divided into half so that the double square can be used by swinging an arc from the centre edge of the large square with a compass measurement equal to the diagonal of the 2 rectangles that are formed of double squares that also make up the largest square onto the extension of the line. Remember that a square is also made up from 4 smaller squares. The height of a Pentagon can also be found if a Golden rectangle is constructed from the centre of the circle that contains the Pentagon with the shorter edge of the Golden rectangle being equal to half the radius of the circle that contains the Pentagon.

• Constructing the shape of the Great pyramid.

Constructing the Great pyramid isosceles triangle from 2 Golden rectangles”:

The slope that is located in the centre of the 4 triangular faces of the largest of the 3 of the Egyptian Great pyramids of Giza can be constructed from 2 vertical golden rectangles. When constructing the central slope of the largest of the 3 Great Egyptian Pyramids an arc that has equal measure to the longest length of any of the 2 golden rectangles can be used and swung unto the centre of the rectangle that is made from the 2 Golden rectangles. The result will be a triangle that has a base length that is the longest edge and equal to 11 units of equal measure while the height of the triangle will be 7 units of equal measure relating the edge length that is the base of the triangle.

“Constructing the Great pyramid triangle from 2 over lapping circles called the Vesica Pisces”:

The triangle that forms the 11 and 7 ratio triangle also forms the central slope of the largest of the 3 Great Egyptian Giza Pyramids and can also be obtained from 2 over lapping circles of equal size that have circumferences that touch each other’s centers. The height of the Pyramid’s triangle is 7 equal units while the base is 11 equal units and the height of the Pyramid’s triangle shares the area called the Vesica Pisces. When the Vesica Pisces is used to construct the shape of a Pyramid that has a square base width of 11 equal units of measure and 7 equal units of measure for the height of this Pyramid triangle then the height of the this pyramid triangle will be equal to the edge of an equilateral triangle that is contained with in the circumference of any of the 2 circles that are used to make the Vesica Pisces.

“Constructing the shape of the Great pyramid triangle upon the diameters of a circle:”

To construct the Great pyramid isosceles triangle upon any of the diameters of a circle the radius of the circle must be multiplied around the circumference of the circle 2 times from any of the circles poles that are on the end of the circles diameters. The result of multiplying the radius of the circle 2 times around the circumference of the circle from any of the circle’s pole diameters is the creation of a measure that is equal to one third of the circle’s circumference. From each end of the measure that is equal to one third of the circle’s circumference swing arcs down on the opposing diameter resulting in the creation of the Great pyramid triangle with the radius of the circle being the height of this Great pyramid isosceles triangle. The apex of this created Great pyramid isosceles triangle is also located on the pole of the appropriate diameter of the circle being also located on the circumference of the circle.

• Golden ratio derived from constructing a scalene triangle based upon the ratio of the moon and Earth touching each other ‘s circumferences:

The ratio of the radius and diameter of the Earth compared to the radius and diameter of the moon is 11 equal units of measure compared to 3 equal units of measure for the diameter of the moon. So the radius and Diameter of the Earth is 11 equal units of measure and from this measure of 11 equal units of measure 3 units of equal measure can be used to obtain the diameter of the moon. 11 divide by 3 produces the ratio: 3.6666666666667. Now if a Scalene triangle is constructed from the circumference of the moon touching the circumference of the earth the apex of the Scalene triangle will be located on the centre of the moon with the second shortest length of the Scalene triangle being equal to the radius of the Earth plus the radius of the moon. The smallest edge of the scalene triangle will only be equal to the radius of the Earth. The largest part of the scalene triangle will be the larger part of the Golden ratio when compared to the radius of the Earth that is the smaller part of the Golden ratio. If the radius of the Earth and the shortest length of the Scalene triangle is 5 and a half of 1 equal units of measure then the second longest edge of the Scalene triangle that is derived from the Earth and moon ratio will be 7 units of equal measure. Since the shortest edge of the Scalene triangle and the radius of the Earth is 5 and a half of 1 equal units of equal measure the longest edge of the Scalene triangle will be 8 point 9 units of equal measure. 8 point 9 divided by 5 and a half of 1 is: 1.618181818181818818181818181818. (An approximation of the Golden ratio).

The second longest length of the Kepler scalene triangle divided by the shortest length of the Kepler scalene triangle is the square root of the Golden ratio: = 1.272727272727273. 1.272727272727273 squared is also an approximation of the Golden ratio of 1.619834710743802.

The Scalene triangle that is used above in the Earth and moon ratio to find the Golden ratio is called a Kepler triangle. The Kepler triangle is also found in a Pyramid that has a square base width of 11 equal units of measure and 7 equal units of measure for the height of the Pyramid and the Great Egyptian pyramid of Giza is such as Pyramid.

• Constructing a Great pyramid Isosceles triangle inside the circumference of a circle, How to construct a isosceles triangle that is made from 2 Kepler scalene triangles inside the circumference of a circle:

If the diameter of a circle is divided into the Golden ratio then the larger part of the measure of the circle’s diameter that has been divided into the Golden ratio can be equal to the height of an isosceles triangle that is made from 2 Kepler Scalene triangles. Also the height of the Isosceles triangle that is made from 2 Kepler Scalene triangles is equal to the distance between the pole of the circle’s diameter that is opposite to the apex of this Isosceles triangle and any of the 2 base points of this Isosceles triangle that can touch the circumference of the circle. If the diameter of the circle that contains this isosceles triangle that is made from 2 Kepler scalene triangles is 140 equal units of measure then the height of this isosceles triangle will be 86.52 equal units of measure, meanwhile the base of this isosceles triangle will be 135.96 equal units of measure. Half of the base width of this isosceles triangle can be 67.98 equal units of measure while the length of the 2 longest edges can be 110.004 equal units of measure. If any of the longest edges of this isosceles triangle that is made from 2 Kepler scalene triangles is divided by half of the base width of this Isosceles triangle then again the result will be the approximation for the Golden ratio known as 1.618181818181818. Also if the diameter of the circle is divided by the height of this Isosceles triangle that is made from 2 Kepler Scalene triangles and is also contained with in the circle then the result will be the approximation for the Golden ratio known as 1.618122977346278.

Please remember that this isosceles triangle that is made from 2 Kepler scalene triangles can be used to create a circle that has a circumference equal to the perimeter of a square if both the circle and square share the same centre and the base width of this isosceles triangle that is made from 2 Kepler scalene triangles is equal to the width of the square and also the height of this isosceles triangle that is made from 2 Kepler scalene triangles is equal to the radius of the circle.

• Combined diameters of Earth and moon forming the second shortest length of a Kepler Scalene triangle that also includes the Golden ratio:

If the measure for the diameter of the moon is added to the measure for the diameter of the Earth then this combined new measure can be used as the second shortest length of a Kepler Scalene triangle that includes the Golden ratio. While the diameter of the Earth alone can be used as the shortest length of the Kepler Scalene triangle. The diameter of Earth again is 7920 statute miles and the diameter of Earth’s moon is 2160 statute miles. 7920 plus 2160 = 10080. If 10080 statute miles is the second shortest length of a Kepler Scalene triangle then 7920 is the shortest length of this Kepler Scalene triangle, while the longest length of this Kepler scalene triangle is 12816 statute miles. 12816 divided by 7920 = 1.618181818181818. 1.618181818181818 is an approximation of the Golden ratio of 1.618 and 1.618181818181818 can also be obtained if 8.9 is divided by 5.5.

Examples for applications of the Golden ratio are abundant through out nature including the formation of galaxies and the spiral curve of the human ear. : [en.wikipedia.org]

From: Liddz.

72.81 divided by 45 = 1.618. Also 40.45 divided by 25 = 1.618. Also. 9.708 divided by 6 = 1.618. The Golden ratio can also be obtained in Trigonometry through the formula Cosine (36) = 0.809016994374948 multiplied by 2 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (36) = 0.809016994374948 x 2 = 1.618033988749895.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (72) = 0.309016994374948 multiplied by 2 = 0.618033988749895 plus 1 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (72) = 0.309016994374948 x 2 = 0.618033988749895 + 1 = 1.618033988749895. The Golden ratio can also be obtained in Geometry through Cosines of both multiples of 36 and 72 and remember to multiply the result twice to get the Golden ratio. The Golden ratio in Trigonometry can also be obtained through the formula 1+SIN (18) = 1.309016994374947 x 2 = 2.618033988749895 – 1 = 1.618033988749895. Please remember that 2.618033988749895 is The Golden section or Golden ratio squared and also has many applications. The Golden ratio squared can also be obtained in Trigonometry through the formula Cosine (36) x 2 + 1 = 2.618033988749895. The Golden ratio squared can also be obtained in Trigonometry through the formula 2+SIN (18) x 2 = 2.618033988749895.

Numerically with a digital calculator the Golden ratio is expressed as 1.618. The Golden ratio is connected to the square root of 5 and can also be obtained from the Square root of 5 by adding 1 to the Square root of 5 and then dividing the Square root of 5 plus 1 into 2. An example of obtaining the Golden ratio is 2.236 plus 1 equals 3.236. 3.236 divide by 2 equals 1.618. The Golden ratio can also be approximated from a mathematical infinite progression called the Fibonacci series with the remainder being the result of adding together the previous 2 numbers in the sequence. The Fibonacci progression is as follows: 1plus1 equals 2, 2 plus 1 equals 3, 3 plus 2 equals 5, 5 plus 3 equals 8, 8 add 5 equals 13, 13 add 8 equals 21, 21 add 13 equals 34, 34 add 21 equals 55, 55 add 34 equals 89. 89 divide by 55 is: 1.618181818181818818181818181818. Pi approximated to 864 divided by 275 = 3.141818181818182 divided by 1.2 = 2.618181818181818. 2.618181818181818 subtract = 1.618181818181818. 3.141592653589793 subtract 1.5235 = 1.618092653589793. An approximation of Pi can be obtained from the Golden ratio also if the Golden ratio is applied to Square root and the result of applying the Golden ratio to square root is the ratio 1.27 and if 4 is divided by 1.272727272727273 the result is Pi approximated to 22 divided by 7 = 3.142857142857143. The square root of the Golden ratio approximated to: 1.619834710743802 is 1.272727272727273. Cosine (36) multiplied by 2 is the Golden ratio of 3.144605511029693and if 1.618033988749895 is applied to square root the result is 1.272019649514069. 4 divided by 1.272019649514069 is Golden Pi = 3.144605511029693. Cosine (36) = the Golden ratio of 1.618033988749895. If the Golden ratio of 1.618033988749895 is divided by Pi approximated to 22 divided by 7 = 3.142857142857143 then the result is 0.514828996420421 and if 0.514828996420421 is multiplied by 5.28 then the result is an approximation for the mathematical constant that is known as E = 2.718297101099824. The sequence can also work with the numbers above being doubled such as 2,4,6,10,16,26,42,68,110,178. 178 divide by 110 is 1.618181818181818818181818181818. The further the progression is extended the closer to view the Golden ratio becomes.

Also 72.81 divided 45 = 1.618. Also 40.45 divided by 25 = 1.618. 9.708 divided by 6 = 1.618.

The Golden ratio can also be approximated very closely if 121393 divided by 75025 = 1.618033988670443. If 121393 is added to 75025 the result is 196418. If 196418 is divided by 121393 the result is 1.618033988780243.

If 196418 is added to 121393 the result is 317811. 317811 divided by 196418 = 1.618033988738303. If 317811 is added to 196418 = 514229. 514229 divided by 317811 = 1.618033988754323. If 514229 is added to 317811 the result is 832040. 832040 divided by 514229 = 1.618033988748204.

Please note the repeating numbers 1.6180339887….. after 196418 has been divided by 121393. So the Golden ratio can be approximated to 1.6180339887 plus an infinite amount of decimal point numbers that never repeat. So the Golden ratio can be reduced to 1.6180339887 because after 1.6180339887 there is an infinite amount of decimal point numbers that are never repeated but 1.6180339887 is repeated after 196418 is divided by 121393.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (36) = 0.809016994374948 multiplied by 2 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (36) = 0.809016994374948 x 2 = 1.618033988749895.

The Golden ratio can also be obtained in Trigonometry through the formula Cosine (72) = 0.309016994374948 multiplied by 2 = 0.618033988749895 plus 1 = 1.618033988749895. The formula for the Golden ratio in Trigonometry can be written in short as Cosine (72) = 0.309016994374948 x 2 = 0.618033988749895 + 1 = 1.618033988749895. The Golden ratio can also be obtained in Geometry through Cosines of both multiples of 36 and 72 and remember to multiply the result twice to get the Golden ratio. The Golden ratio in Trigonometry can also be obtained through the formula 1+SIN (18) = 1.309016994374947 x 2 = 2.618033988749895 – 1 = 1.618033988749895. Please remember that 2.618033988749895 is The Golden section or Golden ratio squared and also has many applications. The Golden ratio squared can also be obtained in Trigonometry through the formula Cosine (36) x 2 + 1 = 2.618033988749895. The Golden ratio squared can also be obtained in Trigonometry through the formula 2+SIN (18) x 2 = 2.618033988749895.

Many mathematicians have called the Golden ratio the most pleasing ratio in existence.

Constructing Golden logarithmic spirals that all share the same centre in a circle:

A circle can be constructed with opposing logarithmic spirals originating from the centre of the circle. To divide a circle into spirals construct a rectangle that is based upon the Golden ratio with the longer length of the Golden rectangle with equal length to the radius of the circle. Now construct a second Golden rectangle that has its longest edge the same as the shorter edge of the larger Golden rectangle. Both of the large and small Golden rectangles must have an edge that emanates from the centre of the circle. The diagonals of the 2 Golden rectangles must then be divided into the golden ratio the same way that 3 divisions of the longest width of a Pentagram is divided. One rectangle must have a diagonal that is 31.71 degrees and the other rectangle must have a diagonal of 58.28 degrees or if the choice is to draw the rectangles tilted then another combination of 2 angles that add up to 90 degrees can be used just like both 31.71 and 58.28 add up to 90. 31.71 degrees and 58.28 degrees must come from the centre of the circle and can expand to infinity by the Golden ratio. Any 1 of the 2 longer divisions of the diagonals that are located on the sides of the smaller middle section must then be used to extend the diagonals through the centre of the circle, allowing 2 new larger golden rectangles to be constructed. Please remember for the purpose of accuracy that when a smaller Golden rectangle shares it’s longest edge length with the shorter edge length of a larger Golden rectangle the full length of the whole rectangle will have a ratio of the square root of 5. The longer length of the shorter Golden rectangle is the shorter length of the Square root of 5 rectangle. Remember again that the longest length of the larger Golden rectangle is equal to the radius of the circle so that the spirals that are drawn from the squares that make the Golden rectangle can all originate from the centre of the circle. Now Golden spirals can be constructed from the largest Golden rectangles with the spirals attempting to emanate from the centre of the circle. The circumference of the circle can be divided into equal sections with the process of constructing Golden spirals that attempt to emanate from the centre of the circle being repeated to infinity if desired by expanding the radius of the circle to form new circles that share the same centre with the method of the Fibonacci progression.

The radius of the circles that all share the same centre must expand by the Golden ratio if the sizes of the spirals are to be increased. If opposing spirals are drawn from the centre of the circle and are on opposite sides and above each other then they can be interpreted as the spirals of a Galaxy. Please note that an unfortunate fact exists and that is the spiral or spirals that can be created from the squares that form a Golden rectangle cannot completely touch the centre of the circle even though the spirals do come close to touching the centre of the circle. The reason why using squares that form the Golden rectangle to create spirals that are desired to touch the centre of the circle cannot be achieved 100% is because the Squares that form the Golden rectangle are created in ratio to each other based upon the Fibonacci sequence and the Fibonacci sequence begins with 1 plus 1 then 2 plus 1 equaling 3, so to obtain the size of the next square the number that proceeded the current number must be added to the current number.

If the Geometer is determined to make the Golden spiral proceed to the centre of the circle that must have both 31.71 degrees and 58.28 degrees emanating from the centre of the circle to indicate the point for the spiral to emanate then the only solution is to divide 1 of the 2 smallest squares into a Golden rectangle. The Golden rectangle that allows the logarithmic spiral to reach the centre of the circle must have its longest edge length equal to 1 of the 2 smallest squares that make up the whole largest Golden rectangle. The Golden rectangle that has its longest edge length equal to 1 of the 2 smallest squares that make up the largest Golden rectangle must also be divided into squares if possible.

Constructing a Golden rectangle inside of any random circle:

Please remember that a Golden rectangle can be constructed inside any random circle when a double square-rectangle is constructed inside of that circle that has the diameter of the circle being equal to the diagonal of the rectangle that is made of 2 squares and is constructed inside of the circle. Please remember that if a Golden rectangle is placed on the 2 parallel edges of a square then the total height of the new rectangle is Square root of 5. This knowledge is also used to construct an Icosahedron inside of a Sphere.

• The Golden ratio:

The Golden section is a ratio that has a line with the small section being the same as the larger section is to the whole length of the line. The Golden ratio first appears when a circle is divided into 5 equal parts producing a Pentagon with the Pentagon’s diagonals being connected to the 5 vertices of the Pentagon inside the circle touching the circle’s circumference. Applications for the Golden ratio are many and the author of this book does NOT have enough space to list all of them here. Some of the applications for the Golden ratio include, constructing a Pentagon inside a circle.

“Using the Golden ratio to construct a Pentagon and also a Decagon”:

The Golden ratio must be used to construct a Pentagon because the edge of a Pentagon in relation to it’s diagonal is the Golden ratio. Here is another method for constructing a Pentagon inside of a circle .The Golden ratio can also be used to divide the circumference of a circle into 10 equal parts because one tenth of a circle’s circumference is the smaller section of the Golden ratio in relation to the radius of a circle. Remember that a Pentagon is half of a Decagon. The radius of a circle can be divided into the Golden ratio Geometrically with just the use of a compass and a straight edge and obviously a stylus by first dividing the radius of a circle into half by constructing a Vesica Pisces with the base being equal to the radius of the circle, the base of the Vesica Pisces and the radius of the circle must share the same base. Next connect a line between the 2 opposing apexes of the Vesica Pisces so that the radius of the circle can be divided into 2. Next construct a semi-circle or a circle with the radius of the large circle that will contain the Pentagon being the diameter of the smaller circle, so that a rectangle made of double squares can be constructed on the radius of the large circle. The radius of the large circle must be the long edge of the rectangle that is made from 2 squares, the short length of the rectangle that is made from double squares is equal to half of the radius of the large circle that contains the Pentagon. Next swing an arc with a compass with a measurement of the short part of the rectangle that is made from double squares onto the diagonal of the rectangle that is made from double squares, next swing another arc from the larger division of the diagonal of the rectangle that is made from double squares onto the radius of the large circle that contains the Pentagon. The radius of the circle that is to contain the Pentagon has been divided into the Golden ratio, take the larger part of the circle’s radius that has been divided into the Golden ratio and multiply it 2 times from the point on the circumference of the circle that has the radius of the circle touching it so that one fifth of the circle’s circumference can be obtained. Now that one fifth of the circle’s circumference has been obtained by dividing the radius of the circle into the golden section and multiplying the larger part twice, next multiply one fifth of the circle’s circumference until the Pentagon is completed and connect the diagonals of the Pentagon up to the 5 vertices of the circle’s circumference so that a 5 pointed star that is based upon the Golden ratio can be observed.

“Inscribing a Pentagon in a circle when only the measure for the radius or diameter of the circle is known or finding the measure for the Pentagon’s radius”:

A method for determining the measure for the edge of a Pentagon when only the measure for the radius or diameter of the Pentagon’s circle is known is to add the result of the circle’s radius squared to the result of the circle’s decagon squared and then apply the result of adding together both the results of the circle’s radius squared and the circle’s decagon squared to square root. Remember the edge of a decagon is obtained by dividing the circle’s radius into the Golden ratio of 1.6180339887498. The edge of a decagon can easily be obtained by also dividing the circle’s circumference into 10 equal units of measure.

Another solution for determining the edge length of a Pentagon when only the radius and diameter of the Pentagon’s circle is known is to take the measure of the diameter of the Pentagon being added to the edge of a decagon that has a radius equal in measure to the Pentagon’s radius must be squared and added to the measure for the Pentagon’s diameter squared and then the result must be applied to square root and the result will be the hypotenuse of a scalene triangle with its shortest length being equal to the diameter of the Pentagon and the second longest length of this scalene triangle is equal in measure to the measure for the Pentagon’s diameter being added to and combined with the edge of a decagon with a radius equal in measure to the Pentagon’s radius. The hypotenuse of the scalene triangle with its shortest length being equal in measure to the Pentagon’s diameter while the second longest length of the scalene triangle is equal in measure to the Pentagons diameter plus the edge of a decagon with a radius equal in measure to the Pentagon’s radius must be divided into the square root of 3 = 1.732050807568877 and the result of dividing the hypotenuse of this scalene triangle into the square root of 3 = 1.732050807568877 and the larger part of the division of the hypotenuse into the square root of 3 must be divided into the Golden ratio and the larger part of that new division from the hypotenuse being divided into the Golden ratio is the measure for the edge length of the Pentagon. The equation above can be simply written as:

1. Diameter of circle added to the length of the edge of the circle’s decagon.

2. Diameter of circle added to the length of the edge of a decagon squared and the results remembered.

3. Diameter of circle squared and the results remembered.

4. The result of the circles diameter squared added to the length of the circle’s diameter combined with the circle’s decagon edge length squared

5. The sum of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root.

6. The result of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root results in a measure that must be divided by the square root of 3 = 1.732050807568877.

7. The result of both the circle’s diameter squared and the circle’s diameter plus the edge length of the circle’s decagon squared and the result of both added together now being applied to Square root results in a measure that must be divided by the square root of 3 = 1.732050807568877 and then divided by the Golden ratio of 1.6180339887498 and the result is the measure for the edge of the circle’s Pentagon.

Another method for determining the measure for the edge of a Pentagon when only the measure for the radius or diameter of the Pentagon’s circle is known is to multiply the measure for the radius of the Pentagon by: Cosine (54) multiplied by 2 So if the radius of the Pentagon is 55 equal units of measure then multiply 55 by Cosine (54) multiplied by 2 and the result for the edge of the Pentagon with a radius of 55 equal units of measure is 64.656377752172052. Remember to reduce the results to 6 or 5 or 4 or 2 decimal places then the measure for the edge of the Pentagon can also be known in addition to the radius and diameter for the Pentagon’s circle already being known. The ratio 1.175570504584946 is the result of dividing the edge length of a Pentagon by the radius of a Pentagon and can be obtained in Trigonometry through the formula Cosine (54) = 0.587785252292473 multiplied by 2 = 1.175570504584946. The ratio 1.175570504584946 can be achieved through the short written formula in Trigonometry as Cosine (54) = 0.587785252292473 x 2 = 1.175570504584946. The ratio 1.175570504584946 reduced to 4 decimal places reads as 1.175.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter a solution is to divide the measure for the edge of the Pentagon by Cosine (54) and the result can be the diameter for the Pentagon and the radius of the Pentagon is obtained by dividing the diameter of the Pentagon by 2 equal halves. So if the edge length of the Pentagon has 34 equal units of the measure and the desire is to know the diameter and radius of a Pentagon with a edge length of 34 equal units of measure the solution can be written as 34 divided by Cosine (54) = 57.844254967938708 equal units is the measure of a diameter for a Pentagon with 34 equal units of measure and 28.922127483969354 equal units is the measure for the radius of a Pentagon with a edge length of 34 equal units of measure. Written shorter the formula for obtaining the radius of a Pentagon with an edge length of 34 equal units of measure is 34/Cosine (54) = diameter of Pentagon = 57.844254967938708. Diameter of Pentagon = 57.844254967938708 divided by 2 = radius of Pentagon: 28.922127483969354.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter a solution is to divide the measure for the edge of the Pentagon by 0.6180339887498 or multiply the measure for the edge of the Pentagon by 1.6180339887498 and the result is the length of the Pentagon’s diagonal. Multiply the Pentagon’s diagonal by the Square root of 3 = 1.732050807568877. Remember the result of the Pentagon’s diagonal being multiplied by the Square root of 3. Divide the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 by the Square root of 5 = 2.23606797749979. Remember the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979. The result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = must now be squared and the result must also be remembered. The result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 must now be divided into the Golden ratio of 1.6180339887498 and the result must also be squared and remembered. The results of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then squared must then be added to the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then being divided by the Golden ratio of 1.6180339887498 and then squared and then applied to Square root. The results of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then squared must then be added to the result of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then being divided by the Golden ratio of 1.6180339887498 and then squared and then applied to Square root and then divided by the Square root of 3 = 1.732050807568877 and the new result can now be the radius of the Pentagon. So a simple formula for the equation above can be written as:

1. Edge of Pentagon divided by 0.6180339887498 or Edge of Pentagon multiplied by 1.6180339887498 resulting in the measure for the Pentagon’s diagonal.

2. Pentagon’s diagonal multiplied by the Square root of 3 = 1.732050807568877.The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 must then be divided by the Square root of 5 = 2.23606797749979.

3. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 must now be squared and remembered.

4. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then divided into the Golden ratio of 1.6180339887498.

5. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then be divided by the Square root of 5 = 2.23606797749979 and then divided into the Golden ratio of 1.6180339887498 and then squared and remembered.

6. The result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then be divided by the Square root of 5 = 2.23606797749979 and then squared and remembered must now be added to the length of the Pentagon’s diagonal being multiplied by the Square root of 3 = 1.732050807568877 and then divided by the Square root of 5 = 2.23606797749979 and then being divided into the Golden ratio of 1.6180339887498 and squared.

Now that the length of the Pentagon’s diagonal has been multiplied by the Square root of 3 = 1.732050807568877 and the result of such has been divided by the Square root of 5 = 2.23606797749979 and then squared and remembered. The result of the length of the Pentagon’s diagonal multiplied by the Square root of 3 = 1.732050807568877 has been divided by the Square root of 5 = 2.23606797749979 and has be divided by the Golden ratio of 1.6180339887498 and squared and remembered. The sum of the result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 and then divided by the Golden ratio of 1.6180339887498 and then squared and remembered must now be added to the result of the length of the Pentagon’s diagonal being divided by the Square root of 3 = 1.732050807568877 and then being divided by the Square root of 5 = 2.23606797749979 squared and remembered and the result is the measure for the length of the Pentagon’s radius.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter of the Pentagon another solution is to:

1. Have half of the edge of the Pentagon squared and the result remembered.

2. Half of the edge of the Pentagon multiplied by the ratio 1.376381920471173 and the result also squared and remembered. The ratio 1.376381920471173 can be obtained on a digital calculator through the Trigonometric formula TAN (54). .

3. Add the result of half of the edge of the Pentagon squared to the result of half of the edge of the Pentagon multiplied by 1.376381920471173 and then apply the result of both combined to Square root and the final result is the measure for the radius of the Pentagon.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter of the Pentagon another solution is to:

1. Multiply half the edge length of the Pentagon by TAN (54) = 1.376381920471173

2. Divide the result of multiplying the edge length of the Pentagon by TAN (54) = 1.376381920471173 by the Golden ratio of 1.618033988749895.

3. The result of the edge length for the Pentagon being multiplied by TAN (54) = 1.376381920471173 and then divided into the Golden ratio of 1.618033988749895 must now be multiplied by 2 resulting in the measure for the radius of the Pentagon.

“How to obtain the height of a Pentagon”:

The height of a Pentagon can be obtained by:

1. Multiplying half the edge of the Pentagon by the ratio 3.077683537175253 and the result will be the height of the Pentagon. The ratio 3.077683537175253 can be obtained digitally on a calculator through the Trigonometric formula TAN (72).

2. The height of a Pentagon can also be obtained if half of the edge of the Pentagon is multiplied by the ratio 1.376381920471173 and the result then added to the radius of the Pentagon from the centre of the circle. The ratio 1.376381920471173 can be obtained from a calculator through the Trigonometric formula TAN (54).

3. If the radius of a circle is divided in half and then divided by 0.6180339887498 or multiplied by the Golden ratio of 1.6180339887498 and then the result then added to the radius of the circle from the centre of the circle the total length is equal to the height of a Pentagon that can be created to fit in the circle.

“How to obtain the radius of a Pentagon with Trigonometry if the height of the Pentagon is already known”:

So the height of the Pentagon has already been determined by multiplying half the edge of the Pentagon by TAN (72) = 3.077683537175253 and the desire is to know the radius of the Pentagon and a simple solution is to multiply half the edge of the Pentagon by TAN (54) = 1.376381920471173 and subtract the result of multiplying the edge of the Pentagon by TAN (54) = 1.376381920471173 from the height of the Pentagon and the result is the radius of the Pentagon.

If only the measure for the edge of the Pentagon is known and the desire is to also know the measure for the Pentagon’s radius and diameter another solution is to divide the measure for the edge of the Pentagon by 1.170 or 1.175 or 1.175570504584997 or 1.175851692873989 or 1.175656984785615 or 1.1756124220922 or 1.175454545454545 or 1.175636363636364 or 1.175505023709698 or 1.175570504584965 and the result will be an approximation for the radius of the Pentagon and the result must be reduced to 6 or 5 or 4 or 2 decimal places for better accuracy. Remember that if the radius of a circle is divided into the Golden ratio the result is the larger part of the division being equal to the measure for the edge of a decagon that has a radius equal in measure to the radius of the circle.

Geometrically the radius of a Pentagon can become visible if all the 5 edges of the Pentagon are divided in half with lines that pass through the centre of the Pentagon’s circle that also touch the centre edges of the Pentagon.

It should become evident from reading above that any line can be divided into the Golden ratio when a double square is constructed over that line and then swing an arc with a compass measurement equal to half of the double square on to the diagonal of the double square and then swinging an arc with compass measurement of the larger division of the diagonal of the rectangle that is made from double the squares. If the desire is to find the shorter part of the Golden ratio when you have a line and your desire is to make the line longer the use of a square that is divided into half so that the double square can be used by swinging an arc from the centre edge of the large square with a compass measurement equal to the diagonal of the 2 rectangles that are formed of double squares that also make up the largest square onto the extension of the line. Remember that a square is also made up from 4 smaller squares. The height of a Pentagon can also be found if a Golden rectangle is constructed from the centre of the circle that contains the Pentagon with the shorter edge of the Golden rectangle being equal to half the radius of the circle that contains the Pentagon.

• Constructing the shape of the Great pyramid.

Constructing the Great pyramid isosceles triangle from 2 Golden rectangles”:

The slope that is located in the centre of the 4 triangular faces of the largest of the 3 of the Egyptian Great pyramids of Giza can be constructed from 2 vertical golden rectangles. When constructing the central slope of the largest of the 3 Great Egyptian Pyramids an arc that has equal measure to the longest length of any of the 2 golden rectangles can be used and swung unto the centre of the rectangle that is made from the 2 Golden rectangles. The result will be a triangle that has a base length that is the longest edge and equal to 11 units of equal measure while the height of the triangle will be 7 units of equal measure relating the edge length that is the base of the triangle.

“Constructing the Great pyramid triangle from 2 over lapping circles called the Vesica Pisces”:

The triangle that forms the 11 and 7 ratio triangle also forms the central slope of the largest of the 3 Great Egyptian Giza Pyramids and can also be obtained from 2 over lapping circles of equal size that have circumferences that touch each other’s centers. The height of the Pyramid’s triangle is 7 equal units while the base is 11 equal units and the height of the Pyramid’s triangle shares the area called the Vesica Pisces. When the Vesica Pisces is used to construct the shape of a Pyramid that has a square base width of 11 equal units of measure and 7 equal units of measure for the height of this Pyramid triangle then the height of the this pyramid triangle will be equal to the edge of an equilateral triangle that is contained with in the circumference of any of the 2 circles that are used to make the Vesica Pisces.

“Constructing the shape of the Great pyramid triangle upon the diameters of a circle:”

To construct the Great pyramid isosceles triangle upon any of the diameters of a circle the radius of the circle must be multiplied around the circumference of the circle 2 times from any of the circles poles that are on the end of the circles diameters. The result of multiplying the radius of the circle 2 times around the circumference of the circle from any of the circle’s pole diameters is the creation of a measure that is equal to one third of the circle’s circumference. From each end of the measure that is equal to one third of the circle’s circumference swing arcs down on the opposing diameter resulting in the creation of the Great pyramid triangle with the radius of the circle being the height of this Great pyramid isosceles triangle. The apex of this created Great pyramid isosceles triangle is also located on the pole of the appropriate diameter of the circle being also located on the circumference of the circle.

• Golden ratio derived from constructing a scalene triangle based upon the ratio of the moon and Earth touching each other ‘s circumferences:

The ratio of the radius and diameter of the Earth compared to the radius and diameter of the moon is 11 equal units of measure compared to 3 equal units of measure for the diameter of the moon. So the radius and Diameter of the Earth is 11 equal units of measure and from this measure of 11 equal units of measure 3 units of equal measure can be used to obtain the diameter of the moon. 11 divide by 3 produces the ratio: 3.6666666666667. Now if a Scalene triangle is constructed from the circumference of the moon touching the circumference of the earth the apex of the Scalene triangle will be located on the centre of the moon with the second shortest length of the Scalene triangle being equal to the radius of the Earth plus the radius of the moon. The smallest edge of the scalene triangle will only be equal to the radius of the Earth. The largest part of the scalene triangle will be the larger part of the Golden ratio when compared to the radius of the Earth that is the smaller part of the Golden ratio. If the radius of the Earth and the shortest length of the Scalene triangle is 5 and a half of 1 equal units of measure then the second longest edge of the Scalene triangle that is derived from the Earth and moon ratio will be 7 units of equal measure. Since the shortest edge of the Scalene triangle and the radius of the Earth is 5 and a half of 1 equal units of equal measure the longest edge of the Scalene triangle will be 8 point 9 units of equal measure. 8 point 9 divided by 5 and a half of 1 is: 1.618181818181818818181818181818. (An approximation of the Golden ratio).

The second longest length of the Kepler scalene triangle divided by the shortest length of the Kepler scalene triangle is the square root of the Golden ratio: = 1.272727272727273. 1.272727272727273 squared is also an approximation of the Golden ratio of 1.619834710743802.

The Scalene triangle that is used above in the Earth and moon ratio to find the Golden ratio is called a Kepler triangle. The Kepler triangle is also found in a Pyramid that has a square base width of 11 equal units of measure and 7 equal units of measure for the height of the Pyramid and the Great Egyptian pyramid of Giza is such as Pyramid.

• Constructing a Great pyramid Isosceles triangle inside the circumference of a circle, How to construct a isosceles triangle that is made from 2 Kepler scalene triangles inside the circumference of a circle:

If the diameter of a circle is divided into the Golden ratio then the larger part of the measure of the circle’s diameter that has been divided into the Golden ratio can be equal to the height of an isosceles triangle that is made from 2 Kepler Scalene triangles. Also the height of the Isosceles triangle that is made from 2 Kepler Scalene triangles is equal to the distance between the pole of the circle’s diameter that is opposite to the apex of this Isosceles triangle and any of the 2 base points of this Isosceles triangle that can touch the circumference of the circle. If the diameter of the circle that contains this isosceles triangle that is made from 2 Kepler scalene triangles is 140 equal units of measure then the height of this isosceles triangle will be 86.52 equal units of measure, meanwhile the base of this isosceles triangle will be 135.96 equal units of measure. Half of the base width of this isosceles triangle can be 67.98 equal units of measure while the length of the 2 longest edges can be 110.004 equal units of measure. If any of the longest edges of this isosceles triangle that is made from 2 Kepler scalene triangles is divided by half of the base width of this Isosceles triangle then again the result will be the approximation for the Golden ratio known as 1.618181818181818. Also if the diameter of the circle is divided by the height of this Isosceles triangle that is made from 2 Kepler Scalene triangles and is also contained with in the circle then the result will be the approximation for the Golden ratio known as 1.618122977346278.

Please remember that this isosceles triangle that is made from 2 Kepler scalene triangles can be used to create a circle that has a circumference equal to the perimeter of a square if both the circle and square share the same centre and the base width of this isosceles triangle that is made from 2 Kepler scalene triangles is equal to the width of the square and also the height of this isosceles triangle that is made from 2 Kepler scalene triangles is equal to the radius of the circle.

• Combined diameters of Earth and moon forming the second shortest length of a Kepler Scalene triangle that also includes the Golden ratio:

If the measure for the diameter of the moon is added to the measure for the diameter of the Earth then this combined new measure can be used as the second shortest length of a Kepler Scalene triangle that includes the Golden ratio. While the diameter of the Earth alone can be used as the shortest length of the Kepler Scalene triangle. The diameter of Earth again is 7920 statute miles and the diameter of Earth’s moon is 2160 statute miles. 7920 plus 2160 = 10080. If 10080 statute miles is the second shortest length of a Kepler Scalene triangle then 7920 is the shortest length of this Kepler Scalene triangle, while the longest length of this Kepler scalene triangle is 12816 statute miles. 12816 divided by 7920 = 1.618181818181818. 1.618181818181818 is an approximation of the Golden ratio of 1.618 and 1.618181818181818 can also be obtained if 8.9 is divided by 5.5.

Examples for applications of the Golden ratio are abundant through out nature including the formation of galaxies and the spiral curve of the human ear. : [en.wikipedia.org]

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