Golden Ratio = Mind Blown!

Golden Ratio = Mind Blown!


When I learned about this math ratio, it
changed my life. Okay, so I’m going to explain in this video a math and design
phenomena called the Golden Ratio. It’s also referred to as “Phi.” So what is the
Golden Ratio? Well, to help explain it, I’m going to put out the sequence of numbers
called the Fibonacci sequence, which is really just the Golden Ratio in a
sequenced, numeric form. Now to arrive at this Golden Ratio sequence of numbers,
we just need to follow a basic math formula. And I’m not a math person, so
we’re just going to keep this very simple. Okay, so you just start with 0+1=1. And now to get to the next number in the sequence, you take the
sum of that simple equation and add it to the last number in the equation. So 1+1=2 and then 1+2=3. And it’s
around this point that the ratio actually starts showing up. As we
continue to do this formula, we start arriving at a set of numbers: 2, 3, 5, 8…
and you see what we’re doing– we’re adding the sum of the equation to the
last number in the equation, and we arrive at this sequence of numbers.
It’s interesting that this sequence and ratio actually remains consistent no
matter how long you follow this mathematical formula. And so this
sequence continues to expand outward around the rate and the ratio of 1 to
1.6. Now to help give you an idea of what a 1 to 1.6 ratio is compare it to a 1 to 1
ratio so to make a 1 to 1.6 ratio, you just envision a little more than half of the
initial line added to the line of the other side. Alright so this ratio 1 to 1.6 This is the ratio that’s called
the Golden Ratio a 1 to 1.6 ratio. So this is what the Golden Ratio looks like as a
rectangle 1 to 1.6. And if we were to start making incremental Golden Ratio points
within that, we can get an idea of what a spiral looks like when it expands
outward at the same measured sequence. Now this is all well and good, but what
does that have to do with everyday life? Well, a lot actually. And that’s because
when we look to nature, we see that so many things flourish when they go to the
golden ratio design and when they follow this sequence. Growing and expanding to
the rate of the golden ratio spiral allows the maximum amount of rain to be
directed down to the roots of many plants. And remarkably when you study
nature, you see the golden Fibonacci numbers like 3, 5, 8, 13, all of that again
and again in the seed patterns and spirals of plants, as well as in the number of
petals. Next time you’re bored and have a
sunflower, try counting the number of seeds in the sunflower spiral– the Golden
Ratio! Or maybe try something else with a spiral, like a pinecone or pineapple. So
we can spend all day counting the seeds of flowers, and plants, and fruit… I mean,
you get the idea. But we don’t just see this Golden Ratio sequence on a small
scale. This ratio is the mathematical sequence in the spirals of our storms.
Tornadoes, hurricanes– these all spin in this golden sequence 1 to 1.6! Even the
waves can be measured using this ratio. But it doesn’t stop there– modern
technology continues to be mystified by the far-reaching scope of the Golden
Ratio. From the alignment of the planets to the spirals of our Milky Way galaxy,
as well as the spirals of other ratios of planets.
Now let’s take this closer to home, I want you to hold out your arm and
look at the distance between your shoulder and your elbow. And then from
your elbow to your fingertips. Notice how your elbow from your shoulder
to your elbow is “1” and then from your elbow to your fingertips is “1.6.” Pretty crazy huh? but doesn’t stop there.
Now take that further the distance from your fingertips to your wrist is “1” and
from your wrist to your elbow is “1.6” Keep it going– from the
furthest tip of your finger to the bottom of your fingers is “1” and from
the bottom of your fingers to your wrist is “1.6.” Now check out the
spacing of your knuckles 1 to 1.6. Are you getting freaked out yet? Think about this:
the head to your belly button “1” and then belly button to your feet “1.6.” From your toes to your knees, and from your knees to your hips– Golden
Ratio! Ever wonder why your two front teeth are so much bigger? Golden ratio!
The pupils of your eyes– I mean, you name it, your whole body is the symphony of
the Golden Ratio! In fact, as you are listening to this video, the sound waves
are passing through your ear in a perfect golden spiral. Scientists have
discovered that the golden ratio pattern is necessary for the human brain, the
neural system, our sense organs, and our lung system. The golden ratio sequence is
even in the helix of our DNA, and it forms the very rhythm of our heartbeat
pattern! This is pretty amazing if you ask me! The
universe is an incredible place, and to think that these beautiful intricacies
of the world all hang on a stacked, razor edge with the incomprehensible
fine-tuning of all these precise constants and quantities which allow the
universe to begin to exist. I mean it’s unfathomable! No wonder we
are a naturally bent to worship a higher creative power. I mean, we are fearfully
and wonderfully made, and we live in a universe that is designed and hardwired
to be life permitting. And speaking of which, I think it’s interesting that the
Golden Ratio also comes up in the ancient Biblical texts. Scholars have
noted that wouldn’t you consider design measurements of things like
Noah’s Ark and the Ark of the Covenant, the Golden Ratio is an exact match to
the measurements. Pretty interesting… Okay, so we’ve seen how the Golden Ratio is
embedded into our life and even into our own heartbeat. So naturally, it’s going to
affect our aesthetics and the arts. It is believed that the Greeks used the
Golden Ratio to achieve ideal acoustics, and many instruments are actually
designed with the ratio. When you look at the amount of black keys and white keys
on a piano, it should be no surprise that you see
the Golden Ratio. And that’s because the musical scales and notes align
with the Fibonacci sequence. So scientists, mathematicians, and artisans
have been aware of the Golden Ratio and have been incorporating it into their
architectural and artistic designs throughout history. For some reason,
designs tend to look better when composed and designed with the Golden
Ratio. Look at iconic ideal Greek temples like the Parthenon in Athens– you can see
that they use the ratio again and again. Painters have also been incorporating
the golden ratio into their artistic designs because it gives the composition
a “je ne sais croix” (“I don’t know what.”) Leonardo da Vinci was obsessed with it,
so was Michelangelo. Even in the 20th century, you see the ratio being used in
painting– whether it be in the dimensions of the canvas, or the placement of the
focal point. And by the way, the Golden Ratio is one of the reasons artists
don’t like to line everything up in the center. But you don’t have to go to
Greece or to the Louvre to appreciate the Golden Ratio in human design. I mean,
just look around your house or go to the grocery store.
The golden ratio is used in product design, logos, and in branding all the
time. It’s a ratio that, for some reason, is pleasing on the eye, and it can be
a great template for solving multiple design problems. I remember when I first
started learning about the golden ratio, I began to see it everywhere, and it’s
really remarkable with how it comes up in nature and in the design world, and
and how artists can greatly improve their designs by using its sequence. So
now that you know about this mysterious sequence that pops up again and again,
now go out, and I want to challenge you to go and look and find places where you
see this ratio. You don’t necessarily have to take around a measuring
stick, but but take pictures, and notice where you see this ratio come up
in design and in nature. And I think you will be surprised– and perhaps, even, this
newfound awareness it might even change your life.

100 Comments

  1. Everyone go watch Theoria Apohaphis on YouTube The explanation's coming out of that guy's mouth with make your head flop on the floor and roll across the hall.

  2. All of this is TRUTH…but, religions aren't TRUTH…Source…YES…just not what religions say is a God…NO! Also why the TRUTH is inside of us…not outside! To know thyself do not worship the outside…nothing outside…that's not where the TRUTH is!

  3. The things she is showing sometimes are with 1:2 ratio
    She is making fool of u, there is no ratio, nature is out of ratio
    The mathematical series which she talking about is even wrong, the way she is explaining the series, with that way the series should be 1,1,2,3,5,8,13,21….so on

  4. Watching this video I am finished trying to get my BP systolic to diastolic yes it is. Who are you Golden lady? love your work. God bless you.

  5. the golden ratio is proportional balance, nothing can exist without balance. There is an obvious intelligence behind all life. WHAT A BEAUTIFUL MIND! Great Video

  6. I've watched a few videos on the golden ratio/Fibonnacci sequence with a view to improving the composition of my oil paintings. It's an amazing subject, and very useful to me as an artist. But to use it as evidence for the existence of God? Beauty and order can be explained without reference to a god.

  7. heard about this before.. but never understood before this.. thank you so much for making it so easy to understand..

  8. The universe is great of course, but in 50 years no one will know of this wonderful video or of my comment; or, some of the people that tell me how horrible I am. 😉

  9. In my 2 d design class it was barely mentioned in the book and only said like "it's pleasing for the eye". It doesn't really go further than that. To be honest this seems almost like direct proof of intelegent design whether it be a Creative Divine Source (God) or a mathematical programed simulation, or both.

  10. Do you know why the biological orgamisms follow growth in Golden ratio? Probably because the cells divide and grow in Fibonacci sequence and reverse Fibonacci sequence. The same probably applies to particle physics as well.

  11. The most trippy thing is this… almost every song I’ve tried, this works with. 9/10 I’d say.
    Take the length in seconds and multiply it by 0.618. At that point in the song, there likely is a climax, or important notably different portion. They call it a phi moment.

  12. Great video! You explained it so well and with such enthusiasm 😉….and when it led you to attributing credit to God…AWESOME! I already knew God was amazing, but this is so cool!
    I have just recently come to understand this formula myself, and had a similar reaction to yours. Thank you for doing such a great job on your explanation and for sharing. 👍

  13. Once you take that one and the 1.6 side and you cut it into 1/3 every time it turns to a rectangle cut 1/3 equals Pi every time, that number goes on forever as well. Easy explanation of the golden ratio definitely for all people understand. The Fibonacci sequence is located everywhere throughout our entire existence into our lives. Sea shells, hurricanes tornadoes, water going down a drain, usually a sunflower head the seeds are in a spiral, we see it everywhere the spiral is every where it's the oldest symbol on Earth. Great job.

  14. It was TOLD BY HINDUS 2500yrs ago, 2nd Century BC. AND EXISTED SINCE THE BIRTH OF THIS UNIVERSE
    Fibonacci an Italian thief shamelessly branded it to himself

  15. @6:23 it's the SHIVA LINGA(rotate 90 degrees clockwise) which the HINDUS worship and which the MUSLIMS broke in MECCA😎 and Christians are now discovering.

  16. found everywhere in nature discovered by our true ancients, long before Egyptians, Mayans etc etc …Pyramids, still standing structures attributed to Romans all are built with this ratio as well as the ratio of the human body…this woman gets it and is honestly excited about the beauty of nature!

  17. India is shaped like a conch.

    Just before the Mahabharata war, Lord Krishna blew on his Panchajanya shankh, striking terror in the Kaurava camp hearts.  
    This sound went all around the world like a soliton carrier wave , using the ionosphere as a wave guide.  

    The sound of the conch with its inherent 7.83 hertz can be heard by the ears and well as the human heart.  
    The human ear can hear only frequencies greater than 20 hertz.
    The conch can sound at the fundamental frequency 7.83 hertz of OM , the Hindu king mantra.  OM opens up quantum tunneling for the wormholes to happen , where the speed of light is NOT the limiting factor. 

    If you make the sound of OM in front of a drop of liquid, it will transform itself into a Sri Yantra which is very specific visual form which is symmetrical and also holographic,  in that every bit of it contains all of it.  

    The Hindu Sri Yantra divine geometry contains the Theory of Everything.  
    The Sri Yantra was revealed to Maharishis with 12 strand DNA and king sized pineal glands in 8000 BC, and it construction is based on thegolden mean of 1.618.   
    The pyramids of Egypt was made using this golden mean of 1.618 and angle of  51 deg 49 minutes 38 seconds.  

    The sign of Hindu king mantra OM is not a ordinary mortal inscription. 

    When Om is chanted the proper way the inscription can been seen in the mind. 

    786 the divine Islamic number is what you see when OM is seen as a reflection in the mirror. Christians say AMEN  a lifted version and Jews say SHALOM another lifted version, which does zilch good, as they do NOT even release Nitric oxide,  leave alone cause Quantum tunneling .

    7.83 hertz  is also known at the Schumann’s resonance or the heart beat frequency of the earth.   The conch can also create harmonics of this 7.83 hertz, which can be heard by the human ear.

    The inner curvature of the air passage inside the conch is based on the Fibonacci series. Fibonacci was an Italian thief.

    Fibonacci was in Bejaya Algeria , when he came to know about Indian Vedic Mathematics, from the Arabs who had translated these texts .  He took this knowledge to Italy. Rest is history.

    Thief becomes a Mathematical genius.

  18. Look for the golden ratio in the world. And miracles of kaba. Golden ratio of the world is Mecca. Look for YouTube video https://youtu.be/CHNFmKFBBtw

  19. I must state that I love the Fibonacci Sequence. A brilliant observation…and yet so incredibly simple. The out-coming ratio has led to an understanding of our World and our Universe that had previously been unfathomable. I have, however, one small problem with it….
    The whole thing about drawing a Golden Rectangle and all of the subsequent rectangles and squares grid therein and then ascribing out the "Golden Spiral" from such leaves me gritting my teeth. Don't get me wrong here…. I, like all of you, see the beauty of the inevitable outcome. What bothers me is that this ISN'T the representation of a true spiral. In fact…what this is is a chain linking a series of quarter circular arcs. Whereas…a true spiral is a line drawn on a continuously varying radius (varying radii). Every arc second should have a slightly variable radius.
    I know…that's just "nit-picking" the whole issue. History has shown that the standard Golden Ratio and Golden Spiral have proven to be an overwhelming influence on architecture and design.

  20. I couldn't understand the golden ratio even after watching plenty of videos,
    But finally you are the one who explained it clearly, now I started observing everything around me and comparing with golden ratio.
    Thanks for the video.

  21. I love your visual explanations for this tricky piece of mathematics. I use the golden ratio for my art, but I never figured the Fibonacci sequence was its 'relative'. I'm going to go & measure my paintings now to see if I had got it right. Your excitement is infectious ♥

  22. Golden Section is: 1.618 not 1.6. Explaining the human proportion without giving credit to Vitruvius is inappropriate. The Golden Section or Golden Ratio was always used in all my books called: "Architectural Design Sketchbook" (The Systems of Proportion) Please give credits to Vitruvius as he was and still is the ultimate authority on classical proportion… "Treatise De Architectura," and Palladio's 4 Books on Architecture. Thanks!

  23. I'm an artist and this Golden Ratio explanation helped me. And dunno maybe golden Ratio is some clue to answering the very existence? Like one of the keys?🤔

  24. OMG, her face is the Golden Ratio! I just checked! And the shadow between her nostrils! And when she makes that little honking noise, if you plot it upsidedown on a curve transposing the signal amplitude with time??? You guessed it… The Golden Ratio!!!

  25. Thank you for this information… So enlightening… Learn to watch the likes of Swami Rama, Shadhguru, Mooji ect… You will learn why this is… God's fingerprint

  26. What about hands and feet of a giraffe 😉 , I'm sure the neck isn't ratio proportioned ,what if you go opposite side and substract plastic bags from peoples behaviour , what about nuclear weapons and zebra patterns and finally what about whiskers of English queen

  27. Some of these made sense, then it kinda went off a cliff. What is it called when you start to notice things only because you're focused on it? Do you remember that Jim Carey movie – "(the number) 23"? This is kinda like that.

    IM NOT SAYING this is fake or false. Im saying you had a point in the beginning , then everything became 1:1.6 (Your arm! Your finger! your knuckles!, then you do an actual measurement and you realize its not correct (for those)).

    Still, the rest of examples were pretty cool.

  28. God's fingerprint, found EVERYWHERE in nature biology space human body DNA galaxy EVERYTHING in universe and is 100% scientific proof of a creator science, ACTUAL SCIENCE proves existence of god by its own paradigms, to deny existence of creator is to deny SCIENCE science by its own paradigms and rules.

  29. Pi multiplied by 4 = 4/√φ multiplied by 4 = 12.578422044118773:

    If the second longest edge length of a Kepler right triangle is divided into 16 equal parts then the shortest edge length of the Kepler right triangle is equal to Pi multiplied by 4 = 12.578422044118773. Also if the second longest edge length of a Kepler right triangle is divided into 16 equal parts and the shortest edge length of the Kepler right triangle is divided by 1 sixteenth of the second longest edge of the Kepler right triangle the result is the ratio Pi multiplied by 4 = 12.578422044118773. Also if the hypotenuse of a Kepler right triangle is divided into 16 equal parts and the second longest edge length of the Kepler right triangle is divided by 1 sixteenth of the hypotenuse of the Kepler right triangle the result is the ratio Pi multiplied by 4 = 12.578422044118773.

    A Golden Pi multiplied by 4 = 12.578422044118773 rectangle with angles of 85.45447565570349 degrees and 4.54552434429651 degrees can be constructed if the shortest edge length of a Kepler right triangle and also 1 sixteenth of the second longest edge length of a Kepler right triangle are placed perpendicular to each other at right angles.

    Also A Golden Pi multiplied by 4 = 12.578422044118773 rectangle with angles of 85.45447565570349 degrees and 4.54552434429651 degrees can be constructed if the second longest edge length of a Kepler right triangle and also 1 sixteenth of the hypotenuse of a Kepler right triangle are placed perpendicular to each other at right angles.

    Minimal polynomial: x^4 + 256 x^2 – 65536:

    https://www.wolframalpha.com/input/?i=4%2F%E2%88%9A+%CF%86+multiplied+by+4

    Pi cubed and the cube root of Pi =

    3.144605511029693144 ^ (1/3) = 1.465059930070137.

    1.465059930070137 ^ 3 = 3.144605511029693144.

    4 ^ (1/3)/ φ^(1/6) = 1.465059930070137.

    Minimal polynomial: x^12 + 16 x^6 – 256

    https://www.wolframalpha.com/input/?i=4+^+%281%2F3%29%2F+%CF%86^%281%2F6%29

    Square root of Pi = 2 divided by 1.127838485561682 = 1.773303558624324:

    https://www.wolframalpha.com/input/?i=2%2F%E2%88%9A%E2%88%9A%CF%86

  30. Pyramid spherical cubit for equal perimeters 1 =

    Concerning the ratio1.162818837094896. The ratio 1.162818837094896 is the cube root of the ratio 1.572302755514847.

    The ratio 1.572302755514847 is half of Golden Pi = 4/√φ = 3.144605511029693. The ratio 1.162818837094896 is the cube root of half of Golden Pi = 4/√φ = 3.144605511029693 = 1.572302755514847.

    If a Phi Pyramid is created and the result of dividing the height of the Phi Pyramid by half the width of the square base of the Phi Pyramid = the square root of the Golden ratio = √φ = 1.272019649514069 and then a sphere is created that has the same volume as the Phi Pyramid that produces = the square root of the Golden ratio = √φ = 1.272019649514069 when the height of the Phi Pyramid is divided by half the width if the square base of the Phi Pyramid then the result of the diameter of the sphere divided by the height of the Phi Pyramid = the cube root of half of Golden Pi = 4/√φ = 3.144605511029693 = 1.572302755514847 ^ (1/3) = 1.162818837094896.

    (2/√φ) ^ (1/3) = 1.162818837094896.

    1.572302755514847 ^ (1/3) = 1.162818837094896.

    (2 divided by the square root of the golden ratio) ^ (1/3) = 1.162818837094896.

    (4/√φ/2) ^ (1/3) = 1.162818837094896.

    (4/√φ divided by 2) ^ (1/3) = 1.162818837094896.

    (3.144605511029693 /2) ^ (1/3) = 1.162818837094896.

    (3.144605511029693 divided by 2) ^ (1/3) = 1.162818837094896.

    (2 (square root of (5) – 1))^(1/6) = 1.162818837094896.

    square root of (2)/(1 + square root of (5))^(1/6) = 1.162818837094896.

    Minimal polynomial:
    x^12 + 4 x^6 – 16

    https://www.wolframalpha.com/input/?i=%284%2F%E2%88%9A%CF%86%2F2%29+^+%281%2F3%29

    Minimal polynomial:

    x^12 + 4 x^6 – 16

    https://www.wolframalpha.com/input/?i=x^12+%2B+4+x^6+-+16

    Pyramid spherical cubit for equal perimeters 2 =

    The cube root of the square root of Phi multiplied by 4 = 1.719958377176499.

    1.272019649514069 multiplied by 4 = 5.088078598056276.

    (4 x √φ) ^ (1/3) = 1.719958377176499.

    (4 multiplied by √φ) ^ (1/3) = 1.719958377176499.

    (4 x 1.272019649514069) ^ (1/3) = 1.719958377176499.

    (4 multiplied by 1.272019649514069) ^ (1/3) = 1.719958377176499.

    4/√φ X φ = 5.088078598056276.

    4/√φ multiplied by φ = 5.088078598056276.

    3.144605511029693144 x 1.618033988749895 = 5.088078598056276.

    3.144605511029693144 multiplied by 1.618033988749895 = 5.088078598056276.

    5.088078598056276 ^ (1/3) = 1.719958377176498.

    √ (2)(1 + √5) ^ (1/6) = 1.719958377176498.

    2 ^ (1/3)/(φ-1) ^ (1/12) multiplied by 2 ^ (1/3)/(φ-1) ^ (1/12) = 1.719958377176498.

    2 ^ (1/3)/(Cosine (72) X 2) ^ (1/12) multiplied by 2 ^ (1/3)/(Cosine (72) X 2) ^ (1/12) = 1.719958377176498.

    2 ^ (1/3)/(Cosine (72) multiplied by 2) ^ (1/12) multiplied by 2 ^ (1/3)/(Cosine (72) multiplied by 2) ^ (1/12) = 1.719958377176498.

    4 ^ (1/3)/(φ-1) ^ (1/6) = 1.719958377176498.

    4 ^ (1/3)/0.618033988749895 ^ (1/6) = 1.719958377176498.

    4 ^ (1/3)/(Cosine (72) X 2) ^ (1/6) = 1.719958377176498.

    4 ^ (1/3)/(Cosine (72) multiplied by 2) ^ (1/6) = 1.719958377176498.

    https://www.wolframalpha.com/input/?i=4+^+%281%2F3%29%2F%28%CF%86-1%29+^+%281%2F6%29

    (φ-1) ^ (1/3) = 0.851799642079243.

    (φ-1) ^ (1/6) = 922929922626438.

    (Cosine (72) X 2) ^ (1/3) = 0.851799642079243.

    (Cosine (72) multiplied by 2) ^ (1/6) = 922929922626438.

    (0.922929922626438 is the square root of 0.851799642079243).
    https://www.wolframalpha.com/input/?i=%28%CF%86-1%29+^+%281%2F6%29
    Minimal polynomial: -1 + x^6 + x^12

    https://www.wolframalpha.com/input/?i=%28%CF%86-1%29+^+%281%2F6%29

    The cube root of 4 divided by the square root of the cube root of Phi minus 1 = 1.719958377176498.

    The square root of = 1.71995837717651 = 5.088078598056376 ^ (1/6) = 1.311471836211708 =
    2 ^ (1/3)/(φ-1) ^ (1/12) = 1.311471836211708.

    https://www.wolframalpha.com/input/?i=2+^+%281%2F3%29%2F%28%CF%86-1%29+^+%281%2F12%29

    2 ^ (1/3)/(0.618033988749959) ^ (1/12) = 1.311471836211708.

    2 ^ (1/3)/(Cosine (72) X 2) ^ (1/12) = 1.311471836211708.

    2^(5/12)/(square root (5) – 1)^(1/12) = 1.311471836211708.

    Minimal polynomial = x^12 – 16 x^6 – 256 = https://www.wolframalpha.com/input/?i=x^12+-+16+x^6+-+256

    Pyramid spherical cubit ratio for equal areas 1 =
    The cube root of 2 = 1.259921049894873.

    2 ^ (1/3) = 1.259921049894873.

    1.259921049894873 ^ 3 = 2.

    Pyramid spherical cubit ratio for equal areas 2 =
    The cube root of 4 = 1.5874010519682.
    2 ^ (2/3) = 1.5874010519682.
    4 ^ (1/3) = 1.5874010519682.
    1.5874010519682 ^ 3 = 4.
    Cubing the sphere ratio 1 for equal surface areas:

    Concerning the ratio 1.381314400949727. The ratio 1.381314400949727 is the square root of 1.5 multiplied by the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 = 1.381314400949727. 1.381314400949727 squared = 1.908029474271104.

    The ratio 1.381314400949727 can be gained also through the formula square root of 6 divided by the square root of Golden Pi.

    √6 divided by √ (4/√ (Cosine (36) times 2)) = 1.381314400949727.

    The ratio 1.381314400949727 applies to the diameter of a sphere divided by the edge of a Cube that has the same surface area as the sphere.
    If a Sphere and a Cube have both been created with the same surface area then the ratio of the volume of the sphere divided by the volume of the Cube is the ratio 1.381314400949727. The ratio 1.381314400949727 is the square root of 1.5 times the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 = 1.381314400949727.
    Construction of the ratio 1.381314400949727:

    If a rectangle is created with its longer edge equal to half the second longest edge length of a Kepler right triangle while the shorter edge of the 1.5 times the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 rectangle is equal to 1 third of the shortest edge length of the Kepler right triangle then the mean proportional of the 1.5 times the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 rectangle that is the square root of the surface area for the 1.5 times the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 rectangle is the edge of a Cube that has the same surface area as a sphere with a diameter that is equal in measure to both the longer edge of the 1.5 times the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 rectangle and half the second longest edge length of the Kepler right triangle that has its shortest edge length divided into 3 equal parts.

    The square root for the surface area of a 1.5 times the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 rectangle is the edge of a Cube that has the same surface area as a sphere with a diameter that is equal in measure to the longer edge of the 1.5 times the square root of the Golden ratio = 1.272019649514069 = 1.908029474271104 rectangle.

    √ (3 times √φ/2) = 1.381314400949727.

    √ (3 times 1.272019649514069/2) = 1.381314400949727.

    √ (1.5 X √φ) = 1.381314400949727.

    √ (1.5 times √φ) = 1.381314400949727.

    √ (1.5 multiplied by √φ) = 1.381314400949727.

    √ (1.5 multiplied by 1.272019649514069) = 1.381314400949727.

    √ (1.5 multiplied by 1.272019649514069) = 1.381314400949727.

    √6 divided by √ (4/√ (Cosine (36) times 2)) = 1.381314400949727.

    √6 divided by √ (4/√ φ) = 1.381314400949727.

    √6 divided by 1.773303558624324 = 1.381314400949727.

    (square root(3) (1 + square root(5))^(1/4))/2^(3/4) = 1.381314400949727.

    https://www.wolframalpha.com/input/?i=%E2%88%9A+%283+times+%E2%88%9A%CF%86%2F2%29

    Minimal polynomial:

    16 x^8 – 36 x^4 – 81

    Cubing the sphere ratio 2 for equal surface areas (Width of the cube divided by the radius of the sphere):

    Concerning the ratio 1.447896292563731. If a Cube and a sphere are both created with the same surface area then the ratio for the edge of the Cube divided by the radius of the sphere is 1.447896292563731.

    √24/3/√√ φ = 1.447896292563731.

    √24 divided by 3 divided by √√ φ = 1.447896292563731.

    √24 = 4.898979485566356 divided by 3 = 1.632993161855452 divided √√φ = the square root of the square root of Phi =

    1.127838485561682 = 1.447896292563731.

    √6 multiplied by 2 divided by 3/ Φ^(1/4) = 1.447896292563731.

    √6 multiplied by 2 divided by 3/1.618033988749895 ^(1/4) = 1.447896292563731.

    Square root of 6 = 2.449489742783178 multiplied by 2 = 4.898979485566356 divided by 3 =
    1.632993161855452/1.618033988749895 ^(1/4) = 1.447896292563731.

    1.632993161855452/1.618033988749895 ^(1/4) = 1.447896292563731.

    https://www.wolframalpha.com/input/?i=%E2%88%9A24%2F3%2F%E2%88%9A%E2%88%9A+%CF%86

    Minimal polynomial:

    81 x^8 + 576 x^4 – 4096

    https://www.wolframalpha.com/input/?i=81+x^8+%2B+576+x^4+-+4096

  31. Locun ratio Pyramid and sphere with the same surface area ratio = 1.263003670594938 according to Pi as 22 divided by 7 = 3.142857142857143.

    Concerning the ratio 1.263003670594938.If a Pyramid is created with a height equal to 7 equal units of measure while the width of the square base of the Pyramid is equal to the square root of 154 equal units of measure then the Pyramid is created with a height that is equal in measure to the radius of a circle that has a surface area equal to the surface area of the square base of the Pyramid. If the height of the Pyramid that is 7 equal units of measure is divided by the ratio 1.263003670594938 the result is 5.542343354158783 equal units of measure. 5.542343354158783 equal units of measure is the radius of a sphere that has the same surface area as a Pyramid with a height of 7 equal units of measure and a square base width equal to the square root of 154 units of measure according to Pi as 22 divided by 7 = 3.142857142857143.

    14 divided by the square root of 154 = 1.128152149635532 divided by 0.893229509858919 = 1.263003670594938.

    The ratio 0.893229509858919 can be gained through the formula the square root of 87.5 = 9.354143466934854 divided by the square root of 38.5 = 6.204836822995428 = 1.507556722888818 plus = 2.507556722888818.

    The square root of 2.507556722888818= 1.583526672616795 multiplied by 14 divided by the square root of 154 = 1.128152149635532 divided by 1.786459019717839 divided by 2 = 0.893229509858919.

    A square base Pyramid with a width equal to the square root of 154 units and a height of 7 equal units has a surface area equal to the ratio 386.163735324878066.

    A sphere with a radius equal to the ratio 5.542343354158783 also has a surface area that is equal to the ratio 386.163735324878066 according to Pi as 22 divided by 7 = 3.142857142857143.

    14 divided by the square root of 154 = 1.128152149635532 is an approximation of √√φ = the square root of the square root of Phi = 1.127838485561682.

    √ ((10√(11))/7-22/7) = 1.263003670594938.

    sqrt(2/7 (5 sqrt(11) – 11) = 1.263003670594938.

    square root(2/7 (5 square root(11) – 11) = 1.263003670594938.

    Minimal polynomial:

    7 x^4 + 44 x^2 – 88

    https://www.wolframalpha.com/input/?i=7+x^4+%2B+44+x^2+-+88

    Equilateral triangle and circle with the same surface area ratio: 1.347419325335723.

    2/(√ (√3) X √√ φ = 1.347419325335723.

    2/(√ (√3) multiplied by √√ φ = 1.347419325335723.

    2/(√ (√3) multiplied by 1.127838485561682= 1.347419325335723.

    2/(square root (square root 3) multiplied by square root square root Phi = 1.347419325335723.

    2 (φ/3)^(1/4)/ √ φ = 1.347419325335723.

    2 (1.618033988749895/3)^(1/4)/ 1.272019649514069 = 1.34741932533572.

    2/(3 X Golden Ratio)^(1/4) = 1.347419325335723.

    2/(3 times Golden Ratio)^(1/4) = 1.347419325335723.

    2/(3 x Cos (36) x 2)^(1/4) = 1.347419325335723.

    2/(3 x Sin (54) x 2)^(1/4) = 1.347419325335723.

    2/(3 multiplied by φ)^(1/4) = 1.347419325335723.

    2 divided by the Golden ratio multiplied by 3 ^(1/4) = 1.347419325335723.

    3 times the Golden ratio = 4.854101966249685.

    1/2 + √ (5)/2 = The Golden ratio = 1.618033988749895.

    https://www.wolframalpha.com/input/?i=2+%28%CF%86%2F3%29^%281%2F4%29%2F+%E2%88%9A+%CF%86
    The ratio for the diameter of a circle divided by the edge of a Pentagon with the same surface area = 1.479351567442321.

    √ (34 times 17 times TAN (54)/2 times 5) times √√φ/34 = 1.479351567442321.

    √ (34 times 17 times TAN (54)/2 times 5) times 1.127838485561682/34 = 1.479351567442321.

    (75/32 + (35 square root (5))/32)^(1/4)= 1.479351567442321.

    1/2 (5/2 (15 + 7 square root (5)))^(1/4) = 1.479351567442321.

    1/2 square root (5) (1/2 (1 + 2/square root (5)) (1 + square root (5)))^(1/4) = 1.479351567442321.

    https://www.wolframalpha.com/input/?i=%E2%88%9A%2834+times+17+times+TAN%2854%29%2F2+times+5%29+times+%E2%88%9A%E2%88%9A%CF%86%2F34

    Minimal polynomial
    256 x^8 – 1200 x^4 – 125

    https://www.wolframalpha.com/input/?i=256+x^8+-+1200+x^4+-+125

    34 multiplied by 17 multiplied by TAN (54) divided by 2 multiplied by 5 = 1988.87187508084575.

    34 X 17 X TAN (54)/2 X 5 = 1988.87187508084575.

    https://www.wolframalpha.com/input/?i=34+multiplied+by+17+multiplied+by+TAN+%2854%29+divided+by+2+multiplied+by+5

    https://www.wolframalpha.com/input/?i=289+sqrt%285+%285+%2B+2+sqrt%285%29%29%29

    Square root of 1988.87187508084575 multiplied by √√φ = the square root of the square root of Phi = 1.127838485561682 divided by 34 = 1.479351567442321.

    √1988.87187508084575 multiplied by √√φ/34 = 1.479351567442321.

    √1988.87187508084575 X √√φ/34 = 1.479351567442321.

    https://www.wolframalpha.com/input/?i=%E2%88%9A1988.871875080845762709417845861232696914924540531789268563+multiplied+by+%E2%88%9A%E2%88%9A%CF%86%2F34

    The sacred cubit corrected according to Golden Pi:

    The sacred cubit is 0.526571522279798296.

    The sacred cubit is NOT 0.523598775598299.

    4/√φ subtract φ ^ 2 = 0.526571522279798296073647509006197600372179051545130088524…

    (0.526571522279798296).

    4/√φ = 3.144605511029693144 subtract The Golden ratio squared = 2.618033988749895 = 0.526571522279798296073647509006197600372179051545130088524…

    (0.526571522279798296).

    x^4 + 6 x^3 + 27 x^2 + 134 x – 79

    https://www.wolframalpha.com/input/?i=4%2F%E2%88%9A%CF%86+subtract+%CF%86+^+2

  32. Cubing the sphere ratio for equal volume 1(Diameter of sphere divided by the width of the cube) = 1.240304615214716.
    (3 times √φ/2) ^ (1/3) = 1.240304615214716.

    (3 times 1.272019649514069 /2) ^ (1/3) = 1.240304615214716.

    (1.5 times √φ) ^ (1/3) = 1.240304615214716.

    (1.5 times 1.272019649514069) ^ (1/3) = 1.240304615214716.

    https://www.wolframalpha.com/input/?i=%283+times+%E2%88%9A%CF%86%2F2%29+^+%281%2F3%29

    Minimal polynomial:

    16 x^12 – 36 x^6 – 81

    https://www.wolframalpha.com/input/?i=16+x^12+-+36+x^6+-+81

    The omose rectangle:

    √ (1 + 1/2 3^(2/3) (1 + √ (5))^(1/3)) = 1.593221748069905.

    √ (1 plus 1/2 3^(2/3) (1 plus √ (5))^(1/3))

    Minimal polynomial:

    16 x^12 – 96 x^10 + 240 x^8 – 356 x^6 + 348 x^4 – 204 x^2 – 29

    Cubing the sphere ratio for equal volume 2 (Width of Cube divided by the radius of the sphere) = 1.61250710145408.
    (16/(3 x √φ)) ^ (1/3) = 1.61250710145408.

    (16/(3 multiplied by √φ)) ^ (1/3) = 1.61250710145408.

    (16/(3 x 1.272019649514069)) ^ (1/3) = 1.61250710145408.

    (16/(3 multiplied by 1.272019649514069)) ^ (1/3) = 1.61250710145408.

    (4/3 X 3.144605511029693144) ^ (1/3) = 1.61250710145408.

    (4/3 X 4/√φ) ^ (1/3) = 1.61250710145408.

    (4 divided by 3 times 4/√φ = 3.144605511029693144) ^ (1/3) = 1.61250710145408.

    (4/3 multiplied by 4/√φ) ^ (1/3) = 1.61250710145408.

    (2 (2/3)^(1/3))/φ ^ (1/6) = 1.61250710145408.

    (2 (2/3)^(1/3))/Golden Ratio^(1/6) = 1.61250710145408.

    (2 (2/3)^(1/3))/(COS (36) x2)^(1/6) = 1.61250710145408.

    (2 (2 (√ (5) – 1))^(1/6))/3^(1/3) = 1.61250710145408.

    1.7471609294725978/1.083505882173848 = 1.61250710145408.

    1.083505882173848 is the square root of 1.173984996705329.

    1.173984996705329 is the cube root of the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895.

    1.7471609294725978:
    https://www.wolframalpha.com/input/?i=%282+%282%2F3%29^%281%2F3%29%29

    Minimal polynomial:
    3 x^3 – 16
    https://www.wolframalpha.com/input/?i=3+x^3+-+16

    26314247 divided by 16318841 = 1.61250710145408.

    https://www.wolframalpha.com/input/?i=%284%2F3+X+4%2F%E2%88%9A%CF%86+%29+^+%281%2F3%29

    Minimal polynomial:
    81 x^12 + 2304 x^6 – 65536

    https://www.wolframalpha.com/input/?i=81+x^12+%2B+2304+x^6+%E2%80%93+65536

    The ratio 4.192807348039591 can be gained if 4 is divided by 3 = 1.333333333333333 and then multiplied by Golden Pi = 4/√ φ = 3.144605511029693144 = 4.192807348039591.

    The ratio 4.192807348039591 can be gained if 4 is divided by 3 = 1.333333333333333 and then multiplied by Golden Pi = 4/√φ = 3.144605511029693 = 4.192807348039591.The ratio 4.192807348039591 can also be gained if Golden Pi = 4/√φ = 3.144605511029693144 is divided 0.75. Golden Pi = 4/√φ = 3.144605511029693144 divided by 0.75 = 4.192807348039591.The ratio 4.192807348039591 can also be gained if Golden Pi = 4/√φ = 3.144605511029693144 is divided by 6 = 0.524100918504949 and then multiplied by 8. Golden Pi = 4/√φ = 3.144605511029693144 divided by 6 = 0.524100918504949 and then multiplied by 8 = 4.192807348039591.

    The ratio 4.192807348039591 can be constructed geometrically if one third of the shortest edge length of a Kepler right triangle is multiplied by 16 equal parts and connected to the second longest edge length of the Kepler right triangle resulting in a rectangle that has measuring angles of 76.58535752773064 degrees and 13.41464247226936 degrees.

    3 multiplied by √φ = 1.272019649514069 = 3.816058948542207.
    √φ = 1.272019649514069 multiplied by 3 = 3.816058948542207.
    16 divided by 3.816058948542207 = 4.192807348039591.

    A circle with a circumference of 12 equal units of measure has a diameter with a measure of 3.816058948542207 equal units of measure according to Golden Pi = 4/√φ = 3.144605511029693144.

    Also a straight line that is divided into The ratio 4.192807348039591 can be constructed if the shortest edge length of a Kepler right triangle is divided into 3 equal units of measure and 1 equal unit of measure is taken from the shortest edge length of the Kepler right triangle that has been divided into 3 equal parts and multiplied 16 equal times and added to the second longest edge length of the Kepler right triangle.
    The longer measure of the straight line divided by the shorter measure of the straight line produces the ratio 4.192807348039591.

    Please remember that if the diameter of a circle is divided by quarter of the perimeter of a regular sided polygon with 8 or more edges contained inside of a circle the result is the √φ = 1.272019649514069 and that means that the ratio 4.192807348039591 can also be constructed geometrically if one 12th of the perimeter of a dodecagon that is contained inside of a circle is multiplied 16 equal times perpendicular to any of the poles for the vertical diameter of the circle that has a circumference of 12 resulting a rectangle that has measuring angles of 76.58535752773064 degrees and 13.41464247226936 degrees.

    The ratio 1.61250710145408 can also be gained if Golden Pi is divided by 6 and then the cube root of dividing Golden Pi by 6 is multiplied by 2. Golden Pi = 4/√φ = 3.144605511029693 divided by 6 = 0.524100918504949. The cube root of 0.524100918504949 = 0.80625355072704. 0.80625355072704 multiplied by 2 = 1.61250710145408.

    (81 x^4 + 2304 x^2 – 65536).

    4.192807348039591^ (1/3) = 1.61250710145408.

    4/3 multiplied by 4/√ φ = https://www.wolframalpha.com/input/?i=4%2F3+multiplied+by+4%2F%E2%88%9A+%CF%86

    4.192807348039591^ (1/3):

    https://www.wolframalpha.com/input/
    i=4.192807348039590859037645791162447624123317641801190600879+^+%281%2F3%29

    16/(3 multiplied by √ φ) = 4.192807348039591.

    16/(3 x by √ φ) = 4.192807348039591.

    16/(3 times √ φ) = 4.192807348039591.

    16/(3 times 1.272019649514069) = 4.192807348039591.

    16/(3 multiplied by 1.272019649514069) = 4.192807348039591.

    16 divided by (3 multiplied by √ φ) = 4.192807348039591.

    16 divided by (3 x by √ φ) = 4.192807348039591.

    16 divided by (3 times √ φ) = 4.192807348039591.

    16 divided by (3 times 1.272019649514069) = 4.192807348039591.

    16 divided by (3 multiplied by 1.272019649514069) = 4.192807348039591.

    16 divided by 3.816058948542207 = 4.192807348039591.

    https://www.wolframalpha.com/input/?i=16%2F%283+multiplied+by+%E2%88%9A+%CF%86%29

    Phi Pyramid and sphere with the same surface area ratio 1:

    1.434632715112648. Diameter of sphere divided by the height of the Phi Pyramid with the same surface area as the sphere.

    √(√(√(5) plus 2)) = 1.434632715112648

    (2 + √ (5))^(1/4) = 1.434632715112648.

    (2 + square root (5))^(1/4) = 1.434632715112648.

    √ 5 plus 2 = 4.23606797749979.

    φ ^ 3 = 1.618033988749895 ^ 3 = √ 5 plus 2 = 4.23606797749979.

    √4.23606797749979 = 2.058171027271492.

    φ = 1.272019649514069 ^ 3 = 2.058171027271492.

    √2.058171027271492 = 1.434632715112648.

    √√φ = 1.127838485561682 ^ 3 = 1.434632715112648.

    √√φ = 1.127838485561682 multiplied by √φ = 1.272019649514069 = 1.434632715112648.

    https://www.wolframalpha.com/input/?i=%E2%88%9A%E2%88%9A%CF%86+^+3

    Minimal polynomial:

    x^8 – 4 x^4 – 1

    https://www.wolframalpha.com/input/?i=x^8+-+4+x^4+-+1

  33. Phi Pyramid and sphere with the same surface area ratio 2: 1.394085035794655. Height of Phi Pyramid divided by the radius of a sphere with the same surface area as the Phi Pyramid.

    2(φ-1) ^ (3/4) = 1.394085035794655.

    2(0.618033988749895) ^ (3/4) = 1.394085035794655.

    2(Cosine (72) X2) ^ (3/4) = 1.394085035794655.

    2(Cosine (72) multiplied by 2) ^ (3/4) = 1.394085035794655.

    Minimal polynomial:
    x^8 + 64 x^4 – 256

    https://www.wolframalpha.com/input/?i=x^8+%2B+64+x^4+-+256

    The ratio 0.572838088083194 applies to the edge of a cube divided by the edge of the square base of a Pyramid that has the same volume as the cube that creates the ratio √√φ = the square root of the square root of Phi = 1.127838485561682 when the height of the pyramid is divided by half width of the square base of the Pyramid.

    1/(6^(1/3) (ϕ – 1)^(1/12)) = 0.572838088083194.

    https://www.wolframalpha.com/input/?i=1%2F%286^%281%2F3%29+%CE%A6^%281%2F12%29%29

    Pyramid with a height of 7 and a square base width of 11 with the ratio of the height of the Pyramid divided by the radius of a sphere that has the same surface area as the Pyramid according to Pi as 22 divided by 7 = 3.142857142857143 = 1.394324599916682.

    1/7√(14(√ (317)-11)) = 1.394324599916682.

    Minimal polynomial:

    7 x^4 + 44 x^2 – 112

    https://www.wolframalpha.com/input/?i=7+x^4+%2B+44+x^2+-+112

    Pyramid with a height of 7 and a square base width of 11 with the ratio of the height of the Pyramid divided by the radius of a sphere that has the same volume as the Pyramid according to Pi as 22 divided by 7 = 3.142857142857143 = 1.720277255074436.

    (4X14/11) ^ (1/3) = 1.720277255074436.

    2/11 7^(1/3) 11^(2/3)= 1.720277255074436.

    Minimal polynomial:

    11 x^3 – 56

    https://www.wolframalpha.com/input/?i=11+x^3+%E2%80%93+56

    Algebraic confirmation: https://www.wolframalpha.com/input/?i=is+%28%284+14%29%2F11%29^%281%2F3%29+algebraic%3F&lk=2
    .
    Locun ratio Pyramid and sphere with the same surface area ratio 1: 1.583452560877265.

    Diameter of sphere divided by the height of the Locun ratio Pyramid that has the same surface area s the sphere.

    √ (√ (√ (φ) plus 1) plus 1) = 1.583452560877265.

    √ (√ (√ (Cosine (36) times 2) plus 1) plus 1) = 1.583452560877265.

    √ (√ (√ (Sin (54) times 2) plus 1) plus 1) = 1.583452560877265.

    √ (√ (√φ + 1) + 1) = 1.583452560877265.

    √ (√ (√φ plus 1) plus 1) = 1.583452560877265.

    √ (√ (1.272019649514069 + 1) + 1) = 1.583452560877265.

    √ (√ (1.272019649514069 plus 1) plus 1) = 1.583452560877265.

    √√φ = 1.127838485561682.

    (√ (√ (φ) plus 1) = 1.507322012548768.

    (√ (√ (φ) plus 1) plus 1) = 2.507322012548768.

    (√ (√ (φ) plus 1) plus 2) = 3.507322012548768

    √ (√ (√ (φ) plus 1) plus 2) = 1.87278456116788.

    https://www.wolframalpha.com/input/?i=%E2%88%9A+%28%E2%88%9A+%28%E2%88%9A%CF%86+plus+1%29+plus+1%29

    Minimal polynomial:

    x^16 – 8 x^14 + 24 x^12 – 32 x^10 + 15 x^8 + 4 x^6 – 4 x^4 – 1

    https://www.wolframalpha.com/input/?i=x^16+-+8+x^14+%2B+24+x^12+-+32+x^10+%2B+15+x^8+%2B+4+x^6+-+4+x^4+-+1

    Locun ratio Pyramid and sphere with the same surface area ratio 2: 1.263062784079846.

    Height of Locun ratio Pyramid divided by the radius of a sphere with the same surface area as the Locun ratio Pyramid.
    The square root of the square root of Phi = 1.127838485561682 divided 0.892939369109291 = 1.263062784079846.

    The ratio 0.892939369109291 can be gained through the formula the square root of 2.272019649514069 = the Locun ratio 1.507322012548768 plus 1 = 2.507322012548768.

    The square root of 2.507322012548768 = 1.583452560877265 multiplied by √√φ = the square root of the square root of Phi = 1.127838485561682 divided by 1.785878738218582 divided by 2 = 0.892939369109291.
    209850139 divided by 166143870 = 1.263062784079846.

  34. Pentagonal Pyramid proportions:

    Edge of the base of the Pentagonal Pyramid = 34.

    Height of Pentagonal Pyramid = 46.796985296019899.

    Surface area of the base of the Pentagonal Pyramid = 1988.871875080844985.

    Volume of Pentagonal Pyramid = 31024.402631275276045.

    “Ratio for the height of a Pentagonal Pyramid with a Pentagonal base divided by the radius of a sphere with the same volume as the Pentagonal Pyramid with a Pentagonal base” = 2.401513521690561.

    Planet moon ratio = 3.676205016021387.

    To discover the Planet Earth moon ratio = 3.676205016021387 subtract 1 quarter of the circle’s circumference from the diameter of the circle and then divide 1 quarter of the circle’s circumference by the result of subtracting 1 quarter of the circle’s circumference from the diameter of the circle.

    Also the Planet Earth and moon ratio = 3.676205016021387 can be discovered if one 8th of the circle’s circumference is subtracted from the measure for the radius of the circle and then one 8th of the circle’s circumference is divided by the result of subtracting one 8th of the circle’s circumference from the radius of the circle.

    Divide 1 quarter of the circle’s circumference by the Planet Earth and moon ratio = 3.676205016021387 and then add the result of dividing 1 quarter of the circle’s circumference by the Planet Earth and moon ratio 3.676205016021387 to 1 quarter of the circle’s circumference to get the measure for the diameter of the circle.

    Please remember that the diameter of the circle can also be gained by multiplying 1 quarter of the circle’s circumference by the square root of Phi = 1.272019649514069.

    Divide one 8th of the circle’s circumference by the Planet Earth and moon ratio = 3.676205016021387 and then add the result of dividing one 8th of the circle’s circumference by the Planet Earth and moon ratio 3.676205016021387 to one 8th of the circle’s circumference to get the measure for the radius of the circle.

    Please remember that the radius of the circle can also be gained by multiplying one 8th of the circle’s circumference by the square root of Phi = 1.272019649514069.

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