Cube a sphere?

Hello TheVoices:

A GA on its way Sir!

I had to read your description several times, and it is only just 'laying on the surface' have not quite penetrated!

Any Maths I learned was after leaving school at 15.....JUST-15! My problem is, (letting out out my innermosts, here) is I understand the theory. It is understanding the different Maths units and terms which are used on-line I can't grasp all the time. My memory is truly terrible and I just simply forget what the units are if I have not used them for a day or so..............

I know what I would use as units here, writing it on paper looking at the screen as I look back and forth trying to grasp something.But they are not the same units, or perhaps more properly 'deciphers' I see used here on CR4. I know deciphers is not the word I should have used as a description, but cannot think of the word at this moment.

The furthest I got was 'long devision' at school. I can draw you a wonderful graph! But actually remembering what these /,^, signs means is taking some getting used to. Ever tried finding 'Psychologist' in the Dictionary?...... I do not know the names so can't always remember what to search for. (I do not look under 'S')! I think you get my drift.

I have given you a GA because it sounds 'feasible', (if that is not too rude)?

Take care..........

We conclude a cube having exactly the same volume as a sphere of radius 1 unit has the dimensions 1.611991954... x 1.611991954... x 1.611991954... to any degree of precision we like, and we didn't have to construct a darned thing.

The fact that we didn't (and couldn't) construct a darn thing clearly illustrates why your solution is incorrect (as an answer to a question in geometry).

The problem, as usually stated, has nothing to do with our ability or inability to calculate the volumes of a sphere and cube of equal volume to some limit of precision. The problem (when presented in 2 dimensions) has to to with constructing a circle and square of equal area. The impossibility of doing that has to do with the fact that pi is 3.1415... , (just as the cube root of 4/3 pi is 1.6119...) Pi is both irrational and transcendental.

Our student's answer is as good as any other, if we are simply wanting to demonstrate that a sphere can have about the same volume as some cube. John's comment (in post #1) applies equally to your answer and the student's:

Your method is equivalent to rolling up a ball of clay (until it's spherical, to some arbitrary degree), measuring the diameter (circumference if you prefer), then squishing it into a cube.

The classic problem of squaring a circle (or cubing a sphere, which just clutters the problem) is an illustration of the difficulty of thinking about (rationalizing ) transcendental numbers.

A giant GA to you sir...

MR. GUY

What do you think is good about this answer?

John's, EMC's, and TVP's answers are all correct and address the impossibility of "squaring a circle" (which applies to cubing a sphere). To me, Voice's answer appears just plain wrong. Our student is no doubt pretty bright, and has probably long known that any number we write for pi is an estimation. Obviously, then, the cube root of 4/3 pi is also an estimation, (as shown by Voices' notation...) and doesn't bring our student any closer to the real answer.

Saying that the side of a cube (equal in volume to a sphere with radius 1) is 1.6119... doesn't tell the student anything he does not already know: in 9th grade, I am sure he knows the the volume of a sphere is 4/3 pi r³. That is mathematically equivalent to saying that the side of the cube is 1.6119... or 1.61... etc. But the student has already correctly said that we can produce a cube and sphere of equal volumes down to whatever precision we are willing to work. The geometrical constuction of such a cube (using just compass and straight edge) however, is not possible. Nor is the squaring of a circle. The student's method and Voices' method both really on measurement -- an measurement is verboten in Geometrical proofs.

Trisecting an angle is another tricky task in Geometry that is easy as pie in (real world) geometry.

Emc brought up the excellent point that we can construct a line with a length of 2^.5. The square root of 2 is irrational. So the problem is not that we cannot construct lines that represent irrational numbers. The problem is that we cannot construct a square of pi^.5 on an edge. Pi, being transcendental, has special properties.

We can calculate areas and volumes to any level of precision we want. Making a Geometric construction is quite a different challenge.

Hi Ken,

I agree that TV's post does not answer the OP's question as it might be framed by a geometer wondering if indeed a circle can be squared. But TV's post seems to have answered others' questions about the relative dimensions of cubes and spheres having the same volume, so I'd say all is not lost, judging from the number of GAs TV's reply has garnered. The ultimate test, of course, of whether TV's post deserves a GA or not is if the OP believes it does. It was the OP's question, after all. The ambiguity of natural language leaves this possibility open. For my part I'm gonna reserve judgment.

Concerning the subject itself, I find it wryly ironic (in a funny kind of way) that the Lindemann-Weierstraus theorem showing that a circle cannot be squared is purely algebraic and does not involve one of those darned geometrical constructions! To wit:

Algebraists: 1

Geometers: 0

PS: For my part, I tried and tried to square a circle but I could never find a pencil sharp enough.

Of course, only geometers claim that you cannot "square a circle." For 9,999 out of 10,000 people, "squaring a circle" would involve cutting off four roundish parts to leave a square with diagonals (if they are careful) about equal to the original circle's diameter.

When I took "shop" in high school, we actually made some kinda neat stuff. But when my dad took shop, he spent almost the whole term making a precise cube by careful planing of a block of wood. Each student had to do this, apparently because the shop teacher wanted to discourage woodworking as a hobby. The shop teacher required a high level of precision.

What might make geometry interesting to our student (or conversely what might make the whole course seem silly and pointless) is the irony that the crude "constructions" made are representations of things perfectly precise that can exist only in our imaginations: there is no need to endlessly plane your cube. When you place the compass point at the "end" of a line segment, you place it with reasonable care, but cannot possibly place it exactly at the end of the segment. The hole in the paper and the squiggly lines drawn (which are, in fact three dimensional, not one dimensional, and which are anything but straight) are a language for representing absolutes. In geometry, squares have angles of exactly 90 degrees even if they are drawn as 89.

We can grab a protractor or a cad system an trisect any angle easily, to some level of precision. But try to do it with a compass and straight edge, and you have to think... and think, and think some more. With a protractor, we can see that the trisection is inherently inaccurate: the width of the pencil mark, parallax errors in placing the pencil tip where you want it, slippage of the protractor, etc. But were you "able" to trisect an angle by construction, you could be moderately sloppy and show your teacher (or search the world for someone else who cares) the construction and the teacher could say "Yes, that is exactly correct... you've gone through the right process conceptually. You've made history by trisecting an angle by construction."

(I'd have to think about this a little longer) but Geometry might be the first time our student takes a fundamentally different view of mathematics. Prior to Geometry, we are aware that some numbers are irrational (and that loads of people are) and we are certainly aware that 1/3 is tricky to write as a decimal. We might (or might not) have learned about implied precision, but understand that some (many) calculations do not produce exact answers. Then along comes Geometry, and flips that around: our sloppy drawings represent something exact. Our teacher may coach us that we will be more accurate in bisecting a line segment with a perpendicular if we set the compass to significantly more than half the segment length, but that at the same time there are an infinite number of compass settings that will work. You can make it look as precise as you want, but the apparent precision has nothing to do with whether you have perfectly bisected the line, conceptually. You can perfectly bisect an angle, but you cannot perfectly trisect an angle, or "square a circle".

There is a mystery there that an algebraic answer misses entirely, I think. It's not that there is anything "wrong" with TV's answer, it's just that John's, TVP's and emc's (etc.) get to the heart of the issue, which is that Geometry looks at things from a different perspective. Algebraically, the claim that one cannot "square a circle" (meaning to calculate the length of a side of a cube that equals, in volume, any sphere) is nonsensical -- of course you can. Algebraically, I can trisect an angle: a 40 degree angle is one third of a 120 degree angle... didn't even have to fire up my calculator.

"... we can construct a line with a length of 2^.5. The square root of 2 is irrational. So the problem is not that we cannot construct lines that represent irrational numbers."

But can we? Are we sure?

Imagine pasting a number line to the wall in your garage (mine is full) and throwing a dart at it. Let's use a number plane, because I'm hell with darts and lucky to hit the broad side of a barn.

We have, in fact, an infinite number of darts and we throw them all at it (this will take awhile, so please be patient ). We will never strike an irrational number. Ever. Our chances of missing them all is 100%. So let's nix this business of constructing things having irrational dimensions and just stick with Approximations; that stuff Intel uses to make FPUs. And dart boards. You can be damn sure that if we can't hit an irrational number with a dart, we sure as hell can't hit it with a pencil!

We will never strike an irrational number. Ever.

Or perhaps we will hit nothing but. Any point we hit falls on an arc with a radius r from whatever we pick as the origin. All the infinite points along that arc must be some distance along the arc from any other point...

And of course, any point on a plane is 2^.5 times L from any other point, were L is the length of each leg of a right triangle in which the distance between the subject points is the hypotenuse.

I feel the need for some transcendental meditation.

Hello Blink,

I do not pretend to follow this entire thread with the various, some would say slightly different, and, other would so opposing ideas and explanations.

But the OP did not mention (maybe because he did not know) about how can you square a circle using the most accurate and precise way possible. He just said is it possible.

Without getting into theory of whether the cube/sphere can be so precise, utterly full and complete (because Pi seems to be infinitely arbitrary in the way it is applied. Some method must be used in everyday life which allows obscure shapes to be quantified. My first thought here is Perfume, and liquid soap containers, they come in all sorts of sometimes really abstract shapes, which seem at first glance at least, to have no relation to any 'straight and flat' sided container. But these containers must have been estimated to have to hold a certain amount of liquid, to have been designed anyway.

Can we not look at it from that perspective, rather than using ways to quantify a cubed sphere down to the 'nth' degree? Please forgive the 'nth' spelling. I know what I mean I just cannot recall how to write it).

Take care.................

Hi babybear,

But the OP did not mention (maybe because he did not know) about how can you square a circle using the most accurate and precise way possible. He just said is it possible.

That we can create a square with the same area as a given circle is obviously true. The calculations required to show that one has accomplished the task are easily handled by an eight or ninth grader, such as our OP. So no one with a grasp of basic math says "squaring a circle" is impossible in that sense. If you do a Google search on "square a circle" (in quotes) you come up with a long list of articles (starting with the Wikipedia article "Squaring the Circle") that talk about the impossibility of "squaring the circle." How can something so simple be impossible?

The answer is that "squaring the circle" has a specific meaning to geometers as described by the Wikipedia article. Interestingly, the geometric construction method need not be terribly precise or particularly accurate, to retain its symbolic accuracy. In geometry, when you draw a square using obviously imperfect tools, that crude drawing is the symbol for a perfect square. Likewise, a drawn line segment is a symbol for a perfect line segment. So when a geometer bisects a line segment, his success in doing so is not measured with a precise ruler but by the evidence of a process: typically two arcs and a line.

Some method must be used in everyday life which allows obscure shapes to be quantified.

While that is true, it has nothing to do with geometric constructions as practiced by geometers. We know that the area of a circle is pi r². We simply do the calculation using as precise a value of pi as we need. "Squaring the circle" is not about quantifying (which everyone agrees we can do very well) but about conceptualizing. It is in that conceptual sense that "squaring a circle" and "trisecting an angle" are both impossible, following the rules that Euclid set down.

If you read through this thread on trisecting an angle, you will see that Fyz comes up with an approximation of a trisection in many steps, so he has not, as he will happily admit, "trisected an angle" in the geometer's sense. (Fyz says: It is known that it is not possible exactly to trisect an angle using just compasses and an unmarked straight edge. However, there is no theoretical limit to the achievable level of accuracy.) On the other hand, it is possible to exactly bisect an angle, using the rules and tools of a geometer, even though the pencil mark widths are so wide that they could make the bisector perhaps 1/4 degree off from what it should be. With just three arcs and a line, one can exactly bisect an angle. Fyz's approximate trisection required 18 steps, and to reduce the error, would require more steps. His work is excellent and precise, but it does not accomplish in 18 steps the analog to "bisecting an angle" in the geometers world (nor did he set out to do so).

None of this has anything to do with pragmatics. Pragmatically, trisecting an angle is just as easy as bisecting one, and calculating the area of squares and circles is simple. Calculating the volume of an odd shape is also simple, and made simpler and more precise by computers which can slice a shape into many small pieces. No pragmatist says "you cannot square a circle". But many geometers have tried (using their peculiar rules) and not one has succeeded.

So no, for the sake of the student's question, we cannot look at this from the pragmatic perspective, because, in that view, the statement he referenced is just silly -- of course we can make a square and circle of equal area, or a cube and sphere of equal volumes. To answer his question, we have to reference the Euclidian approach, which has little to do with quantifying things down to the "nth" degree. (I think your spelling is fine.)

Cheers, Ken

Hello ken,

I really thank you for trying to expand in simple terms what you have explained in more detailed Math in other posts. I find this very interesting and am drawn to the idea. Euclid was an amazing fellow! He must have been pretty bright and perhaps surrounded by like minds?

I have a memory problem, and can often if not always 'lose my way' or forget what line I was reading, which makes following the more esoteric explanations rather difficult. I manage and understand but the 'Euclid version', and the perhaps more useful pragmatic approach. I guess I am a 'hands on guy' and unless it is specifically written, I do not always 'get' the airy, 'ethereal' threads.

I hope you can understand my clumsy explanation and ideas. And, I really appreciate your reply post. Geometry was one of my favourite subjects at school but, will ill health, it meant I was in the bottom class simply because I was not at school much, and it seems, decisions were taken not to 'go too deep' in any particular subject, as, in my mind anyway, we were not worth teaching. I will never forgive the Maths teacher, as he taught us the same stuff over and over for three years. I had grasped all the first time round, you know? I am still angry.

Learn and figure things out is my constant quest. And I can usually follow a subject to a correct or 'decent' conclusion.

This is the first time I have talked about this subject, and at first thought it is something that looks obvious, ............of course you can do it like this.............then I fall flat on my face! I fully understand your ref' to the thickness of the pencil line taking a few degrees, which is why it always pays to keep a sharp point on your pencil! Because you may be trying to draw this in '2D', you have to imagine it in 3D while you are doing it?

I will have to leave it there, my brains hurting!

What a clever and brilliantly exiting thread this has been?

I am without a doubt 'out of my league' with this I am afraid, but, as with everything else I have thought about and succeeded with, the 'bulb' is beginning to glow.

Take care and happy holiday..........

Nice explanation

In school it is only required from a leaner to understand the formula and principles and not to calculate exact answers. The school system cannot discriminate on the calculator that can be afforded.

If you take the value of pi as 22/7 or 3.14 (or even 3) your answer will be wrong but you will still score full marks.

Guest,

You can, in fact, construct a real square to many places of accuracy, but you cannot construct an ideal square to exact accuracy. There is a difference, important to geometers and only mildly interesting to engineers. It's a grand exercise for your mind to work through the proof of why it can't be done; you might want to ask your teacher about doing that.

Great question. It shows real thought and maturity of insight.

because of pie

Cubing a rational number describes the computation of the volume, by unit, of a certain geometrical shape, previously defined as a cube. Of course, there are mathematical relations to a sphere of the same volume or diminsion. But the sphere has a different definition.

You clearly have given this some thought.

The answer to the difficulty in "squaring a circle" may lie in a course you have not yet taken: Geometry. In Geometry (the course not the real world) you work only with a straight edge and compass. In Geometry, you will construct many plane figures in such as way as to be able to say, for example, that you have perfectly bisected a line or an angle... even if your pencil is not perfectly sharp. No measurement is required (your straight edge does not need to be labeled or marked off in any way).

For engineers, classical Geometry is of limited use -- in other words we do not spend any time doing geometry "proofs." (there are however some Geometry techniques that are handy -- such as being able to precisely find the center of a board without measuring it.) Classical Geometry is extremely very useful in understanding number theory.

The Wikipedia articles on squaring a circle and transcendental numbers will get you thinking about such stuff, but some of this might make little sense until after you taken Geometry.

Take a thread ,make it endless by joining the ends by knotting. Using this endless thread,the total length of which is equal to the circumference , construct a circle.

Divide this endless length by 4 and using that value for the sides,construct a square.

Never bother about the irrational number you get when you do the calculation to find the diameter of the circle. But , practically,it is possible to make a circle as well as square using same dimension.

1. This will give a square with the same perimeter as a circle - "squaring the circle" refers to constructing a square with the same area as the circle.

2. "Using this endless thread ... construct a circle" - how?

Just to restate what others (especially Ken) have said: saying that it is impossible to square a circle is just referring to ones inability to do it perfectly using the specific rules of geometry (i.e compasses and a straight edge). In the same way it is impossible to trisect an angle.

If you are interested in this sort of thing you may like to look at Fyz' challenge: which was to get very close to trisecting an obtuse angle in as few moves as possible:-

http://cr4.globalspec.com/blogentry/5532/Trisection-CR4-Challenge-04-15-08

This culminated in his own incredible solution at post #89:-

http://cr4.globalspec.com/comment/221138/Re-Trisection-and-the-answer-should-have-been

Note that he had not anticipated this result when he posed the challenge.

I wish I had studied this stuff when I was young because I don't know if I don't understand the question or if I don't understand the answer. Since I know I can divide a line into 3, I don't see why I can't divide angle in 3. I guess since I didn't study geometry I don't understand the restrictions and what constitutes proof. I've vented enough for now! -- JHF

I don't have a clue about cubing a sphere. To me that is a bit like up-ing a down or redding a blue. But, this is not your average "please do my homework for me so I won't have to read the book" student guest.

I'd like to suggest that this "guest student" register and continue to be part of the CR4 community. He/She will probably have a lot of good postings in the future that will remind the rest of us of the days when we had lots of good questions. Few things limit our potential as much as knowing answers and setting aside questions.

Interesting concept. however, would it not be easier to take the calculated volume of the sphere nad solve the equation for the volume of a cube, inserting the volume of the sphere and then working backwards?

Please see Post #2.

There is an old story about Achilles chasing a hare. Achilles being the fastest man in all of Greece was halving the distance between himself and the hare every minute. All the Greek mathematicians, who were the best in the world, concluded that it was impossible for Achilles to catch the hare because he could halve the distance forever and never be able to draw even with the hare. About that time Achilles reached down and snatched the hare's head in his hands and broke it's neck, skinned it, built a fire and ate the hare.

The moral of the story is that perfection is not required in the real world, you only have to get close enough to get what you want. The trick is to determine what constitutes "close enough".

It seems to me that you have to define the terms "square a circle" and "cube a circle". Having adjustable sides, that is changing the dimensions of the square or sphere dose not meet the criteria of squaring a circle as I have understood it. Once agreement is reached on the definitions, then possibility or not can be determined. -- JHF

I get the feeling everyone is missing the point here. The problem is one of semantics and logic, not geometry and numerical analysis. The question is an example of an oxymoron, or contradiction of terms.

Semantically, you cannot square a circle because the moment you start to change the geometric shape we call a circle to that of one which we call a square; it is no longer a circle. It is now something we give another name. Probably the sequence of things would go first to ellipse, then oval, then rectangle, and finally square. This only proves that you physically manipulate an object from a circular shape to a square shape.

Plato defines this as the form of an object. Concepts that exist outside of the physical universe we can accept and understand, yet not necessarily physically exist. The object we call a circle exists, but the concept we call a circle exists outside of the physical universe. Plato called this place the formos, a conceptual reality we can all relate to. For more on this, read Plato's The Cave.

We can accept the concept of an ellipsoid circle or oval circle, since these are not contradictory. They are like terms that refine the description of the geometric form we call a circle. However, we linguistically do not accept contradictory forms.

We know contradictory forms as an oxymoron. The very word oxymoron is itself an oxymoron. It comes from the Greek roots oxy, sharp; and moros; dull. Can something be both dull and sharp at the same time? No. Can a circle be a square at the same time. No. This form does not exist in both the real universe or conceptual.

In short, what you have described is a procedure to change a circle into a square, or a sphere into as cube. It should be called by what it is: "thing that was formally a circle but is now a square." But both the concept, or form, and reality of a "squared circle" simply does not exist and you cannot create it.

What ever happened to the concept of process? You know, the stuff that happens between the time you take your laxative and the time you become..er..unpuckered?

Hello Steamerst:

Interesting post! I admit, though I have not had much input to this thread, it is another way to look at things.

"thing that was formally a circle but is now a square", I have some of his CD's!

Take care and happy holiday!

Cubing the sphere. A cube and a sphere can be created with approximate equal volumes: How to determine the volume of a Cube: The volume of a cube can be determined if the surface area of any of the cube's 6 square face is multiplied by the edge length of the cube. Alternatively the volume of a cube can be determined with the ^3 symbol on an electronic calculator. Method 1: 1. Edge of the cube divided by the ratio 1.61199195401647= the radius of the sphere with the same volume or similar volume. 2. The volume of the cube can be known if the edge of the cube is cubed with the ^3 symbol. 3. The volume of the sphere can be known and is standard as 4 divided by 3 multiplied by Golden Pi = 3.1446055110297 multiplied by the radius of the sphere cubed with symbol 3. Example A: Edge of Cube is 12. 12 Cubed is 1728. Edge of Cube divided by the ratio 1.61199195401647 = 7.444205890792799 the radius of the sphere. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 7.444205890792799 cubed = 412.529612494192434 = 1729.657190549575106. So the volume of a sphere with a radius of 7.444205890792799 equal units of measure is 1729.657190549575106. Example B: Edge of Cube is 4. 4 Cubed is 64. Edge of Cube divided by the ratio 1.611911911911912 = 2.481525181643172 the radius of the sphere. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 2.481525181643172 cubed = 15.281150741854433 = 64.070921116947914. So the volume of a sphere with a radius of 2.481525181643172 equal units of measure is 64.070921116947914 Example C: Edge of Cube is 5. 5 Cubed is 125. Edge of Cube divided by the ratio 1.611911911911912 = 3.101906477053965 the radius of the sphere. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 3.101906477053965 cubed = 29.845997542684438 = 125.138517806538877. So the volume of a sphere with a radius of 3.101906477053965 equal units of measure is 125.138517806538877. Example D: Edge of Cube is 6. 6 Cubed is 216. Edge of Cube divided by the ratio 1.611911911911912 = 3.722287772464758 the radius of the sphere. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 3.722287772464758 cubed = 51.573883753758708 = 216.239358769699189. So the volume of a sphere with a radius of 3.722287772464758 equal units of measure is 216.239358769699189. Method 2: 1. Edge of the cube divided by the Golden ratio of Cosine (36) multiplied by 2 = 1.6180339887499 = the radius of the sphere with the similar volume. 2. The volume of the cube can be known if the edge of the cube is cubed with the ^3 symbol. 3. The volume of the sphere can be known and is standard as 4 divided by 3 multiplied by Golden Pi = 3.1446055110297 multiplied by the radius of the sphere cubed with symbol 3 plus Golden Pi = 3.144605511029. Golden Pi = 3.144605511029 may have to be added to the original result for the volume of the sphere until the Cube and the sphere appear to have equal volume. Example: Edge of Cube is 6. 6 cubed is 216. The volume of a Cube with an edge length of 6 is 216. 6 divided by the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895 = 3.708203932499369. Radius of Sphere is 3.708203932499369. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 3.708203932499369 cubed = 50.990683139954571 = 213.794110950760009 plus Golden Pi = 3.144605511029693 = volume of a sphere with a radius of 3.708203932499369 = 216.938716461789702. The volume of a sphere with a radius of 3.708203932499369 equal units of measure = 216.938716461789702. (The ratio 1.6119….. or alternatively the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895 are the formulas for creating a Cube and a Sphere with approximate volume of measure with the edge of the Cube being divided by the ratio 1.6119…. or alternatively the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895 to get the measure for the sphere with a volume approximately equal to the volume of the sphere ). <"The ratio for the volume of a sphere divided by the volume of a cube that has a height equal to the diameter of a sphere">: If the diameter of a sphere is equal in measure to the edge of a cube and then the volume of the sphere is divided by the volume of the Cube then the ratio is Golden Pi = 3.144605511029693 divided by 6 = 0.524100918504949. Example: The edge of the Cube is 6. The diameter of the sphere is also 6. The surface area of a square with width of 6 equal units of measure is 36. 6 squared is 36. The Cube is made of 6 squares. 36 multiplied by 6 = 216. The volume of a Cube with an edge of length 6 equal units of measure is 6. The diameter of the sphere is 6. The radius of the sphere is 3. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 3 cubed = 113.205798397068957.The volume of a sphere with a radius of 3 equal units of measure is 113.205798397068957 equal units of measure. 113.205798397068957 divided by 216 = 0.524100918504949. 0.524100918504949 multiplied by 6 = 3.144605511029693. The volume of the Cube divided by the sphere with a diameter equal in measure to the edge of the Cube = 6 divided by Golden Pi = 3.144605511029693 = 1.908029474271104. The volume of a Cube with an edge length of 6 = 216 equal units of measure. The volume of a sphere with a radius of 3 = 113.205798397068957 equal units of measure. 216 divided by 113.205798397068957 = 1.908029474271103. " Ratio of the volume of a Sphere and a Cube with equal perimeters": If the circumference sphere's circle is equal to the perimeter of any of the 6 squares of a cube then the square root of the Golden ratio = 1.272019649514069 is produced when the diameter of the sphere is divided by the edge of the Cube. If the diameter of a Sphere divided by the edge of a Cube produces the square root of the Golden ratio = 1.272019649514069 then the ratio between the volume of the sphere divided by the volume of the Cube is equal to 1 third of the square root of 5 = 2.23606797749979 plus 1 = 3.23606797749979 = 1.078689325833263. Example: The edge of the Cube is 3. The surface area of a square with width of 3 equal units of measure is 9. 3 squared is 9. The Cube is made of 6 squares. 9 multiplied by 3 = 27. The volume of a Cube with an edge of length 3 equal units of measure is 27. The diameter of the sphere is 3.816058948542207. The radius of the sphere is 1.908029474271104. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 1.908029474271104 cubed = 29.124611797498108. The volume of a sphere with a radius of 1.908029474271104 equal units of measure is 29.124611797498108 equal units of measure. 29.124611797498108 divided by 27 = 1.078689325833263. 1.078689325833263 multiplied by 3 = 3.23606797749979. 3.23606797749979 divided by 2 = the Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895. : https://grahamhancock.com/onstotts1/

Gesundheit!

Take china clay and make a cube. Now press the edges and and form it into sphere.

A circle can be squared and a Sphere can be Cubed: Squaring the circle by creating a circle and a square with equal areas: Squaring the circle involves creating a circle with a circumference equal to the perimeter of a square. Also squaring the circle can involve creating a circle and a square with equal areas or approximate equal areas. Squaring the circle can also include harmonious relationships such as the part of the square that intersects the circle's circumference can be similar to the radius of the circle or the same as the radius of the circle or equal to half of the square's edge length. Squaring the circle with the area of the square being equal to the area of the circle usually cannot be achieved with 100% accuracy because traditional Pi 3.141592653589793 has been proven to be Transcendental in addition to being irrational. Traditional Pi 3.141592653589793 is Transcendental because Traditional Pi 3.141592653589793 does not fit any polynomial equations. Squaring the circle becomes possible and easy after traditional Pi 3.141592653589793 has been rejected and replaced with other values of Pi that are NOT transcendental. Golden Pi = 3.144605511029693 is irrational but Golden Pi is NOT transcendental because Golden Pi = 3.144605511029693 is the only value of Pi that fits the following polynomial equations: 8th degree polynomial for Golden Pi: π8 + 16π6 + 163π2 = 164. 4th dimensional equation/polynomial for Golden Pi = 3.144605511029693 (x4 + 16x2 - 256 = 0). A polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry. Both Golden Pi = 3.144605511029693 and Pi accepted as 22 divided by 7 = 3.142857142857143 can be used to create a circle and a square with equal areas of measure involving 100% accuracy. Example 1 of circle being squared with 100% accuracy: "The ancient Egyptian square root of Pi rectangle and the ancient Egyptian square root for the Golden root rectangle and also the Pythagorean theorem": https://en.wikipedia.org/wiki/Pythagorean_theorem Is it possible to create a circle with a surface area of 154 equal units because if we can create a circle with a surface area of 154 equal units of measure then we can also create a square with a surface area of 154 equal units of measure by creating a scalene triangle with the second longest edge length as 12 equal units of measure taken from the diameter of the circle that has a surface area of 154 equal units of measure, while the shortest length of the scalene triangle has 4 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 12 equal units of measure and the shortest length of the scalene triangle as 4 equal units of measure is equal in measure to the width of a square that has a surface area of 154 equal units of measure. We can use the theorem of Pythagoras: to prove that the square with a width equal to the longest length of the scalene right triangle also called the hypotenuse also has a surface area of 154 equal units of measure. Area of circle = 154. Diameter of circle = 14. Circumference of circle = 44. Ancient Egyptian Pi = 22 divided by 7 = 3.142857142857143. 12 squared = 144. 3 squared = 9. 1 squared = 1 144 + 9 + 1 = 154. A square with a surface area of 154 equal units of measure can be created if the second longest edge length of a scalene right triangle has 12 equal units of measure while the shortest edge length of the scalene right triangle has 4 equal units of measure. According to the Pythagorean theorem if a square has a width that is equal to the hypotenuse of a scalene right triangle that has its second longest edge length as 12 equal units of measure while the shortest edge length for the scalene right triangle has 4 equal units of measure then the surface area of the square that has a width equal to the measure of the hypotenuse for the scalene right triangle that has its second longest edge length as 12 while its shortest edge length is 4 is 154 equal units of measure. If the width of the square that has a surface area of 154 equal units of measure is then accepted as the longer length of a square root of ancient Egyptian Pi = 1.772810520855837 rectangle then a circle can be created with the shorter edge length of the square root of ancient Egyptian Pi = 1.772810520855837 rectangle being equal in measure to the radius of the circle with a surface area equal to the surface area of the square that has a surface are of 154 equal units of measure. According to ancient Egyptian Pi = 3.142857142857143 if the radius of a circle has 7 equal units of measure then the surface are of the circle is 154 equal units of measure. The measuring angles for a square root of ancient Egyptian Pi = 1.772810520855837 rectangle are 60.57369496075449 degrees and 29.42630503924551 degrees. 60.57369496075449 degrees can be gained when the square root of ancient Egyptian Pi = 1.772810520855837 is applied to the inverse of Tangent in Trigonometry. 29.4263050392455 degrees can be gained when the ratio 0.564076074817766 is applied to the inverse of Tangent in Trigonometry. If a circle with a diameter of 14 equal units of measure has already been created so that the surface area of the circle can have 154 equal units of measure according to ancient Egyptian Pi = 3.142857142857143 and the desire is to have a square that also has a surface area equal to the circle's surface area of 154 equal units of measure then a solution is to add 1 quarter of the circle's circumference hat is 11 to the diameter of the circle with 14 equal units of measure and at the division point where 14 is subtracted from the diameter line of 25 equal units of measure draw right angles that can touch the circumference of a circle or a semi-circle if the diameter of 25 equal units of measure is divided into 2 halves. A rectangle with its longest length as 14 while its second longest length is the square root of 154 has the ratio for the square root of the Golden root as 1.128152149635533. 1.128152149635533 is the square root of 1.272727272727273. 4 divided by 1.272727272727273 is ancient Egyptian Pi = 3.142857142857143. So the longer length of the ancient Egyptian square root for the Golden root = 1.128152149635533 rectangle is 14, the diameter of the circle with a surface area of 154, while the shorter length of the ancient Egyptian square root for the Golden root = 1.128152149635533 rectangle is the square root of 154 = 12.40967364599086, the width of the square. 1.128152149635532 squared is 1.272727272727272 and 1.272727272727272 squared is the Golden ratio of 1.619834710743799. The ancient Egyptian square root for the Golden root = 1.128152149635532 is important. Example 2 of circle being squared with 100% accuracy: My question is it possible to create a circle with a surface area of 106 equal units because if we can create a circle with a surface area of 106 equal units of measure then we can also create a square with a surface area of 106 equal units of measure by creating a scalene triangle with the second longest edge length as 9 equal units of measure taken from the diameter of the circle that has a surface area of 106 equal units of measure, while the shortest length of the scalene triangle has 5 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 9 equal units of measure and the shortest length of the scalene triangle as 5 equal units of measure is equal in measure to the width of a square that has a surface area of 106 equal units of measure. We can use the theorem of Pythagoras: to prove that the square with a width equal to the longest length of the scalene right triangle also called the hypotenuse also has 106 equal units of measure. Area of circle = 106. Rational measure for the diameter of circle = 11.62. Irrational measure for the diameter of the circle = 11.61180790611399 according to Golden Pi = 3.144605511029693. Irrational measure for the diameter of the circle = 11.61180790611399 divided by the width of the square the square root of 106 = the square root of the Golden root = 1.127838485561683. The Golden root = 1.272019649514069.The Golden root = 1.272019649514069 is the square root of Golden ratio of Cosine (36) multiplied by 2 = 1.618033988749895. Irrational measure for the circumference of the circle = 36.514555134584213 according to Golden Pi = 3.144605511029693. Square root of Golden Pi = 3.144605511029693 = 1.773303558624324 9 squared = 81. 5 squared = 25. 81 + 25 = 106. Most values of Pi can confirm that if a circle has a rational measure for the diameter as 11.62 equal units of measure then the surface area of the circle with a rational measure for the diameter of 11.62 equal units of measure is 106 equal units of measure. 2 Examples of a Sphere being Cubed : Cubing the sphere. A cube and a sphere can be created with equal or approximate equal volume: How to determine the volume of a Cube: The volume of a cube can be determined if the surface area of any of the cube's 6 square face is multiplied by the edge length of the cube. Alternatively the volume of a cube can be determined with the ^3 symbol on an electronic calculator. Method 1: 1. Edge of the cube divided by the ratio 1.612507097953907 = the radius of the sphere with the same volume or similar volume. 2. The volume of the cube can be known if the edge of the cube is cubed with the ^3 symbol. 3. The volume of the sphere can be known and is standard as 4 divided by 3 multiplied by Golden Pi = 3.1446055110297 multiplied by the radius of the sphere cubed with symbol 3. Example A: Edge of Cube is 12. 12 Cubed is 1728. Edge of Cube divided by the ratio 1.612507097953907 = 7.441827707441828 the radius of the sphere. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 7.441827707441828 cubed = 412.134369126344826 = 1728.000011252599603. So the volume of a sphere with a radius of 7.441827707441828 equal units of measure is 1728.000011252599603. Example B: Edge of Cube is 4. 4 Cubed is 64. Edge of Cube divided by the ratio 1.612507097953907 = 2.480609235813943 the radius of the sphere. 4 divided by 3 = 1.333333333333333. 1.333333333333333 multiplied by Golden Pi = 3.144605511029693 = 4.192807348039591. 4.192807348039591 multiplied by the radius of the Sphere 2.480609235813943 cubed = 15.264235893568333 = 64.000000416762973. So the volume of a sphere with a radius of 2.480609235813943 equal units of measure is 64.000000416762973.