@@balintkovacs9996 He has not made a mistake, he has used a ruler instead of a straightedge to construct OE, so it is still impossible to construct with just a compass and straightedge. It is absolutely possible with other tools like a ruler, hatchet, or Archimedean spiral.
What if I told you that I found a unified way to divide any angle into any possible number, but not just with a ruler and compass? Can this be considered a solution to this problem? (Any angle can be accurately divided into any possible number of divisions 3,4,5,7,9, etc.) and not an approximate division
The impossibility for the three Greek's problems is absolutely true by any tools or any means, tools are only for understanding, they are so irrelevant to any theoretical problems in principle since tools have a perpetual error of measurements, Ref: Google search for (Karzeddin non-existing angles) Good luck
It would involve a marked ruler, as he says. Altho this isn't perfectly clear, I imagine that this would allow you to know where exactly to place OE so that DF, FE, and OA are equal. For the Greeks, this is likely equivalent to a neusis.
Is it possible to use a compass, an unnumbered ruler, and (rope, string, etc. any means to measure the arc length of an angle) to triangulate the angle?
The number 360 makes for a handy means of stating angles in numerical values. Unfortunately it is incommensurate with the division of a circle. Well actually, in some ways maybe that is good.- Some power of 2 could work, say 512 degrees.
At some point there would have to be the trisection of an angle. Anyway, degrees were devised as a method of measurement defined by the circle. So one cannot define a circle by 360 degrees. The number 360 conveniently has many devisors, making it useful. Ninety degrees is by definition a right angle. The number 90 is arbitrary; and it cannot be bisected by pure constuction: 90/2 is 45, then 45/2, 45/4, 45/8, 45/16, 45/32, 45/64,.. skipping 1. A protractor is a mechanically-printed, empirical measuring tool used to approximate angle magnitude. It cannot be constructed by compass and straightedge. In pure geometry there is the acute angle, right angle, obtuse angle, and so on. Numbers are seldom involved. Euclid showed that in any triangle the sum of the angles is equal to two right angles.
π = 2 in Riemann Paradox and Sphere Geometry System Incorporated... So Square ⬜ the Circle 🔴 is Possible 🙂😄 💯 In English and German... University 🎓🏫, Universal ♾️💍 as Same as Universe That Means π = 2 in Universe is Indeed Correct 😄💯
I tried to solve this problem for more than thirty-five years, adhering to the old rules of geometry. I only used calipers and an unnumbered ruler, and then you used algebra. Algebra is not included here. The rule is clear. You should not use other than the ruler and the compass. I consider algebra as a tool that should not be used here .. ...Thank you
Hi Assaf, if you continue to watch the series of videos, you'll see that we are not using algebra as an extra tool --- rather we use algebra to express what we can construct with just compass and straightedge. I hope you find it helpful.
Squaring the circle has ALREADY been solved by KNOWING the TRUE value of pi. The CORRECT, EXACT, TRUE value of pi is 3.144605511029693144 or 4/sqrt(golden ratio). This is what happens when CLOWNS don't physically measure the circumference of at least a 100m in diameter circle, doesn't check for error where the circumscribed segment is LONGER than the arc of the circle, which is obvious at 0.002 degree.
Other than straight edge and compass, if allow me also can use a pencil ,a cutter and few pieces of cardboard, I can trisect any arbitrary angle by geometry method only. I invite challenge who can do that. Dominic Chan of Hong Kong. 2022.Oct 23.
Your geometric construction for trisecting the angle is correct when AO=DF=EF but cannot be accomplished with ruler and compass because what you have done is randomly draw line OE yet the included angle is automatically = to 1/3 of the angle in question. Rubbish! Your reason for the inability to square the circle because pi is transcendental is also flawed despite it's being the acclaimed proof. Numbers have nothing to do with geometry and ruler and compass construction. If you roll a circle on a line for 1/2 it's circumference it will generate a line equal to pi which can then be used to square the circle. See 'Squaring the Circle by Rolling the Circle' on Manoj Dhakal's channel. It's brilliant.
I encourage you to watch the rest of this series if you'd like to see these arguments developed in more detail and understand why we can't construct a transcendental number, etc. Rolling a circle on a line is not an allowed move. th-cam.com/play/PLOROtRhtegr5r-nmEZ7nbvXupW51P2VhH.html
@@MathatAndrews Now that's one way of getting the viewing numbers up. All I need to know is that 1/2 the circumference = pi and the difficulty is then to transform that curve into a line using compass and ruler and rolling the circle to flatten the curve can be done using ruler and compass only so NO ONE CAN SAY THAT IT IS NOT AN ALLOWED MOVE. If I roll a wheel the circumference of which is half painted red and the other half blue then on rotating the wheel just over one turn i will get a small patch of red, a block of blue, a block of red, a small patch of blue. I know the block of red and blue will both be exactly pi in length relative to a wheel radius of exactly 1 and no one can tell me otherwise. The numerical construct of pi is totally irrelevant.
@@alastairbateman6365 "can be done using ruler and compass only": In this series I prove that the operation you describe cannot be done with straightedge and compass. This was proved in the 1800s using field theory; it is no longer up to debate. You are, of course, welcome to introduce variations of the problem, permitting new moves that can allow for new constructions. But you should know that the problem raised by the Greeks has been fully resolved by modern algebra.
@@MathatAndrews Only in your eyes, not mine or I suspect Manoj's either. My perspective is this. What ever we can do with compass and ruler in the real world, no matter how imperfect, we can perform in the world of the mind with perfection provided the moves are provable as in Euclid's 'Elements' or are rational, logical and indisputable /self evident like rotating a wheel. I uploaded a video the other day 'Making the Impossible Possible : Squaring the Circle'. I give the means for rotating a circle using ruler and compass only that fit the criteria I have given. The Constructions of Manoj Dhakal and myself are to all intents and purposes identical to each other and to Euclid's 'Element's' book 2 prop 14 and to Rene Descartes constructions for the square root and quadratics. Our offerings are far more positive and constructive where one can see a valid geometrical diagram than a bland no it can't be done because pi is transcendental. We have shown that the statement ' ... roots are algebraic and therefore constructable whilst transcendental numbers are not algebraic and therefore not constructable'' to be FALSE. Problems or deductions in modern algebra are not necessarily applicable to bog standard ruler and compass constructions
All of these have solutions, just not algebraic solutions For trisecting angles you just need some basic understanding of shapes and how they interact with other shapes, hell, you were nearly there with your trisection The squared circle is a matter of convergence with a pattern, and realizing that pi is rational geometrically (via a consistent variable ratio), but not algebraically The doubled cube is harder, but that’s doable if you think outside of the ‘box’ in terms of what defines a cube I’ll have a video up about this eventually, but I don’t have a timeframe for that Either way, algebra isn’t the limiter of geometry, if anything it’s the other way around, and that’s all that’s been misunderstood with these supposed impossible problems If you think I’m wrong just consider the Möbius Strip
Hi Orion, thanks for commenting! The problem the Greeks were concerned about is if it is possible to achieve these constructions with *just* straightedge and compass. Modern algebra allows us to prove this is not possible. However, if one uses other tools, such as a ruler instead of a straightedge, then some of them do become possible (such as trisecting the angle). You can see this explored further in the lectures in this series. All the best exploring these themes in your upcoming video.
@@MathatAndrews thanks for responding! I think you missed my point, I understand the basis of the argument, I’m saying algebra isn’t the deciding factor in what can and can’t be done in geometry, as it’s a reduction of complexity in values That makes it much easier to compare those values, absolutely, else we wouldn’t know that Pi goes on forever after it’s decimal But geometry allows for that infinite to end via a circle, without losing any value If anything it’s geometry that determines what is possible within algebra, not the other way around I’m saying all of those are solvable with a compass and straightedge, and you don’t need to measure or define a value to do so, the compass can create comparative lengths well enough for any of those anyway You might want a ruler and protractor handle to check afterward, but those can sit to the side during the drawing portion, the outcome is consistent I’m serious about this, geometry is another beast entirely compared to algebra
@George Armstrong simple bruh These don’t have mathematical solutions They have geometric solutions instead It’s really that simple, but if you want the more complex answer: Dimensions don’t reduce as well as they expand, and certain constructs are better understood in higher dimensions Take a circle, just a simple circle, the ratio between the circle’s circumference and diameter creates a value that never ends, even though both those values reference 2D constructs that do end If that’s too much for ya to start with let me know, cause I have a hell of a lot more, this is a thing I’ve studied into and researched myself, I’ve got a books worth of sketches and notes on it
@George Armstrong I can’t type out a picture, but I’ll try to explain it First step: recognize that the value you are trying to reach is geometric in the first place, that being π. Since that number has no exact value, we’ll be reaching a squared circle via a convergent pattern To start simply draw a circle, then bisect it twice, creating a crosshair pattern (if you can’t do this with a compass and straightedge then don’t bother reading the rest or responding) Next at each four interactions of the circle and crosshair create a circle of equal size to the first, this will give you twelve points around your circle Next is the tricky part to explain, but if you followed along you should see the beginning of the square. Connect the points adjacent to each cross point with a line perpendicular to the line creating a cross point, which should look like this: (+ Continue each of those lines outward and congrats, you’ll have the beginning of a pattern that converges on both a circumference based squared-circle, and an area based squared-circle Sorry if you can’t follow what I’m saying, but that’s not my problem
@George Armstrong the next steps require you to be looking at the first shape, and is better shown in video or person, but since you seem to think I’m demeaning, I’ll avoid giving you the homework line No, clearly you know everything, and I am the fool, unable to prove a pointless conjecture to the master of mathematics, geometry, and the TH-cam comments section Seriously tho, not providing proof is one thing, but not even putting in the basic effort of understanding the argument? Then demanding I make you see it? Nah homie, disprove my conjecture, I’m saying you can converge on a perfect squared circle using a geometric pattern, go ahead and pull up that 200 year old ‘proof’ written by a brilliantly foolish late teenager, who died shortly after publishing his work by a goddamn gun duel Do it, actually put in the effort or screw off
I'll simply said : Trisect an angle? = Sure! that's why crepes exists! Double the cube? = Two combined cube = a log/logtangle! CASE CLOSED! Squaring the Circle? = Hmmm, this is HARD for you~ Call your circle(friends,Gf,etc of yours) and then makes all tell all of their secrets and eventually you all SQUARED! 😂✌🏼
Thank you, it really helps me to understand my report in Abstract algebra. 😊👍
Does the narrator mean to say that he has succeeded in trisecting any angle by pure construction?
that is impossible. He made a mistake by "constructing" OE
@@balintkovacs9996 He has not made a mistake, he has used a ruler instead of a straightedge to construct OE, so it is still impossible to construct with just a compass and straightedge. It is absolutely possible with other tools like a ruler, hatchet, or Archimedean spiral.
What if I told you that I found a unified way to divide any angle into any possible number, but not just with a ruler and compass? Can this be considered a solution to this problem? (Any angle can be accurately divided into any possible number of divisions 3,4,5,7,9, etc.) and not an approximate division
great and wonderfull
The impossibility for the three Greek's problems is absolutely true by any tools or any means, tools are only for understanding, they are so irrelevant to any theoretical problems in principle since tools have a perpetual error of measurements,
Ref: Google search for (Karzeddin non-existing angles)
Good luck
Easy to double a cube if you don’t mind turning it into a rhombic dodecahedron. :)
5:55 How are you constructing line OE?
Exactly, how does he cacluate D? It just appears :)
It would involve a marked ruler, as he says. Altho this isn't perfectly clear, I imagine that this would allow you to know where exactly to place OE so that DF, FE, and OA are equal. For the Greeks, this is likely equivalent to a neusis.
Is it possible to use a compass, an unnumbered ruler, and (rope, string, etc. any means to measure the arc length of an angle) to triangulate the angle?
The number 360 makes for a handy means of stating angles in numerical values.
Unfortunately it is incommensurate with the division of a circle.
Well actually, in some ways maybe that is good.-
Some power of 2 could work, say 512 degrees.
24 likes is a disservice to this amazing video
Is it possible to divide a circle into 360 degrees without using a protractor, calculator, or graduated ruler?
Is a protractor not essentially just a round, graduated ruler?
At some point there would have to be the trisection of an angle.
Anyway, degrees were devised as a method of measurement defined by the circle. So one cannot define a circle by 360 degrees. The number 360 conveniently has many devisors, making it useful.
Ninety degrees is by definition a right angle. The number 90 is arbitrary; and it cannot be bisected by pure constuction:
90/2 is 45, then 45/2, 45/4, 45/8, 45/16, 45/32, 45/64,..
skipping 1.
A protractor is a mechanically-printed, empirical measuring tool used to approximate angle magnitude. It cannot be constructed by compass and straightedge.
In pure geometry there is the acute angle, right angle, obtuse angle, and so on.
Numbers are seldom involved.
Euclid showed that in any triangle the sum of the angles is equal to two right angles.
the greeks were supposed to trisect arbitrary angles with only straightedge and compass and in a finite number of steps
he did not prove that GF is perpendicular to AE.
The circle one requires pi, unfortunately
π = 2 in Riemann Paradox and Sphere Geometry System Incorporated...
So Square ⬜ the Circle 🔴 is Possible 🙂😄 💯
In English and German...
University 🎓🏫, Universal ♾️💍 as Same as Universe That Means π = 2 in Universe is Indeed Correct 😄💯
Tau = 2^2 = π^π = 2 × π = 4 in the Riemann Paradox and Sphere Geometry System Incorporated...
You can trisect an angle of 1 degree. But other than that ???
*In order to trisect an angle of 1 degree, you would first need to construct a 1 degree angle.*
I tried to solve this problem for more than thirty-five years, adhering to the old rules of geometry. I only used calipers and an unnumbered ruler, and then you used algebra. Algebra is not included here. The rule is clear. You should not use other than the ruler and the compass. I consider algebra as a tool that should not be used here .. ...Thank you
Hi Assaf, if you continue to watch the series of videos, you'll see that we are not using algebra as an extra tool --- rather we use algebra to express what we can construct with just compass and straightedge. I hope you find it helpful.
Thanks .... Yes, I will follow up@@MathatAndrews
Squaring the circle has ALREADY been solved by KNOWING the TRUE value of pi. The CORRECT, EXACT, TRUE value of pi is 3.144605511029693144 or 4/sqrt(golden ratio). This is what happens when CLOWNS don't physically measure the circumference of at least a 100m in diameter circle, doesn't check for error where the circumscribed segment is LONGER than the arc of the circle, which is obvious at 0.002 degree.
Archimedes method of exhaustion and infinite series are PSEUDOMATH.
Pi isnt fixed tho, its n-polygon dependent.
Other than straight edge and compass, if allow me also can use a pencil ,a cutter and
few pieces of cardboard, I can trisect any arbitrary angle by geometry method only.
I invite challenge who can do that. Dominic Chan of Hong Kong. 2022.Oct 23.
Your geometric construction for trisecting the angle is correct when AO=DF=EF but cannot be accomplished with ruler and compass because what you have done is randomly draw line OE yet the included angle is automatically = to 1/3 of the angle in question. Rubbish!
Your reason for the inability to square the circle because pi is transcendental is also flawed despite it's being the acclaimed proof. Numbers have nothing to do with geometry and ruler and compass construction. If you roll a circle on a line for 1/2 it's circumference it will generate a line equal to pi which can then be used to square the circle.
See 'Squaring the Circle by Rolling the Circle' on Manoj Dhakal's channel. It's brilliant.
I encourage you to watch the rest of this series if you'd like to see these arguments developed in more detail and understand why we can't construct a transcendental number, etc. Rolling a circle on a line is not an allowed move. th-cam.com/play/PLOROtRhtegr5r-nmEZ7nbvXupW51P2VhH.html
@@MathatAndrews Now that's one way of getting the viewing numbers up.
All I need to know is that 1/2 the circumference = pi and the difficulty is then to transform that curve into a line using compass and ruler and rolling the circle to flatten the curve can be done using ruler and compass only so NO ONE CAN SAY THAT IT IS NOT AN ALLOWED MOVE.
If I roll a wheel the circumference of which is half painted red and the other half blue then on rotating the wheel just over one turn i will get a small patch of red, a block of blue, a block of red, a small patch of blue. I know the block of red and blue will both be exactly pi in length relative to a wheel radius of exactly 1 and no one can tell me otherwise.
The numerical construct of pi is totally irrelevant.
@@alastairbateman6365 "can be done using ruler and compass only": In this series I prove that the operation you describe cannot be done with straightedge and compass. This was proved in the 1800s using field theory; it is no longer up to debate. You are, of course, welcome to introduce variations of the problem, permitting new moves that can allow for new constructions. But you should know that the problem raised by the Greeks has been fully resolved by modern algebra.
@@MathatAndrews Only in your eyes, not mine or I suspect Manoj's either.
My perspective is this. What ever we can do with compass and ruler in the real world, no matter how imperfect, we can perform in the world of the mind with perfection provided the moves are provable as in Euclid's 'Elements' or are rational, logical and indisputable /self evident like rotating a wheel. I uploaded a video the other day 'Making the Impossible Possible : Squaring the Circle'. I give the means for rotating a circle using ruler and compass only that fit the criteria I have given.
The Constructions of Manoj Dhakal and myself are to all intents and purposes identical to each other and to Euclid's 'Element's' book 2 prop 14 and to Rene Descartes constructions for the square root and quadratics.
Our offerings are far more positive and constructive where one can see a valid geometrical diagram than a bland no it can't be done because pi is transcendental.
We have shown that the statement ' ... roots are algebraic and therefore constructable whilst transcendental numbers are not algebraic and therefore not constructable'' to be FALSE. Problems or deductions in modern algebra are not necessarily applicable to bog standard ruler and compass constructions
@@alastairbateman6365 Feel free to check out the series if you're interested in understanding a bit more of the field theory. Regards.
All of these have solutions, just not algebraic solutions
For trisecting angles you just need some basic understanding of shapes and how they interact with other shapes, hell, you were nearly there with your trisection
The squared circle is a matter of convergence with a pattern, and realizing that pi is rational geometrically (via a consistent variable ratio), but not algebraically
The doubled cube is harder, but that’s doable if you think outside of the ‘box’ in terms of what defines a cube
I’ll have a video up about this eventually, but I don’t have a timeframe for that
Either way, algebra isn’t the limiter of geometry, if anything it’s the other way around, and that’s all that’s been misunderstood with these supposed impossible problems
If you think I’m wrong just consider the Möbius Strip
Hi Orion, thanks for commenting! The problem the Greeks were concerned about is if it is possible to achieve these constructions with *just* straightedge and compass. Modern algebra allows us to prove this is not possible. However, if one uses other tools, such as a ruler instead of a straightedge, then some of them do become possible (such as trisecting the angle). You can see this explored further in the lectures in this series. All the best exploring these themes in your upcoming video.
@@MathatAndrews thanks for responding!
I think you missed my point, I understand the basis of the argument, I’m saying algebra isn’t the deciding factor in what can and can’t be done in geometry, as it’s a reduction of complexity in values
That makes it much easier to compare those values, absolutely, else we wouldn’t know that Pi goes on forever after it’s decimal
But geometry allows for that infinite to end via a circle, without losing any value
If anything it’s geometry that determines what is possible within algebra, not the other way around
I’m saying all of those are solvable with a compass and straightedge, and you don’t need to measure or define a value to do so, the compass can create comparative lengths well enough for any of those anyway
You might want a ruler and protractor handle to check afterward, but those can sit to the side during the drawing portion, the outcome is consistent
I’m serious about this, geometry is another beast entirely compared to algebra
@George Armstrong simple bruh
These don’t have mathematical solutions
They have geometric solutions instead
It’s really that simple, but if you want the more complex answer: Dimensions don’t reduce as well as they expand, and certain constructs are better understood in higher dimensions
Take a circle, just a simple circle, the ratio between the circle’s circumference and diameter creates a value that never ends, even though both those values reference 2D constructs that do end
If that’s too much for ya to start with let me know, cause I have a hell of a lot more, this is a thing I’ve studied into and researched myself, I’ve got a books worth of sketches and notes on it
@George Armstrong I can’t type out a picture, but I’ll try to explain it
First step: recognize that the value you are trying to reach is geometric in the first place, that being π. Since that number has no exact value, we’ll be reaching a squared circle via a convergent pattern
To start simply draw a circle, then bisect it twice, creating a crosshair pattern (if you can’t do this with a compass and straightedge then don’t bother reading the rest or responding)
Next at each four interactions of the circle and crosshair create a circle of equal size to the first, this will give you twelve points around your circle
Next is the tricky part to explain, but if you followed along you should see the beginning of the square. Connect the points adjacent to each cross point with a line perpendicular to the line creating a cross point, which should look like this: (+
Continue each of those lines outward and congrats, you’ll have the beginning of a pattern that converges on both a circumference based squared-circle, and an area based squared-circle
Sorry if you can’t follow what I’m saying, but that’s not my problem
@George Armstrong the next steps require you to be looking at the first shape, and is better shown in video or person, but since you seem to think I’m demeaning, I’ll avoid giving you the homework line
No, clearly you know everything, and I am the fool, unable to prove a pointless conjecture to the master of mathematics, geometry, and the TH-cam comments section
Seriously tho, not providing proof is one thing, but not even putting in the basic effort of understanding the argument? Then demanding I make you see it?
Nah homie, disprove my conjecture, I’m saying you can converge on a perfect squared circle using a geometric pattern, go ahead and pull up that 200 year old ‘proof’ written by a brilliantly foolish late teenager, who died shortly after publishing his work by a goddamn gun duel
Do it, actually put in the effort or screw off
I'll simply said :
Trisect an angle? = Sure! that's why crepes exists!
Double the cube? = Two combined cube = a log/logtangle! CASE CLOSED!
Squaring the Circle? = Hmmm, this is HARD for you~ Call your circle(friends,Gf,etc of yours) and then makes all tell all of their secrets and eventually you all SQUARED! 😂✌🏼
It's still great and wonderful 🤍