If the extension is infinite, the volume cannot be figured. You can only figure the volume by assuming that there is an end, a limit, and this is rounding off. But then you are not figuring the volume of an infinite extension.

As everyone on this thread knows, the reason why calculus was reformulated based on limits was to avoid talking about infinitesimals (and actual infinity). Instead of actually computing the volume of an infinitely long horn, we break the calculation down into a potentially infinite process, where we compute the volumes of an endless sequence of horns of increasing length (all of which have a finite volume - and so there is nothing paradoxical about any of these horns). The problem is that we're not interested in any of these horns, we're interested in the horn of actually infinite length. And the paradox (re)surfaces because we're using limits (which were introduced to avoid the paradoxes of actual infinity) to describe something that is actually infinite. I'm inclined to believe that Gabriel's Horn doesn't exist any more than the "number" 1/∞.

If the extension is infinite, the volume cannot be figured. You can only figure the volume by assuming that there is an end, a limit, and this is rounding off. But then you are not figuring the volume of an infinite extension. — Metaphysician Undercover

LOL. Well then how do you know the area under the curve is infinite then?

And the paradox (re)surfaces because we're using limits (which were introduced to avoid the paradoxes of actual infinity) to describe something that is actually infinite. — Ryan O'Connor

Wow, someone who actually understands.

I'm inclined to believe that Gabriel's Horn doesn't exist any more than the "number" 1/∞. — Ryan O'Connor

Of course it's a fictitious object.

Well then how do you know the area under the curve is infinite then? — fishfry

Because it cannot be measured. That's what infinite means.

Whereas one can describe the collection of points in 3-space comprising GH with the zeros of $P(x,y,z)=z^2x^2+y^2x^2−1$ — jgill

And the lazier definition of it typically given, such as the one in the video, is that we start with the curve 1/x starting at x=1 and rotate it about the x axis. Then the resulting object we call GH (with the same "starting at x=1" specification). But that is precisely this object. And we're talking about this object, so it's relevant for that reason.

So what are you on about jgill? And I mean that question literally; I have no idea what you're actually objecting to. Incidentally, no, calculus doesn't give us the GH paradox... broken intuitions do. I also find it a bit strange to claim that calculus is used to define the object; rather, it's used to analyze the object (surface area/volume in this case).

Because it cannot be measured. That's what infinite means. — Metaphysician Undercover

soph·ist
/ˈsäfəst/

* a paid teacher of philosophy and rhetoric in ancient Greece, associated in popular thought with moral skepticism and specious reasoning.
* a person who reasons with clever but fallacious arguments.

Because it cannot be measured. That's what infinite means. — Metaphysician Undercover

The surface area of the horn has no limit - as we consider larger and larger horns that area continues to increase, so we can loosely say that it has 'infinite' area. But as MU notes, I think it's more appropriate to say that we cannot measure the area (or I would argue that the horn simply doesn't exist). What's paradoxical is that the volume does have a limit. But that's not to say that the horn has a definite volume. We cannot complete the potentially infinite process associated with the limit any more than we can explicitly list all decimal digits of pi, so we cannot claim that it has a definite volume. The best anyone can do to offer a measure of the volume is to prematurely terminate the potentially infinite process and output a rational approximation of the volume.

(all of which have a finite volume - and so there is nothing paradoxical about any of these horns). — Ryan O'Connor

...is there? Back to "intuitive paint" based on real paint, 1 cubic foot of paint can paint 3000 square feet of wall. With that in mind, we can construct a Gabriel's horn with units of feet, chopped off at a finite length, such that it holds less than 3.15 cubic feet of paint, but takes over 1000 cubic feet to paint the outside.

The best anyone can do to offer a measure of the volume — Ryan O'Connor

Let me clarify this...there's something better than giving a numerical measure....the best we can do is to stick with an algorithm which defines a potentially infinite process. And that's how we define pi.

we can construct a Gabriel's horn with units of feet, chopped off at a finite length, such that it holds less than 3.15 gallons of paint, but takes over 1000 gallons to paint the outside. — InPitzotl

I don't see the paradox - as long as we're allowed to play with the thickness we can come up any combination of numbers for the finite horns. What's important is that the numbers in your example are all finite.

I don't see the paradox — Ryan O'Connor

I don't see the original paradox; a square foot of area has no meaningful volume.

But the same exact questions arise in the finite scenarios. "If it's holding about 3.15 cubic feet of paint, isn't that already painting the inside? If so how come it takes 1000 cubic feet then to paint the outside?"

THERE IS NO SUCH THING AS A PARADOX!!!

For example, why compare an imaginary horn with real paint?

If it's a real horn, then there must be a certain limit beyond which the internal space cannot be diminished while still maintaining the coherence of molecular and atomic structure of what constitutes the physicality of anything (a horn).

So, why compare a metaphysical horn with actual paint?

If we postulate paint made up of a 'quantized super fluid with infinite tension/compression (whichever fits) capability' (e.g. the fundamental matter of cosmic space, just an idea), then we can paint infinity.


=> If 'infinite' is compared to 'finite' then obviously, ultimately, there would be incongruence - call it paradox, chaos, whatever...

So what are you on about jgill? And I mean that question literally; I have no idea what you're actually objecting to. Incidentally, no, calculus doesn't give us the GH paradox... broken intuitions do. I also find it a bit strange to claim that calculus is used to define the object; rather, it's used to analyze the object (surface area/volume in this case — InPitzotl

Math 631 (Algebraic Geometry) (U of Mich):

"Intended Level: Graduate students past the alpha algebra (593/594) courses. Students should either already know or be concurrently taking commutative algebra (Math 614). Students should also know the basic definitions of topology — we won't be using any deep theorems, but we will use topological language all the time. Basic familiarity with smooth manifolds will be very helpful, as much as what we do is the hard version of things that are done more easily in a first course on manifolds. Undergraduate students intending to take this course should speak to me about your background during the first week of classes."

This description speaks for itself. Correct me if I am mistaken, but it appears you have tossed in AG to impress the readers of this thread. If you are indeed a mathematics professor and feel AG is necessary, then I would understand. Are you? I was one for many years and we never had an undergraduate course in AG, although some schools do. GH always came up in a standard calculus course. Tell me where you are coming from and why you found it essential to define GH this way.

By the way, you should now go to the Wikipedia article on GH and inject your considered opinion. It's a nice piece and never mentions AG. You apparently think it should. Again, if you are or were a professional math person and have strong feelings about this I will understand.

calculus doesn't give us the GH paradox. — InPitzotl

Jeez I'm with @jgill here. This is a standard example from freshman calculus. The integral of 1/x from 1 to infinity is infinite and the integral of 1/x^2 is finite. Or 1/x is square integrable but not integrable if you prefer. Algebraic geometry is a little high-powered in this context, it's not needed. Your judgment is off from letting yourself be trolled by @Metaphysician Undercover.

Correct me if I am mistaken, but it appears you have tossed in AG to impress the readers of this thread. — jgill

This is a standard example from freshman calculus. — fishfry

I'm open to suggestions, but all I'm after here is a description of the space and the object. This is related to the conversation. This came up several times:

Saying that 1/infinity equals zero is obviously an instance of rounding off. — Metaphysician Undercover

...referring to 6:20 in the video. So MU thinks the process is to substitute infinity in, as one would do in a proper integral. The point being made here is that in contrast to the proper integral limit, the ∞ in the improper integral with infinite limit isn't even a coordinate in the space.

This is not really what I'm saying. What I mean is that if the value on the x axis, or y for that matter, is proposed as infinite, then it is wrong to assign a limit, such as zero, to the perpendicular axis. By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation. In other words it is incorrect to speak of a limit here, and to say that infinity is a limit is nonsense.

What I think is that there is a fundamental incommensurability between two distinct dimensions of space, demonstrated by the irrationality of pi and the square root of two, which indicates that representing spatial forms with perpendicular axis is a sort of misrepresentation. Spatial forms, as we know them, cannot be accurately represented this way.

So if we look at the difference between a straight line and a curved line, we see that one requires two dimensions, fundamentally, while the other defines (or establishes the limits of) one dimension. The curved line, even at an infinitesimally small length requires two dimensions to be mapped by a straight-lined coordinate system. This would mean that even the most simple thing in a spatial reality (a point particle for example), would require a multi-dimensional representation. And, the multi-dimensional representation would get it wrong because of the incommensurability between two distinct dimensions.

Therefore the curved (real) line will never be commensurable with the straight (artificial) line. And, when we map the curved line with a straight-lined coordinate system, the incommensurability shows as infinity. Gabriel's horn is a curved line being mapped by a straight-lined coordinate system, and the incommensurability is evident.

The further question, which comes to my mind, is which is the proper way to represent space. Which way represents how space really is? Is the proposed "real" curvature just an illusion created by deficiencies of observation, and there is no such thing as a true arch, or perfect circle. Or does all of space consist only in multi-dimensional, non-straight relations, and our artificial dimensional straight-lined coordinate systems are incapable of giving precise representations of multidimensional existence? Or is the apparent incompatibility due to something else, which we have not yet grasped? Perhaps we ought not be representing multi-dimensional lines with perfect curves, archs, and circles. Maybe we need to banish this type of object from geometry as not properly representing reality.

This is not really what I'm saying. — Metaphysician Undercover

But that is really what's going on at 6:20; the ∞ there is the ∞ symbol of the upper limit of the integral, and it is a sentinel; a placeholder meaning unlimited. There's a notation here that requires filling in a spot for the lower limit and a spot for the upper limit. "Usually" you would fill that in with something like 1 and 2. To show you're doing this same kind of thing, but there is no upper limit, you put ∞ there. The 1/x comes from that integral.

By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation. — Metaphysician Undercover

You told me that infinite just means unlimited. Try taking that seriously for a moment. Don't say it's infinite, just say it's unlimited.

By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation. — Metaphysician Undercover

But there is a limit to how small the value can be; for any real number > 1, 1/x cannot be less than 0, whereas it can be less than any other positive real number. The limit is saying something similar... that the farther out you go, the closer you get to 0 (and that you can get arbitrarily close). The limit in its definitive form can be used to show that this is only true for 0; it is not true to say that the farther out you go, the closer you get to 1 billionth. It is only true to say that the farther out you go, the closer you get to 0 (arbitrarily so).

What I think is that there is a fundamental incommensurability between two distinct dimensions of space — Metaphysician Undercover

Again, pi and square root of 2 are real numbers. So the spaces involved have those as coordinates.

So if we look at the difference between a straight line and a curved line — Metaphysician Undercover

From here down you're pontificating about physical space, which is not the space being used here.

But that is really what's going on at 6:20; — InPitzotl

Actually I referred to 6:26, when he says one over infinity, that's zero.

But there is a limit to how small the value can be; for any real number > 1, 1/x cannot be less than 0, whereas it can be less than any other positive real number. — InPitzotl

Let's remove this necessity of a "real number", maybe that's what's misleading you. Is there any limit to how small the value can be? No, that's what's meant by infinitely small, we can conceive that there is always a further value, smaller than any value which we put a number to. That's the same as what's meant when we say that the natural numbers are infinite, only in the inverse direction. We can conceive that there is a further value, larger than any value which we put any number to.

Therefore, what is meant is that there is no limit to how small the value can be, just like in the natural numbers there is no limit to how big the value can be. This, I think, is what's misleading you. You keep thinking that there must be a limit to how small the value can be, but what is indicated by "infinite" is that there is no such limit. If a limit was intended, we'd employ "infinitesimal", which indicates that there is a smallest possible. But "infinite" indicates that there is no limit to how small we can go.

The limit in its definitive form can be used to show that this is only true for 0; it is not true to say that the farther out you go, the closer you get to 1 billionth. It is only true to say that the farther out you go, the closer you get to 0 (arbitrarily so). — InPitzotl

What is arbitrary is the choice of "0" here, as the representation of some non-existent limit. There is no limit, the value can keep getting smaller and smaller, always beyond any numerical representation which you might give it, that's what's indicated by "infinite". So there is no point in representing this value as getting close to some imaginary limit, "0". The value keeps changing without ever reaching that proposed limit, so in reality it never gets any closer to that limit. There is always infinite more values to cover before it gets there. The value really never gets any closer to the proposed limit, it's always infinitely far away, so the limit is completely irrelevant. Therefore "the farther out you go, the closer you get" is not an accurate representation at all. because the whole point in saying "infinite" is to say that there is no end, so it's impossible that the end is getting any closer.

But the same exact questions arise in the finite scenarios. "If it's holding about 3.15 cubic feet of paint, isn't that already painting the inside? If so how come it takes 1000 cubic feet then to paint the outside?" — InPitzotl

You are essentially saying that it takes <3.15 ft3 to paint the finite horn AND it takes >3.15 ft3 to paint the finite horn. If film thickness doesn't explain the apparent contradiction, then I would conclude that your problem definition is invalid. Perhaps you cannot claim that 1 ft3 can only paint 3000 ft2 of wall. I think focusing on physical paint is a distraction.

I don't see the original paradox; a square foot of area has no meaningful volume. — InPitzotl

I agree that an area has no volume, so a single drop of mathematical paint could paint any surface of arbitrarily large size. However, we are not justified to claim that it can paint a surface of actually infinite area. That requires a leap of though which we are not in a position to make.

1/x cannot be less than 0, whereas it can be less than any other positive real number. — InPitzotl

Consider the Stern-Brocot tree. If L=1/2, LL=1/3, LLL=1/4, and so on, then L_repeated is a "real number" which it cannot be less than. I suspect you'll argue that L_repeated = 0, which brings us to the classical debate of whether 0.9_repeated=1. I would argue that 0.9_repeated describes a potentially infinite process and is not a number in the same sense that 1 is (dare I say that 0.9_repeated is not a rational number), but that's a debate for another time. In any case, consider this: if a number at the 'bottom' of the Stern-Brocot tree is equivalent to both of its neighbors, are any of the real numbers actually distinct from each other?

What is arbitrary is the choice of "0" here, as the representation of some non-existent limit. — Metaphysician Undercover

Given that y continually approaches 0 as x increases, the limit is 0. We don't need a point at (∞,0) for the limit to be equal to 0. I think the problem is that your definition of limit doesn't match the standard definition. With that said, I sympathize with your view and I think your argument would be stronger if you focused on the limit used to calculate the volume of GH. Such a limit is a whole different beast since it converges to pi - an irrational 'number'. Just because the interval corresponding to V can get arbitrarily small as a approaches infinity, it doesn't necessarily mean that V has a definite value. Just as it is impossible to explicitly compute all decimal digits of pi, the best anyone can do here is either (1) provide a small interval for V or better yet (2) leave V in algorithmic form (i.e. pick your favorite formula for pi and don't bother to compute it).

Let's remove this necessity of a "real number", maybe that's what's misleading you. Is there any limit to how small the value can be? — Metaphysician Undercover

You seem to be imagining a hypothetical number "so big that" 1/x dips below 0. This sounds like speculative fantasy to me. Your reasoning that it dips below 0 could equally be applied to an imagined consequence that it levels off at and stays at 0, or that it rises again. There's basically no meaning to this.

Perhaps you cannot claim that 1 ft3 can only paint 3000 ft2 of wall. — Ryan O'Connor

Those are loosely based on real numbers. A gallon of paint can paint approximately 400 square feet . A gallon is about 1/7.5 square feet.

But thinking like an engineer, our really long Gabriel's horn is "mostly" "essentially" a negligible sized needle... the first portion that's much bigger is essentially a fixed amount of error. If our paint is 1/3000 feet thick (as the volume to area-covered ratio implies), then most of our paint on the outside is a cylinder with radius 1/3000. We could imagine this as if we're squeezing paint out of a tiny hole like a roll of toothpaste; in that sense, there's no limit to the amount of "paste" we can squeeze out of the tube.

Consider the Stern-Brocot tree. — Ryan O'Connor

Already had that thread with MU.

But thinking like an engineer, our really long Gabriel's horn is "mostly" "essentially" a negligible sized needle... the first portion that's much bigger is essentially a fixed amount of error. — InPitzotl

Most of the area is on the "needle" portion but most of the volume is enclosed by the "horn" portion. I find it odd that you're focusing on the needle portion. The typical area and volume calculations for GH are based on limits as a approaches infinity (not the other way around) - in other words, as the horn gets longer and longer. But sure...

We could imagine this as if we're squeezing paint out of a tiny hole like a roll of toothpaste; in that sense, there's no limit to the amount of "paste" we can squeeze out of the tube. — InPitzotl

You lost me here. Are you saying that the paint on the outside is a cylinder of radius 1/3000 and infinite length and asking why that infinite amount of paint doesn't agree with the finite amount of paint needed to fill needle?

Already had that thread with MU. — InPitzotl

Fair enough.

Are you saying that the paint on the outside is a cylinder of radius 1/3000 and infinite length and asking why that infinite amount of paint doesn't agree with the finite amount of paint needed to fill needle? — Ryan O'Connor

Almost; I'm talking about arbitrarily large but finite amounts of paint. And the only point here is that it's not really surprising this "outside" can be indefinitely long with a layer while the inside is limited.

Almost; I'm talking about arbitrarily large but finite amounts of paint. And the only point here is that it's not really surprising this "outside" can be indefinitely long with a layer while the inside is limited. — InPitzotl

It seems like you're implying that finite math is equally paradoxical by comparing the infinitely long needle with an infinitely long horn. If you want to say something about finite math you need to talk about a horn of finite length.

You seem to be imagining a hypothetical number "so big that" 1/x dips below 0. — InPitzotl

I didn't say anything about dipping below zero. Where did you get that idea from? I said it doesn't ever get any closer to zero. There is always an infinitude of values between it and zero, so it's really not ever getting any closer to zero. Zero is off the scale, it's literally not part of the scale, as it is excluded by virtue of being impossible. So there is always an infinity of values between the value of y and zero. Since there is always that infinity of values between any given value of y, and zero, it makes no sense to say that it is getting closer to zero.

Given that y continually approaches 0 as x increases, the limit is 0. — Ryan O'Connor

As I explained y does not in any way approach zero. It is always infinitely far away from zero, no matter what value it has, therefore it never gets any closer to zero, and cannot be said to approach zero. Zero is imposed as an arbitrary limit, on something which, by definition, has no limit. If the line came to an end at a particular value, we could say that value is the limit. But it doesn't, the line continues onward without limit. It doesn't reach zero, yet continues infinitely, so zero is right off the scale, irrelevant as unobtainable.

Consider this example.
First proposition: The natural numbers are infinite, therefore there is no highest number.
Second proposition: 20 is closer to the highest number than 10 is.
Do you see how the second proposition contradicts the first? We have the very same type of contradiction when we say that x can increase infinitely without y ever reaching zero, yet we also say that it is getting closer to zero. Zero has been excluded from the scale as an impossibility, just like the highest natural number. So we can't say that one point is closer to zero than another. How does that make any sense? It's just like saying that one number is closer to the highest number than another when there is no highest number. Here it's the lowest number, we're talking about and that's not zero because there's an infinitude of numbers to go through, making zero impossible, just like the highest number is impossible.

That's not the proper way to speak, to say that one number is closer than another to the highest number. One number is higher than another, but it is not closer to the highest number, because there is no higest number. Likewise, with the value of y, one value is lower, and another higher, but we can't say that one value is closer to the lowest number because there is no lowest number, just like there is no highest natural number. There is just more and more numbers, and zero cannot be posited as the lowest of those numbers, because it is not one of them, as unobtainable, impossible, outside the bounds.

I didn't say anything about dipping below zero. Where did you get that idea from? — Metaphysician Undercover

From this:

Is there any limit to how small the value can be? No — Metaphysician Undercover

There is always an infinitude of values between it and zero, so it's really not ever getting any closer to zero. — Metaphysician Undercover

That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1.

Zero is off the scale, it's literally not part of the scale, as it is excluded by virtue of being impossible. — Metaphysician Undercover

I assume by off the scale you mean 0 will never be reached. But that's not required for 0 to be the limit.

Since there is always that infinity of values between any given value of y, and zero, it makes no sense to say that it is getting closer to zero. — Metaphysician Undercover

Sure it does; but it's a bit more precise than this. The limit specifies that it's possible to get arbitrarily close to 0. "Arbitrarily" here is used in a strong sense that includes all positive distances at once.

That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1. — InPitzotl

That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1. — InPitzotl

I told you, you need to stop thinking about the numbers which make up the coordinate system which produces the shape. They are outside the shape, irrelevant, and insufficient for measuring the shape. There is no zero point to start from in our measurement, nor is there a zero ending point. The value of one dimension is allowed to increase indefinitely such that there is no highest value, and correspondingly, the value of the other dimension is allowed to decrease indefinitely such that there is no lowest value. Zero does not enter the picture. It is excluded, (just like "highest value" is excluded, so is "lowest value", or zero) and therefore cannot be a part of the measurement scheme.

Sure it does; but it's a bit more precise than this. The limit specifies that it's possible to get arbitrarily close to 0. "Arbitrarily" here is used in a strong sense that includes all positive distances at once. — InPitzotl

See, zero is irrelevant. At any point on the shape, the y value is "arbitrarily" close to zero. So it is false to say that it gets any closer, at any point, because it's always the same, "arbitrarily close". This arbitrariness indicates that zero is completely irrelevant to any valid measurement.

It appears to me, like you want to allow the x value to increase indefinitely, without limit, assuming no highest number, but you will not allow the y value to decrease in the corresponding way. You want to limit the y value's decrease with an imposed zero. This removes the symmetry from the shape, and is a false representation of it.

They are outside the shape, irrelevant, and insufficient for measuring the shape. — Metaphysician Undercover

Your entire response is misguided. The limit is not describing a point in the shape. If it were, it would be an empty concept; the limit would just be a function evaluation. Take:
$\hspace{2 em}f(x)= \left\{ \begin{array}{ll} \frac{x^2-1}{x-1} & x \neq 1 \\ 8 & x=1\end{array} \right.$
The value of this piecewise function at f(1) is 8, as explicitly denoted. But:
$\hspace{2 em}\lim_\limits{x \rightarrow 1}{f(x)}=2$
Despite explicitly talking about 1, which is explicitly defined by the bottom piece, the limit is about that top piece, which doesn't even have a value at x=1, nor is there an f(x)=2.

So it is false to say that it gets any closer, at any point, because it's always the same, "arbitrarily close". This arbitrariness indicates that zero is completely irrelevant to any valid measurement. — Metaphysician Undercover

Your notion of 'close' that is based on the number of points between A and B can only have value in a number system which is not dense in the reals, such as the integers. For example, since there are 3 integers between 0 and 4 but 6 integers between 0 and 7, we can conclude that so 4 is closer to 0 than 7. If you want to restrict your mathematics to the integers then your notion of 'close' is suitable. However, such a mathematics is far less powerful than orthodox math so I suspect that you'd have a very tough time convincing anyone to adopt your view.

But as I said before, I sympathize with your position. In one breath we say that we use limits to avoid actual infinity but then in the next we say that the limit can be a number inseparably tied to actual infinity (e.g. pi). But I would argue that the resolution to such contradictory thinking is simple: just don't say that the limit is a number with actually infinite digits. Instead, keep the limit in algorithmic form - say that the limit is a potentially infinite process which is described by [pick your favorite] algorithm for pi, for example, 4(1-1/3+1/5-1/7+1/9-...)

Your notion of 'close' that is based on the number of points between A and B can only have value in a number system which is not dense in the reals, such as the integers. For example, since there are 3 integers between 0 and 4 but 6 integers between 0 and 7, we can conclude that so 4 is closer to 0 than 7. If you want to restrict your mathematics to the integers then your notion of 'close' is suitable. However, such a mathematics is far less powerful than orthodox math so I suspect that you'd have a very tough time convincing anyone to adopt your view. — Ryan O'Connor

You don't seem to quite grasp why I reject "closer". The line is known to never reach the limit, that's the point with "infinite". So it makes no sense to say that it is getting closer to the limit. It is impossible for the line to reach the limit. so it is impossible that one point is closer to the limit than another. The proposed limit is outside the parameters by which the line can be measured.

The issue is how to best measure the shape which is Gabriel's horn. Orthodox mathematics, while it is very good at other things, fails here, as is evident from the appearance of the paradox. The reason it fails, I believe, is because it describes a feature which very clearly cannot be described in terms of limits, as getting closer to a limit, and that's nonsensical.

So I don't agree with your resolution, because you still want to apply limits, where the shape denies the application of limits. If measurement necessarily involves the application of limits, then we regard this as having created a shape which cannot be precisely measured, just like a circle.