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Democritus of Abdera, the ancient philosopher of Greece, reasoned over two thousand years ago:

If a cut were made through a cone parallel to its base, how should we conceive of the two opposing surfaces which the cut has produced - as equal or as unequal? If they are unequal, that would imply (the lateral side of) a cone is composed of many breaks and protrusions like steps. On the other hand if they are equal, that would imply that two adjacent intersecting planes are equal, which would mean a right cone, being made up of equal rather than unequal circles, must have the same appearance as a cylinder, which is utterly absurd.

– The Presocratics, Philip Wheelwright Editor, 1997, p. 183

So, is a cut made in such a way produce two surfaces that, according to Democritus, must be different in size? If the surfaces are exactly the same size, then why?

Please see the following link of calculating the lateral surface area of a cone for additional information: Surface area of a cone

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The answer to this mystery is in the modern mathematical idea of a boundary. The key point is that points on the the boundary of a set need not belong to the set.

In this case, the plane cuts the cone into an upper half and the lower half. It doesn't matter which way the cone's oriented, so you can get any picture you like in your mind.

The intersection of the plane with the cone is the boundary of both halves. Let's arbitrarily assign the boundary points to the lower half cone. Then the upper half cone has the same boundary, but does not include any of those points. The size of the boundary is exactly the same in each case -- there's only one boundary.

That solves the problem. There is no "next" circle up that's either the same size or a different size. That's a false mental picture of what happens when the plane cuts the cone.

If you like, you can think of it this way. When the plane cuts the cone, there are three point sets formed:

  • The lower half cone.

  • The upper half cone.

  • The points in the intersection of the cone and the plane.

Now you can see that any horizontal slice of the upper half cone is smaller (or larger, depending on which way your cone is oriented) than the size of the boundary; and any horizontal slice of the lower cone is larger (or smaller, respectively) than the boundary circle.

There is no paradox.

You can do the same thing in the plane. Take the unit circle. It divides the plane into two regions, the inside and the outside of the circle. The inside disk has the unit circle as its boundary; and the outside shape (the plane minus the unit disk) has the unit circle as its boundary. The unit circle is the boundary of both regions. It doesn't matter how you assign the boundary points, all to the inner region or all to the outer region or some to one and some to the other. No matter how you assign the points, the two regions have the same boundary, namely the unit circle.

  • "Points on the the boundary of a set need not belong to the set" Okay, so the set of all circles on the outside of the boundary are all larger than the set of all circles inside the boundary. While the size of the circle that is the boundary is neither a member of either set. – Michael Lee Jan 23 '15 at 1:29
  • @MichaelLee The OP's question specified that we cut the cone into two parts. So we need to assign the boundary points to one part or the other. But there is still only one boundary, namely the intersection of the cone and the plane; or in my second example, the unit circle. – user4894 Jan 23 '15 at 2:05
  • I've been pondering this problem more. I must confess, I still believe the surface area closer to the apex of the cone has a slightly less diameter than the surface closer to its base. – Michael Lee Feb 1 '15 at 22:46
  • @MichaelLee Consider the analogous setup in 2 dimensions. On the x-y plane draw two rays emanating from the origin of slope 1 and -1, respectively. Then cut this 2D cone with the vertical line x = 1. The cut creates a left-hand piece and a right-hand piece. The boundary of the left hand piece is exactly 2; and the boundary of the right-hand piece is also exactly 2. Draw the picture and you'll see this clearly. The cut boundary of both halves is the vertical line segment from (1, -1) to (1, 1) having length 2. It's independent of whether you assign the boundary points to one half or the other. – user4894 Feb 1 '15 at 22:54
  • @MichaelLee There is the calculus approach which arrives at user4894's boundary approach via your intuition. Consider if you cut the cone with a big saw, with one edge of your saw right on the "cut line." The saw will remove material as it cuts, and when you are done, its clear the bottom of the top piece will be smaller than the top of the bottom piece. Now repeat this with a saw that's half as thick. The bottom of the top piece will still be smaller, but by half as much. Now repeat it by a quarter-thickness saw. With a tenth-thickness saw. With a hundredth. With a thousandth. – Cort Ammon Mar 1 '18 at 18:25
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Its a great question; I hadn't come across this particular one before by Democritus; I had understood that the documentary evidence for his thinking is thin.

It appears part of a whole host of questions about analysing and understanding the behaviour of the very small, along the lines of Zeno.

Democritus is an atomist; and against infinite divisiblity; so for him an atom has a minimal definite width; thus the two 'lateral sides' of the cone should be of different sizes; one could say one is measuring by an 'atomised' real line.

There is a 'logical' correspondence of this notion using the non-standard real line; so that the bottom half is wider by an infinitesimal on each edge.

The usual answer, by using the standard real line, is that the two sizes are exactly the same.

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He writes here that it's 'utterly absurd' that the circles would be equal, because then, if we would split a cone up in as many partitions as possible, and reconstruct it, we would have a cylinder - which would then mean that a cylinder and a cone are essentially equivalent, which they are not.

Since the circles can not be equal, they must be unequal. From the first part of the quote, it can be understood that therefrom follows that 'a cone is composed of many breaks and protrusions like steps'.

The underlying problem here is that math doesn't talk about the real world.

Suppose we have a function d : [0, 10] → R  with  d(h) = h  for the diameter of the cone on a specific height. Then cutting the cone on height h1 would mean looking at d' : [0, h1]   and   d'' : (h1,10]   (note the open-closed interval here), where again d'(h)=h   and   d''(h)=h. Surely the highest point of d' is not equal to the lowest point of d'': d' includes h1 in the domain, while d'' doesn't.

However, the conclusion Democritus draws from this, that 'a cone is composed of many breaks and protrusions like steps', is incorrect: the function d is a smooth function. This is because Democritus mixes up math and the real world.

  • The height (H) of a right cone divided by its slant height (S) is exactly equal to the ratio of the differentials of its height (call it dh) to its lateral side (call it ds), or simply dh/ds = H/S It makes sense to think of the side of a cone being composed of many breaks and protrusions. It's easy to calculate the surface area of a cone thinking this way. – Michael Lee Jan 23 '15 at 0:54
  • I was only giving an example with 'd(h)=h'. Thinking of the side being composed of breaks and protrusions may be easy and correct in the physical world, it is incorrect in euclidean geometry. d(h) is a smooth function. – user2953 Jan 23 '15 at 0:58
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From the perspective of nonstandard analysis, one can come up with a different answer that is in some ways more direct. In this context, the two opposing surfaces (circles) that are formed when the cone is cut are NOT in fact equal, but differ by an infinitesimal amount.

Here is this analysis in more detail. I paraphrase Democritus' paradox as follows:

  1. A cone is a sum of infinitely many cross sections.

  2. If we cut the cone at one cross section, the cross section above it and the cross section below it must be either the same size, or different.

    (2a) If they are different, then the cone must change size significantly at this exact cut, i.e. it has a "step", which seems false.

    (2b) If they are the same, then every cross section is the same as the next one, so no cross section can differ in size from any other cross section, and so the cone is indistinguishable from a cylinder. This seems false as well.

Premise (1) may seem to be preposterous from the outset to many traditional mathematicians and laymen. But in the context of nonstandard analysis, premise (1) is in fact perfectly reasonable and true. Likewise, (2) is certainly true. The problem is in the meaning of "different".

  • If "different" means "differing by a nonzero but possibly infinitesimal amount", then the two cross sections are indeed different, so (2a) holds. But it is no problem for us to consider the cone as having an infinitesimal "step".

  • If "different" means "differing by more than an infinitesimal amount", then (2b) holds. But there is again no contradiction, for each cross section being the same as the next up to an infinitesimal is not sufficient for all cross sections to be the same. Adding up the differences over a positive real distance along the cone's height we will be adding up infinitely many infinitesimal numbers and the result will be a positive, larger-than-infinitesimal change in the size of the cross section.

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The answer to the main question is that the surfaces (circles) are the same size. This is a direct consequence of a plane having (by definition) zero thickness. Thus, only one surface (circle) is (mathematically) created. This answer does not lead to Democritus second conclusion because:
1) the statement that "two adjacent intersecting planes are equal..." is wrong. If two adjacent intersecting planes are equal, then "they" are one (and the same) plane.
2) since a plane has no thickness, any information perpendicular to it is not "available", so whether the circle came from a cone or a cylinder, is no longer known, so the object can not be "reconstructed."

On the other hand, in the "real" world, a saw blade does have thickness, so if you cut a cone with it, each of the two surfaces created will have a different size.

  • "On the other hand, in the "real" world, a saw blade does have thickness, so if you cut a cone with it, each of the two surfaces created will have a different size." -- Just curious as to how you account for the sawdust in this thought experiment. How can you claim to do this experiment at all and arrive at any kind of sensible answer? – user4894 Jan 29 '15 at 20:09
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math.stackexchange/.../setting-up-an-integral-to-find-a-cones-surface-area

Setting Up an Integral to Find A Cone's Surface Area I tried proving the formula presented here by integrating the circumferences of cross-sections of a right circular cone:

Your link was to the math StackExchange, and this infers a postulate which is, in this case, a 3-dimensional Euclidean space. Mathematical notions such as that of a "limiting case" for thin slices, are used to obtain an answer.

Euclidean space is fictional, but a good approximation to real space. It is good enough to make predictions for everyday life on an earthly scale.

Cones are fictionnal. Nobody reading this has ever seen a cone (or a sphere, or a cube...), but just rough appoximations.

A "Limiting case" for slices is also fictional because particle sizes here are finite. (in fact, the volume occupied by a particle)

As implied in other answers, the error comes from assuming a perfect two-way transposition linking things in our space to things in Euclidean space.

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Above, somebody said, "Nobody reading this has ever seen a cone (or a sphere, or a cube...), but just rough approximations."

I think he's got it backwards. Nobody reading this has ever seen a mathemagical cone (or sphere, or cube); they've seen actual cones, spheres, and cubes which can be seen and felt and used in real life; it's the imaginary mathemagical versions that are the approximations.

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