Have a cone and divide it into infinately small slices

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Discussion Overview

The discussion revolves around the concept of dividing a cone into infinitely small slices and the implications of this division on the geometry of the cone versus a cylinder. Participants explore the mathematical and conceptual nuances of infinitesimals, integration, and the nature of areas of the slices.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that if a cone is divided into infinitely small slices, both sides of each slice would have the same area, suggesting that the cone could be considered a cylinder.
  • Others argue against this, stating that the assumption of equal areas is a mathematical convenience for integration and does not reflect the reality of a perfect cone.
  • A participant questions the nature of infinitely small slices, suggesting that if they are infinitely small, they could never complete the cone or cylinder.
  • There is a discussion about the relationship between infinite points and non-zero lengths, raising questions about how infinitesimals relate to actual measurable quantities.
  • Some participants inquire whether there is a difference between "infinitely small" and "zero," with one noting that in non-standard analysis, infinitesimals can be distinct from zero.

Areas of Agreement / Disagreement

Participants express differing views on the implications of dividing a cone into infinitely small slices, with no consensus reached on whether this leads to a cylinder or retains the cone's properties. The nature of infinitesimals and their relation to zero also remains a point of contention.

Contextual Notes

The discussion involves assumptions about the nature of infinitesimals and the mathematical framework being used, which may not align with traditional interpretations of geometry.

Lorentz
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If I have a cone and divide it into infinately small slices. Wouldn't both sides of one slice have the same area and wouldn't the next slice (and so on) have the same area as the slice before. So wouldn't your cone actually be a cylinder?

My answer is no, because the reasoning is wrong. If I had infinately small slices I would never complete the cone/cylinder in the first place. And the assumption of both sides having the same area is an assumption to be able to integrate, but is not reality. If we're talking about the perfect cone then both sides of the slices should have different areas even if the slices were infinately small.

What's your view on this?
 
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Lorentz said:
If I have a cone and divide it into infinately small slices. Wouldn't both sides of one slice have the same area and wouldn't the next slice (and so on) have the same area as the slice before. So wouldn't your cone actually be a cylinder?
It would something more like, as the number of slices approaches infinity, the width of each slice approaches zero, and the areas on the two faces of the slice approach each other.

If I had infinately small slices I would never complete the cone/cylinder in the first place.
Here's something to think about. You should know that a line is made up of infinite points. Each point has zero size. So, given an infinite number of zero-sized points put together, how is it that you get a line with non-zero-size?

And the assumption of both sides having the same area is an assumption to be able to integrate, but is not reality.
Well, be careful here, because you can't really make any arguments "from reality" when dealing with math. Math is a useful tool in modelling reality, but that doesn't mean that it is based on reality (it is based on its own axioms, some of which seem rather unnatural), nor does it mean that it is an accurate tool in modelling reality.
If we're talking about the perfect cone then both sides of the slices should have different areas even if the slices were infinately small.
What does it mean to be infinitely small? Can something be smaller than infinitely small? What would the difference in area be between the two faces?
 
AKG said:
Can something be smaller than infinitely small? What would the difference in area be between the two faces?

erm... the difference would be infinitely small? But that would still be a difference which makes it possible to glue the slices together and get the cone back again rather then a cylinder. If the difference would be zero we would get the cilinder.
 
This question just popped into my mind: Is there a difference between infinitely small and zero?
 
Lorentz said:
This question just popped into my mind: Is there a difference between infinitely small and zero?

Yes, IF you are working in "non-standard analysis" and, by "infinitely small", you mean "infinitesmal". Otherwise "infinitely small" is just a (misleading) shorthand for "in the limit".
 

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