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Hoe many times can you stack 2D objects before it becomes 3D? I assume stacking 2-dimensional planes alone the 3rd dimension would never actually stack, as along the 3rd dimension, the 2-dimensional planes have a thickness of 0 so i assume adding 2 dimensional objects is like adding 0s so is it technically possible for a set of 2-dimensional objects to be qualitatively similar in size/volume to a 3d object? would an uncountably infinite amount of 2D be enough to reach 3D? or is it simply not possible?

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Mar 7, 2023 at 10:00
  • $\begingroup$ Im asking for how many 2D objects would fit into a 3D one for example, would invite 2D objects stacked become equivalent to 3D in size $\endgroup$
    – Adithya
    Mar 7, 2023 at 10:16
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    $\begingroup$ Any 3D shape can be seen as a stack of uncountably infinitely many 2D shapes. Whether one can actually stack uncountably many 2D shapes is more of a philosophy question and depends on what you exactly mean by stack as an action. $\endgroup$ Mar 7, 2023 at 10:35
  • $\begingroup$ You can fill a 2D region of space with curve (a 1 parameter object) so I guess you may be able to fill a 3D region with a two parameter 'sheet'. This 'sheet' would still exist as a set of points in 3 dimensions though. I would guess the answer is no to stacking (whatever that might mean) 2D objects. $\endgroup$
    – Paul
    Mar 7, 2023 at 11:20
  • $\begingroup$ @Paul by “stacking” imagine placing a 2D plane over another one on a 3rd dimensional direction (like a plane that is perpendicular to the z axis being placed on top of a plane along another identical one along the z direction (assume an x, y, z 3D vector space) i guess its analogous to adding infinite 0s, or how many 0s add up to 1. from what i can see its impossible, but people say uncountably infinite 2D objects would be qualitatively similar to a 3D object in size, i just needed some clarification $\endgroup$
    – Adithya
    Mar 8, 2023 at 18:04

1 Answer 1

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In pure measure terms, the thickness of a 2D object is zero, and stacking any finite (or countably infinite) number of them together does not change this.

Integral calculus sort of treats a 3D object as being made of (uncountably) infinitely many 2D slices, just as it treats 2D graphs as being made of 1D line segments, but this mostly comes from starting with slices of non-zero thickness and looking at what happens as that thickness shrinks to zero.

There's also something called a "space-filling curve", in which you can describe a function that draws a 1D curve that nevertheless fills a 2D space, or any step up from there that you might consider. However, these kinds of curves are usually considered "pathological" because they lack a lot of the nice behaviour that we normally attribute to curves of their dimension.

So to put it simply, "infinitely many, one, and/or the question doesn't actually make sense"

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  • $\begingroup$ thanks for the answer. what im getting at is that no stacking of 2D shapes along a 3rd dimension will ever gain size along that 3rd dimension; its like adding 0s until it makes 1 or infinity- simply not possible. thanks alot! hope i didn’t misunderstand youre explanation $\endgroup$
    – Adithya
    Mar 8, 2023 at 18:31
  • $\begingroup$ Yes, you've got it, although it depends heavily on how you define your objects and how you're allowed to combine them. No normal, intuitive way of layering 2D objects will give you something that has thickness in 3D. $\endgroup$
    – ConMan
    Mar 10, 2023 at 3:24

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