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Can we stack $2D$ sheets to form a $3D$ cuboid?

This makes no sense to me because if each sheet has $0$ thickness, then the cube must also have $0$ thickness, right?

$$0+0+...+0=0?$$

But some of my school math teachers thought otherwise, and more recently again I saw one video on YouTube about this idea.

So, is there some system where it makes sense to say "stacked $2D$ sheets form a $3D$ cuboid"? This would seem like getting an extra-dimension out of nowhere, wouldn't it?

Clayton
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Truth-seek
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  • This might be of interest: space-filling curve. Also similar in flavor is the Banach-Tarski Paradox. – Clayton Feb 20 '18 at 17:10
  • yes, just like a line segment formed out of points that have no length, a cube can be considered as infinite number of squares stacked up. – Vasili Feb 20 '18 at 17:10
  • I don't see how this is any different from an oft-seen question around here "how can a lot of points stacked together make a line?" https://math.stackexchange.com/q/1083841/29335 – rschwieb Feb 20 '18 at 17:10
  • I think that is true in real life because there is a distance (although astronomically small) between each of the atoms and glosbes and everything is created from atoms and glosbes. – user061703 Feb 20 '18 at 17:25

1 Answers1

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It is true that 3-dimensional space is a union of 2-dimensional planes (infinitely many, in fact uncountably many, of them).

In the same way, a line segment of length 1 is a union of infinitely many (uncountably many) points. Measure theory is a rigorous way to quantify "lengths" and "sizes" of geometrical objects. A line segment of length 1 would have measure 1 on the real line. Each individual point making up that line has measure 0. This would appear to give the same contradiction as your plane example... we line up infinitely many things of zero length and suddenly have something of length 1.

Measures have the property that they are countably additive, which means that if you piece together countably many objects and know their measures, then the measure of the new pieced-together object is the sum of the measures of all the pieces. If you don't know what countable means, it roughly means "you can write them all down in a list and number the list 1, 2, 3, ..."

A (perhaps surprising at first glance) fact is that the set of all rational numbers is countable. So if we placed a point at each rational number on the number line, the set of all those points would have measure zero according to countable additivity.

However, the interval of length 1 consists of uncountably many points. So countable additivity does not apply. That is, we cannot say the measure of the interval is $0+0+0+\cdots$ since it consists of uncountably many points. Similarly, we cannot say that the height of uncountably many 2-d planes stacked on top of each other is $0+0+0+\cdots$ either.

kccu
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