3 dimensional volumes of a finite size made up of 2 dimensional shapes made up of lines/curves that are infinitely thin

[Unity+]

It is probably a too long of title, but this question loomed in my mind. We all know that 3 dimensional geometric volumes have a finite size(unless told otherwise), and 2 dimensional shapes are made up of lines that are infinitely thin. However, 3 dimensional volumes such as cylinders are made up(theoretically) of 2 dimensional circles that are stacked upon each other until a given height, yet 2 dimensional shapes with lines that are infinitely thin are able to make up a certain height.

 

How is this possible? I mean the formulas for determining surface area and volume are based on these principles, though I could be wrong.

[Bill Angel]

That's calculus, which facilitates the computation of volumes from thin 2 dimensional slices.

[HalfWit]

It is probably a too long of title, but this question loomed in my mind. We all know that 3 dimensional geometric volumes have a finite size(unless told otherwise), and 2 dimensional shapes are made up of lines that are infinitely thin. However, 3 dimensional volumes such as cylinders are made up(theoretically) of 2 dimensional circles that are stacked upon each other until a given height, yet 2 dimensional shapes with lines that are infinitely thin are able to make up a certain height.

 

How is this possible? I mean the formulas for determining surface area and volume are based on these principles, though I could be wrong.

This is the same idea as a line segment of length 1 being made up of an infinity of points, each of which is of length zero. Calculus or not, it's really a philosophical mystery. I don't think anyone knows the answer.

 

I would go so far as to say that calculus and even advanced mathematics doesn't address this question at all. There is no branch of mathematics that allows us to add up an uncountable infinity of points of length zero to get a line segment of length 1. There is no answer to this riddle as far as I know.

Edited July 29, 2013 by HalfWit

[studiot]

See also peano curves.

https://www.google.co.uk/#sclient=psy-ab&q=peano+curve&oq=peano+&gs_l=hp.1.1.0l4.813.2563.0.4563.6.5.0.0.0.0.312.1124.0j2j2j1.5.0....0...1c.1.22.psy-ab..1.5.1046.bLwUJpatai0&pbx=1&bav=on.2,or.&bvm=bv.49784469,d.d2k&fp=e4b101b41d16f2c0&biw=1024&bih=559

[gabrelov]

It is probably a too long of title, but this question loomed in my mind. We all know that 3 dimensional geometric volumes have a finite size(unless told otherwise), and 2 dimensional shapes are made up of lines that are infinitely thin. However, 3 dimensional volumes such as cylinders are made up(theoretically) of 2 dimensional circles that are stacked upon each other until a given height, yet 2 dimensional shapes with lines that are infinitely thin are able to make up a certain height.

 

How is this possible? I mean the formulas for determining surface area and volume are based on these principles, though I could be wrong.

 

Mostly these formula were derived by integrating differential areas which are very small and we limit the dA into really small values such that it approaches almost zero but not equal to zero. then this minute area is then summed or integrated to create a big part and so on. This is used to determine volume of irregular shapes such as graphs given their equation.

 

Note: most integration produces errors or are not 100% accurate specially if the shape of the object is irregular and cannot be computed by cutting it to regular shapes. Errors which are neglected assuming it is small enough such that it almost approches zero in value which is not added to the actual value. This happens on such formulas on speed of sound and so on which errors are negligible that is why now we use actual apparatus and computer software to those calculations for us.

 

But actually you can compute it manually using other laws or theorems but would take you too long that is why calculus was made, imagine a box and sing a 2d plane for base calculation and adding almost all this infinitely small 2d planes if the area is "a" then we multiply by how many minute slice there is but consider this, the slice is almost zero in thickness let say .0000001 thick and add up all of it. That is very time consuming. Simple reasoning will tell you that when a 4x2x3 box is computed we just need to get the slice or the plane then multiply it by its thcikness to get the volume.

 

eg. A 4 by 3 box has a thickness of 2 so we just multiply two or if you want get small parts of the thickness and add it all up and you will get the same answer.

 

 

4x3 = 12 then the question is how many 12 areas are there in the box the answer is add all the infinitely small 12s using a given thickness such as 2 and you will get the answer.