Is a Sheet of Paper a 2D or 3D Object?

Date: 05/17/2008 at 06:44:45
From: Peter
Subject: is a piece of paper 3D or 2D?

Dear Dr. Math:

I am a Taiwanese fourth grade student. In math class, our teacher said
that a piece of paper is a 2D item and it has no height, but some of
my classmates and I disagree and got punished because we were
"disturbing the class¨. We think a piece of paper still has a height,
even if it is very, very, very, small, like 0.00000000000001cm. That
is my question: can a thing that has very small height like paper be
seen as a 2D item? And can a 2D item exist in a 3D space?
                                               
I tried to explain it with a "3D" germ with a height much smaller than
a piece of paper, but if a piece of paper is 2D, the germ will be,
too. I failed to find anything with a height smaller than paper that
is 3D for sure. Can you also give me an example of small 3D things so
I can explain to my teacher? Thank you.

Date: 05/17/2008 at 12:19:12
From: Doctor Ian
Subject: Re: is a piece of paper 3D or 2D?

Hi Peter,

Well, if you really want to get into it, according to Einstein things
exist in four-dimensional space-time, so maybe a sheet of paper isn't
really 3D, either.  :^D

And string theorists think there may be 11 dimensions, or possibly more.  

And you might want to look into fractals, which use fractional
dimensions to describe things like mountains and sea coasts. 

So here's the thing:  Objects are whatever they are.  (And sometimes
we talk about "objects" that might not be objects at all--is a
mountain or a sea coast properly an "object", with a unique identity?
Or are they just convenient abstractions?)

What we do is create mathematical spaces, with particular numbers of
dimensions (which don't have to be whole numbers), and then we look
around in the world for things that can be described using those spaces. 

But dimensions are properties of the MATHEMATICAL SPACES, and NOT of
the PHYSICAL OBJECTS.

There's a joke I like because it helps me keep this important
distinction in mind.  One man pulls a photo out of his wallet and
shows it to another man, saying "This is my family, my wife and kids."
And the other man says, "Wow, are they really that small and flat?"

Of course, they're not--the photo isn't his family, it's  
REPRESENTATION of his family, and the properties of the representation
are necessarily different from the properties of the things being
represented.  

Similarly, things like planes and spheres are REPRESENTATIONS of
objects in the world, and the properties of those representations are
not the same as the properties of the objects. 

In some contexts (for example, if we're trying to figure out how much
writing we can fit on a page), we're just interested in the surfaces
of pieces of paper, so we treat them as if they're two-dimensional.  

In other contexts (for example, if we're trying to figure out how much
room a bunch of paper will take up in a warehouse), we're interested
in the volumes of pieces of paper, so we treat them as if they're
three-dimensional.  

If we're trying to describe what happens to a piece of paper traveling
at speeds near the speed of light, we treat it as if it's
four-dimensional.  

If we're trying to calculate the gravitational attraction that a piece
of paper exerts on some other object, we might find it convenient to
treat it as a zero-dimensional point. 

But a piece of paper isn't any of these things, inherently.  It's
whatever it is, and we don't really KNOW what it is.  We just have a
lot of mathematical models that are "good enough" for various
situations.  

Does this make sense? 

All of which is to say:  Mathematically, a 2D object can exist in a 3D
space--a plane is just such an object.  But physically, anything made
of atoms must have the same number of dimensions that atoms have--
whether that's three, or four, or eleven, or whatever.  The thing is,
we don't know what that number of dimensions is... or even whether
atoms are a valid description of physical reality.  

The most math can tell you is:

  IF a piece of paper is made of atoms, 
  AND IF atoms have three dimensions, 
  THEN a piece of paper has three dimensions.

That is, if the premises are true, so is the conclusion.  But we don't
know if the premises are true.  :^D

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/