Is a Sphere 2-D or 3-D?

Date: 8/8/96 at 10:38:39
From: Janice Behrens
Subject: Is a Sphere 2-D or 3-D?

Dear Dr. Math: 

Is a sphere a two- or a three-dimensional object? 

Date: 9/1/96 at 17:26:49
From: Doctor Jerry
Subject: Re: Is a Sphere 2-D or 3-D?

Just to be certain we mean the same thing by sphere, a sphere is the 
set of all points in space (or R^3, which is Euclidean three space) 
at an equal distance from a fixed point.  Not a solid ball. 

Although the sphere is a subset of three-dimensional space, it is a 
two-dimensional object.  A circle can exist in either two-dimensional 
or three-dimensional space (or even higher-dimensional spaces), but it 
is a one-dimensional object.  

A line segment, which is one-dimensional, can be deformed into a 
circle.  We can think of the line as elastic.  A circular disk, which 
is two-dimensional, can be deformed elastically into a sphere. 

If you know about describing curves and surfaces parametrically, then, 
with certain restrictions, the dimension of the object is equal to the 
number of parameters required in its description.  For example, a line 
in two space can be described by the equations

     x = 2+3t
     y = 1+5t

The parameter t varies over a line or line segment, both of which are 
two-dimensional.  Hence, the line in two space is one-dimensional.

A sphere of radius a can be described by

     x = a*cos(theta)*sin(phi)
     y = a*sin(theta)*sin(phi)
     z = a*cos(phi)

The parameters are phi and theta.  So, the sphere is two-dimensional.

There are other, more complex definitions of dimension.  These can be 
found in subject called topology, particularly in something called 
dimensional theory.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 9/1/96 at 17:27:39
From: Doctor James
Subject: Re: Is a Sphere 2-D or 3-D?

That's a very tricky question, and I can see exactly where your 
confusion lies.

In one sense, we'd like to think that a sphere is a three-dimensional
object, since we can take a ruler and measure its length, width, and 
height.

But in another sense, it can't be a three-dimensional object because
it has no volume (remember, a sphere is only the "skin of a ball,"
not the inside of the ball too), just like a flat plane has no
volume.  

If you've noticed, I've carefully avoided answering your question. :>.
I can give you a technical answer that might not mean that much to 
you, but I'll try to explain it. A sphere is what is called a 
2-manifold. A 2-manifold is a type of mathematical object, like a 
sphere, that looks like a plane if you zoom in far enough on it.
Some other manifolds are a plane, the surface of a donut (also called
a torus), and a 2-sheeted hyperboloid.  A cone is NOT a 2-manifold,
because it has a pointy part and no matter how much you zoom in on
that point, the point won't start to look like a regular plane.

There are 1-manifolds too, and those look just like lines when you
zoom in on them.  Some examples of 1-manifolds are lines, circles,
knotted circles, and lines that squirm around a bit in space.  Two
intersecting lines do NOT make a 1-manifold, because you can't zoom
in on the intersection point and make it look like part of a regular
line.

Mathematicians like to be formal about defining things sometimes,
and so we define a 2-manifold as an object that is "locally
diffeomorphic to a plane."  This is a formal way of saying that
every piece of a 2-manifold looks like a patch of a plane if you
zoom in far enough.

This kind of math is part of a subject called 'real analysis,' which
is great fun.


-Drs. James and Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 9/1/96 at 17:28:6
From: Doctor James
Subject: Re: Is a Sphere 2-D or 3-D?

I thought about it some more, and thought I didn't do that good a job
explaining what a manifold was to you. Here's another try:

Take a part of a plane (which is two-dimensional, right?). Cut out a
part. Now, this plane is actually made of rubber. So you can pull it
and stretch it and squish it and curve it and do most anything to it.
(but you pop it if you make a sharp point or edge, so you can't do
that - a type of mathematician called a 'topologist' loves to do
this.)  Now, you can twist it around some and get part of a sphere,
right? if you put this sphere in three-dimensional space, that means
you have a 2-manifold in 3-space. 3-space just means three-
dimensional space.  (mathematicians like to sound cool by saying
3-space.)

Similarly, take a long line made of rubber (very thin rubber!). If you
stretch it and curl it and put it in a plane, we call it a 1-manifold
in 2-space. If we put it in a three-dimensional space, we call it a
1-manifold in 3-space.

Now for the really mind-boggling part. Take a portion of three space
(your room, for example), and twist it around and stretch it. Put it
in four-dimensional space. That's what's called a 3-manifold in 
4-space. There's no real way to picture this, which is why 
mathematicians tend to rely on equations, not just on pictures!


-Dr. James,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/