Conference rusure::math

1.  A Sierpinski sponge.  To construct it, start with a solid cube.
    Divide each square on the faces into nine smaller squares, and drill
    a square hole through the middle square on each face.  This leaves
    eight squares on each face -- divide each of these into nine smaller
    squares, and repeat the process to the limit.
    If drilling square holes is a problem, round holes can be used instead.

2.  Prove that no shape can have a finite surface area and an infinite volume.

    For any given (finite) surface area, the shape that maximizes the
    enclosed volume is a sphere (this is a well-known result).  Thus, the
    maximum volume for a finite surface area is constrained to be finite.

When I was in High School my professor showed us a semi-strange solid. We 
were, at that time, taking one-to-one functions and rotating them around the 
x-axis to produce a solid. There was one function that had something like
an inverse logarithmic decay.... needless to say, the function would never
cross the x-axis, and when rotated around the axis produced a finite volume.

I'm not a math oriented person, so I can't be any more detailed. I just thought
I'd pass what little help this is along to others. And if I'm not mistaken
there is a class of functions that all have the same property (i.e. when 
rotated produce infinite surface area yet finite volume)

Let y(x) = 1/x be the curve.  We rotate this about the x axis to get a surface,
and a solid.

The volume of the solid (with x ranging from 1 to infinity) is given by:

     infinity	      infinity
	/		 /			|infinity
	|	 2	 |  pi dx        -pi	|
	| pi y(x)  dx =	 |  -----  =  ( ----- )	|  = 0 - (-pi/3) = pi/3
	|		 |   x^2	3 x^2	|
	/		 /		 	|1
	1		 1

The surface area (with x ranging from 1 to infinity) is less than the integral:

     infinity
	/                             |infinity
	|  2 pi dx                    |
	|  -------  =  ( 2 pi ln(x) ) |  = 2 pi ln(infinity) = infinity
	|     x                       |
	/                             |1
	1

Note that the true integral for the surface area is:

     infinity
	/
	|			   2
	|  2 pi y(x) sqrt(1 + y'(x) ) dx
	|
	/
	1

The cross-sectional area is also infinite.

					- Gilbert

Newsgroups: net.math
Path: decwrl!decvax!linus!faron!pws
Subject: Re: Strange Shapes
Posted: Tue Nov 20 13:18:01 1984

Keywords: SPOILER-SPOILER-SPOILER


In article <ihnet.176> [email protected] (K. A. Dahlke) writes:
>Here is a question which some of you have probably heard before.
>What 3-dimensional shape has a finite volume, but an infinite 
>surface area?
>You can fill it up with paint, but you can't paint it.
>I will give the answer in a week if it has not been spoiled.
>The second part is harder.
>Prove that no shape has a finite surface area 
>and an infinite volume.
>I might give the answer to this part too,
>if I get the chance to think about it during turkey (chomp chomp).
>Enjoy continuously!
>
>Karl Dahlke    ihnp4!ihnet!eklhad


as for the first part, consider the curve y = 1/x (x >= 1)
rotate this around the x-axis. the resulting shape has volume
PI, but its surface area integral:

     /~\ infinity
     |
     |
     |   (1 + 1/x^4)^.5                diverges by inspection. (always 
2*PI |  ---------------- dx            below the curve y = 1/x) 
     |         x
     |
   \_/ 1

BUT, neither you, nor i, nor the Pentagon could possibly keep 
one of these things in their bedroom. so here is something a 
bit smaller: construct a "cylinder" whose cross section is 
a snowflake curve. such a cylinder can be constructed so as 
to fit inside of a normal cylinder of arbitrarily small diameter 
and height, but STILL would have infinite surface area.

                                       happy munching,
                                             -phil

Newsgroups: net.math
Path: decwrl!decvax!genrad!wjh12!foxvax1!brunix!browngr!jfh
Subject: Re: Strange Shapes
Posted: Tue Nov 20 10:23:20 1984


SPOILER (sort of) ****************************************************

The answer to the second part is that it's incorrect: the subset of space
consisting of all points whose distance from the origin is greater
than one is an example: its volume is infinite, but its surface area
is 4 pi.

By the way, the first problem seems to be a paradox unitl you consider the
following algorithm for painting the plane: take 1 gallon of paint. Use the
first half gallon to paint all the points whose distance from the origin
is less than one. Now use the next quarter gallon to paint the points
whose distance form the origin is between one and two. Now continue in
this manner (of course the paint is much thinner as you move away
from the origin).
   Nonetheless, you see that you can 'paint' and infinite area with
a finite quantity of paint. This is just what happens when you fill
the answer to part 1 with paint...

From:	ROLL::USENET       "USENET Newsgroup Distributor" 27-NOV-1984 22:29
To:	HARE::STAN
Subj:	USENET net.math newsgroup articles

Newsgroups: net.math
Path: decwrl!decvax!harpo!whuxlm!whuxl!houxm!ihnp4!ihnet!eklhad
Subject: Re: Strange Shapes
Posted: Sun Nov 25 07:00:20 1984


Thanks for the many responses to "strange shapes".
If you were not familiar with infinite-area finite-volume shapes,
here is one example.
Take the graph  y = 1/x  and rotate it around the x axis.
Let x run from 1 to infinity, and close the shape at x = 1 with a unit disc.
This hyperbola of revolution works because volume drops as the cube of R,
and surface area drops as the square of R.
The difference is just enough to produce a finite volume
and an infinite surface area.  Try the integrals yourself.

One interesting response stated the shapes need not be unbounded.
A snowflake (fractle) kind of shape might work.
I guess I never said the shapes had to be smooth or continuous.
I am a little skeptical, and very interested.
Can anyone give an analytic definition for a bounded
shape with these properties.

As for the converse, I was a bit embarrassed by the appropriate response
"all points whose distance from the origin >= 1"
This shape indeed has infinite volume and finite surface area, but it
wasn't quite what I had in mind.
I better learn to phrase questions unambiguously.
I wanted the infinite volume to remain inside the shape.
Now define inside (aargggg).
Anyways, I didn't think much about this part, too busy eating.
I guess one could flatten non-convex portions, decreasing area and
increasing volume.
The shape must then be convex, and therefore bounded or infinite, and...
(wave wave wave, it's intuitively obvious, won't bore you with details).
-- 

Karl Dahlke    ihnp4!ihnet!eklhad

Newsgroups: net.math
Path: decwrl!decvax!genrad!wjh12!talcott!gjk
Subject: Re: Re: Strange Shapes
Posted: Sun Nov 25 14:26:33 1984


> I have heard this one before - you can't paint it, but can
> fill it with paint, pour out the paint and thus it is painted.
> My only problem is the amount of time it takes to fill it with
> a finite amount of paint.  (Ref: FTL discussion in net.physics.)
> --henry schaffer  north carolina state university

A little thought will give a better answer:

Painting an object means putting a coat of paint *of uniform thickness* on
the object.  All objects with finite volume and infinite surface area have
arbitrarily small cracks and crevices where a coat of paint "does not fit".

As someone aptly pointed out, an example of such an object is a snowflake.
Try putting a coat of paint on a snowflake.

Newsgroups: net.math
Path: decwrl!decvax!genrad!mit-eddie!godot!harvard!seismo!brl-tgr!gwyn
Subject: Re: Re: Strange Shapes
Posted: Sun Nov 25 19:22:51 1984


> Painting an object means putting a coat of paint *of uniform thickness* on
> the object.  All objects with finite volume and infinite surface area have
> arbitrarily small cracks and crevices where a coat of paint "does not fit".
> 
> As someone aptly pointed out, an example of such an object is a snowflake.
> Try putting a coat of paint on a snowflake.

"On" an object in practice means "within cohesive distance" of the object.
There are always tiny cracks in practice but they are covered by paint.
This means that it is easy to paint a snowflake; just immerse it.

And there you have the difference between a physicist and a mathematician.

This is not as facetious as it sounds.  If you admit conventional concepts
of infinity, set theory, and the like, then you support such paradoxes as
the Banach-Tarski dismantling of a sphere into a few congruent parts that
can be reassembled into a smaller sphere, and so forth.  There is more to
reality than that, since in practice no real sphere can be so rearranged.

From:	ROLL::USENET       "USENET Newsgroup Distributor"  4-DEC-1984 22:25
To:	HARE::STAN
Subj:	USENET net.math newsgroup articles

Newsgroups: net.math
Path: decwrl!sun!qubix!ios!oliveb!hplabs!hao!seismo!mcvax!ukc!qtlon!flame!ubu!dmrp
Subject: Re: Re: Strange Shapes
Posted: Fri Nov 30 05:37:35 1984


Am I being simple-minded, or is there not a surface of maximal volume with
a given finite surface area -- a sphere ? So there is no surface with finite
area containing infinite volume. Or is there some subtlety I missed ?

Newsgroups: net.math
Path: decwrl!sun!qubix!ios!oliveb!hplabs!hao!seismo!harvard!wjh12!foxvax1!brunix!browngr!jfh
Subject: Re: Re: Strange Shapes
Posted: Wed Nov 28 06:59:59 1984



   I think that the proposal "Find a two-dimensional surface with finite
surface area which separates 3-space into two regions, each of which has
infinite volume" has some merit.
  Shall we assume that "2-dimensional surface" means a smooth or polygonal
2-dimensional manifold (i.e. a set that is locally homeomorhic to the plane)?
Or is 'dimension' to be measured in some more abstract way?
  I propose that, initially at least, we restrict to sets which are 'parametric'
surfaces, i.e. are the images of nice functions from 2-space to 3-space. (e.g.
the sphere is the image of (x, y) -----> (cos x cos y, cos x sin y, sin x),
where x goes from -pi/2 to pi/2, and y goes from 0 to 2 pi).

   I suspect that in this case, the answer is that there is no such surface.

Newsgroups: net.math
Path: decwrl!sun!qubix!ios!oliveb!hplabs!hao!seismo!harvard!wjh12!foxvax1!brunix!browngr!jfh
Subject: Re: Strange Shapes
Posted: Wed Nov 28 06:53:22 1984


There is a nice picture of a bounded 'shape' with infinite volume inn
the book "Calculus on Mandifolds", by Michael Spivak, pub. Benjamin; the
shape is best described as a failure to extend the usual definitionn of
arclength (sorry about that bouncy 'n' key): one defines arclength
by subdividing with points, and then taking the polygonal curve that
passes through the points in the same order as the curve did. Then one
computes the length of the polygonal path, and takes a limit as the
"gap" between points goes to zero. If you try to do the same thing
for surface area, eg for a cylinder, by inscribing polygos within it,
it turns out that the limit need not exist: the polygons can get arbitrarily
small without the sum of their areas tending toward a constant. In fact
one can make this sum-of-areas go to infinity.
     There are more details (I think) in Spivak's Comprehensive Introduction
to Differential Geometry, Vol. 1 or 2, Publish or Perish Press.

I suspect the first sentence of the previous response should read:

"There is a nice picture of a bounded 'shape' with infinite SURFACE AREA in ..."