🚀 go-pugleaf

In article <5cmpbo$nds@agate.berkeley.edu>,
Louis Hom <lhom@nature.berkeley.edu> wrote:
>So, I'm trying to collect posters' favorite Things in Math & Science.

    I always thought "DAMN, THAT'S COOL!" about things that there are more
    than one but finitely many of:
	* regular polyhedrons
	* digitally invariant pluperfect numbers
	* regular polygons that tile the plane
    
    I always thought "DAMN, THAT'S COOL!" about things that feel abitrarily
    assigned or limited:
	* the four-color map theorem
	* knots can only exist in 3-D (I believe)
	* Vertices - Edges + Faces = 2
	* exp( pi * i ) = -1

    For all of these, both the *fact that they are true* and the *reason
    that they are true* are in the "DAMN, THAT'S COOL!" category.

    Other things that are way in the "DAMN, THAT'S COOL!" category are
    homomorphism, Conway surreal numbers, and the infinite hotel.

    alter,
    pat
-- 
I live in another Dimension, But I Have a Summer Home in Reality

>>>>> "JY" == Jeffery Young <jsy@media04.cray.com> writes:

 JY> In article <01bc0db2$216f18c0$b8ad11cf@lucasea-home>, "Eric Lucas"
 JY> <lucas@superlink.net> writes:

 >> Louis Hom <lhom@nature.berkeley.edu> wrote in article
 >> <5cmpbo$nds@agate.berkeley.edu>...
 >> > For example, one of my favorite things is the solid that you get when
 >> > you rotate y=1/(log x) about the x-axis -- it has finite volume but an
 >> > infinite surface.  So if you had a container that shape, you could fill
 >> > it up with some finite quantity of paint, but it would never be enough
 >> > to actually paint the surface of the thing.

 JY> But if you could fill it up with a finite amount of paint, wouldn't the
 JY> "interior surface area" be painted.  And isn't the "interior" SA equal
 JY> to the "exterior" SA?

 JY> I know the math, but the physical explanation doesn't sit with me.

It depends how you paint your surface. If you allow the paint to have any
arbitrarily small thickness, you need a finite amount of paint to cover the
surface (miracle paint!)

But I think that the original poster assumed the thickness of the paint had
to be a constant on the external surface.

Eric Daniel

Jeffery Young wrote:

> I know the math, but the physical explanation doesn't sit with me.

The physical explanation is the distinction between mathematical surface
area and volume and realistic surface area and volume, when it comes to
painting a surface or filling a volume with paint.

-- 
        Erik Max Francis, &tSftDotIotE / email:  max@alcyone.com
                      Alcyone Systems /   web:  http://www.alcyone.com/max/
 San Jose, California, United States /  icbm:  37 20 07 N  121 53 38 W
                                    \
               "I am become death, / destroyer of worlds."
                                  / J. Robert Oppenheimer (quoting legend)

Louis Hom <lhom@nature.berkeley.edu> wrote in article
<5cmpbo$nds@agate.berkeley.edu>...
> 	For example, one of my favorite things is the solid that you get
> when you rotate y=1/(log x) about the x-axis -- it has finite volume but
an
> infinite surface.  So if you had a container that shape, you could fill
it
> up with some finite quantity of paint, but it would never be enough to
> actually paint the surface of the thing.

Pretty cool.  Simple (obvious) explanation:  as the two branches approach
the asymtote, the distance between them gets infinitely small, leaving
infinitely little space for the layer of paint.  Now, if you could come up
with a paint that covers in an infinitely thin layer, you'd really have
something.

	Eric Lucas

In article <01bc0db2$216f18c0$b8ad11cf@lucasea-home> "Eric Lucas" <lucas@superlink.net> writes:
>From: "Eric Lucas" <lucas@superlink.net>
>Subject: Re: DAMN, THAT'S COOL! (Help a fellow geek)
>Date: 29 Jan 1997 07:01:09 GMT

>Louis Hom <lhom@nature.berkeley.edu> wrote in article
><5cmpbo$nds@agate.berkeley.edu>...
>>       For example, one of my favorite things is the solid that you get
>> when you rotate y=1/(log x) about the x-axis -- it has finite volume but
>an
>> infinite surface.  So if you had a container that shape, you could fill
>it
>> up with some finite quantity of paint, but it would never be enough to
>> actually paint the surface of the thing.

>Pretty cool.  Simple (obvious) explanation:  as the two branches approach
>the asymtote, the distance between them gets infinitely small, leaving
>infinitely little space for the layer of paint.  Now, if you could come up
>with a paint that covers in an infinitely thin layer, you'd really have
>something.

But this result doesn't depend on the dimensions of atoms or anything,  It is 
one of those weird things that calculus does, when you calculate surfaces and 
volumes of revolution.  Even your infinitely thin paint wouldn't cover it.

The lesson is:  If you are a painter, be careful of what you bid on.

Bill


********************************************************
Bill Penrose, President, Custom Sensor Solutions, Inc.
526 West Franklin Avenue, Naperville, IL 60540
630-548-3548, fax: 630-369-9618
email wpenrose@interaccess.com
********************************************************
Applications of gas sensors:  Contract R&D, product
development, and consultation.
********************************************************

In article <wpenrose.354.00550F9D@interaccess.com>,
William R. Penrose <wpenrose@interaccess.com> wrote:
>
>But this result doesn't depend on the dimensions of atoms or anything,  It is 
>one of those weird things that calculus does, when you calculate surfaces and 
>volumes of revolution.  Even your infinitely thin paint wouldn't cover it.
>
>The lesson is:  If you are a painter, be careful of what you bid on.

Hmmm, my real analysis is years old, and I didn't do very well on it
(certainly not at the sci.math level this is crossposted to), but doesn't 
it matter what you mean by "paint"?  

I can take a sphere, i.e., finite surface area, and do a mapping of it
onto a plane.  There's a point missing on the sphere, which would make
the surface unbound; I think you have to be careful with that.  Similarly,
I'd speculate you can map the volume of revolution onto the surface;
is that "painting"?

More interesting artifacts are, say, Peano curves (?): I can take a bound
interval on the number line, say [0,1], and map it to a bound area, say 
a square, such that my curve passes through every point of the square.
This curve, BTW, is everywhere continuous but nowhere differentiable.

Louis Hom wrote:
> 
> So, I'm trying to collect posters' favorite Things in Math & Science.
>         For example, one of my favorite things is the solid that you get
> when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
> infinite surface.  So if you had a container that shape, you could fill it
> up with some finite quantity of paint, but it would never be enough to
> actually paint the surface of the thing.

It would take a while to pump the paint from the top to the bottom of
the container, yes?  Hard to store, too.  And setting the container
upright or tipping it to pour is a little problematical, eh?  y=sin(1/x)
will give you nice accordian pleats in case you need a little container
expansion to hold a little more paint than you anticipated.

>         My interest in starting this collection grew out of a desire to
> write something about the Math/Science Aesthetic and realizing that I
> didn't have a lot of interesting examples readily at hand.  I don't know if
> I will ever write such an essay, but I think it would be fun to gather some
> examples at least.  Maybe I'll archive them on my web page or something.

Bullvalene (as in "bullshit' - there was some scepticism voiced pior to
its remarkable synthesis from cyclooctatetraene), the molecule without a
discrete structure (at least when warm), is the archetype of fluctional
degenerate structures.  Kewl.  Then came barbaralone (named after a
mistress) that does it even better.  Even trimethylsilyl ethers of
1,3-diketones have degenerate cis-enol ethers - flip, flop.

And let us not forget anholonomy in physics (Hannay angle, Berry
phase).  You can get here from here in the simple case, but you must go
around twice to end at the start.

-- 
Alan "Uncle Al" Schwartz
UncleAl0@ix.netcom.com ("zero" before @)
http://www.ultra.net.au/~wisby/uncleal.htm
 (Toxic URL! Unsafe for children, Democrats, and most mammals)
"Quis custodiet ipsos custodes?"  The Net!

In article <5cmpbo$nds@agate.berkeley.edu>,
Louis Hom <lhom@nature.berkeley.edu> wrote:
>So, I'm trying to collect posters' favorite Things in Math & Science.
>	For example, one of my favorite things is the solid that you get
>when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
>infinite surface.  So if you had a container that shape, you could fill it
>up with some finite quantity of paint, but it would never be enough to
>actually paint the surface of the thing.

	Or, a cylindrical can whose base is the Koch snowflake would
do the trick, and would actually fit on a shelf.

	I think it would be more mind-boggling to point out that
the can holds a finite amount of paint, but it takes an infinite amount
of paint to paint THE INSIDE.  Aren't you technically painting the
inside when you fill it up with that finite amount of paint???

	Of course, this problem stems from the abstraction that the 
layer of paint you lay down on a surface is infinitely thin, which 
isn't true in reality, and serves as an amusing reminder that areas
and volumes are not comparable mathematically the way they are in 
the real world.

 ,oooooooo8     o     ooooo@math.niu.edu  --  http://www.math.niu.edu/~caj/
o888'   `88   ,888.    888                                                 
888          ,8'`88.   888   "What he can do with numbers would make a
888o.   ,oo ,8oooo88.  888    thousand-dollar-a-night hooker blush like
`888oooo88 o88o  o888o 888    a nun."             
____________________8o888'______________-Hank Jennings, _Twin_Peaks._ _____

Xcott Craver (caj@holmes.math.niu.edu) wrote:

: 	Or, a cylindrical can whose base is the Koch snowflake would
: do the trick, and would actually fit on a shelf.
 Just curious, but by _one_of_ the definitions I have for a fractal
(i.e. fractional dimension, it has infinite length, but zero area)
wouldn't any fractal base accomplish the same (a,e)ffect?  

: 	I think it would be more mind-boggling to point out that
: the can holds a finite amount of paint, but it takes an infinite amount
: of paint to paint THE INSIDE.  Aren't you technically painting the
: inside when you fill it up with that finite amount of paint???

: 	Of course, this problem stems from the abstraction that the 
: layer of paint you lay down on a surface is infinitely thin, which 
: isn't true in reality, and serves as an amusing reminder that areas
: and volumes are not comparable mathematically the way they are in 
: the real world.
 Yeah, I was just about to bring up the point you brought up above.
 But alas, you beat me to the point.

:  ,oooooooo8     o     ooooo@math.niu.edu  --  http://www.math.niu.edu/~caj/
: o888'   `88   ,888.    888                                                 
: 888          ,8'`88.   888   "What he can do with numbers would make a
: 888o.   ,oo ,8oooo88.  888    thousand-dollar-a-night hooker blush like
: `888oooo88 o88o  o888o 888    a nun."             
: ____________________8o888'______________-Hank Jennings, _Twin_Peaks._ _____

In article <01bc0db2$216f18c0$b8ad11cf@lucasea-home>, "Eric Lucas" <lucas@superlink.net> writes:
> Louis Hom <lhom@nature.berkeley.edu> wrote in article
> <5cmpbo$nds@agate.berkeley.edu>...
> > 	For example, one of my favorite things is the solid that you get
> > when you rotate y=1/(log x) about the x-axis -- it has finite volume but
> an
> > infinite surface.  So if you had a container that shape, you could fill
> it
> > up with some finite quantity of paint, but it would never be enough to
> > actually paint the surface of the thing.
> 
> Pretty cool.  Simple (obvious) explanation:  as the two branches approach
> the asymtote, the distance between them gets infinitely small, leaving
> infinitely little space for the layer of paint.  Now, if you could come up
> with a paint that covers in an infinitely thin layer, you'd really have
> something.
> 
> 	Eric Lucas


But if you could fill it up with a finite amount of paint, wouldn't the
"interior surface area" be painted.  And isn't the "interior" SA equal
to the "exterior" SA?

I know the math, but the physical explanation doesn't sit with me.

Thanks,
Jeff

In article <1997Jan29.215107.11542@news.wrc.xerox.com>, fleck@dirk (Pat
Fleckenstein (x58885)) wrote:

> In article <5cmpbo$nds@agate.berkeley.edu>,
> Louis Hom <lhom@nature.berkeley.edu> wrote:
> >So, I'm trying to collect posters' favorite Things in Math & Science.
> 
structural colors as in the blue of a blue jay's feathers

ecotenes

the structure of BrF5

the inductive effect

aldol condensations

the supressing effect the signal from one ear has on the signal from the
other ear.

bats catching food on the fly in their wing or tail and bending in flight
to bring the food to the mouth. Damn, those movies are beautiful!!

the saddle joint of the thumb

electromagnetic induction

the loop of Henle, Purkinje fibers and the far-flung isles of Langerhans

John L. Park
ChemTeam@clubnet.net
http://dbhs.wvusd.k12.ca.us/ChemTeamIndex.html
----------------------------------------------------------------------
Selected "Top 500 of the Web" by HomePC 
Magazine, December 1996
----------------------------------------------------------------------
Currently recommending: "Beethoven Violin Sonatas: 
Nos. 5 (Spring), 8, 9 (Kreutzer). Artur Rubinstein and 
Henryk Szeryng, RCA Victor Gold Seal 09026-61861-2

In article <1997Jan29.215107.11542@news.wrc.xerox.com>,
	fleck@dirk (Pat Fleckenstein (x58885)) writes:
>In article <5cmpbo$nds@agate.berkeley.edu>,
>Louis Hom <lhom@nature.berkeley.edu> wrote:
>>So, I'm trying to collect posters' favorite Things in Math & Science.

Bucky Balls.  Cool.

============================================
Gardner Buchanan    <gbuchanan@sympatico.ca>
Ottawa, ON

Erik Max Francis wrote:
> 
> Jeffery Young wrote:
> 
> > I know the math, but the physical explanation doesn't sit with me.
> 
> The physical explanation is the distinction between mathematical surface
> area and volume and realistic surface area and volume, when it comes to
> painting a surface or filling a volume with paint.
> 
The problem comes when you take into account the finite size of the
paint molecules. Only a finite amount of the surface is painted and an
infinite amount is left unpainted. The actual volume of paint required
is slightly less than the total volume of the shape as well.
A bad ascii picure hopefully shows this.
\           /
 \         /
 |         |
 \        /
  |       | This volume is filled and the surface 'painted'
  \      /
   |  _  | 
    \/ \/
    || || This paint 'molecule' cannot get any further down.
    \\_//
    |   |
    |   |This volume cannot be filled nor the surface covered.
    \   /
Malcolm.

> Louis Hom <lhom@nature.berkeley.edu> wrote:
> >So, I'm trying to collect posters' favorite Things in Math & Science.
> 
>     I always thought "DAMN, THAT'S COOL!" about the fact that cosmology and particle physics are really the same thing...

On Thu, 30 Jan 1997 taguebw@wfu.edu wrote:

> > Louis Hom <lhom@nature.berkeley.edu> wrote:
> > >So, I'm trying to collect posters' favorite Things in Math & Science.
> > 
> >     I always thought "DAMN, THAT'S COOL!" about the fact that cosmology and particle physics are really the same thing...
> 
I like the way that you can start with the time independent Schroedinger
equation, and work out the energy levels in hydrogen, explaining all those
lines which you saw in chemistry years before.


Anthony Potts

CERN, Geneva

In article <32EF82AE.58F8@ix.netcom.com>,
\"Uncle Al\" Schwartz <UncleAl0@ix.netcom.com> wrote:
>Louis Hom wrote:
>> 
>> So, I'm trying to collect posters' favorite Things in Math & Science.
>>         For example, one of my favorite things is the solid that you get
>> when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
>> infinite surface.  So if you had a container that shape, you could fill it
>> up with some finite quantity of paint, but it would never be enough to
>> actually paint the surface of the thing.
>
>It would take a while to pump the paint from the top to the bottom of
>the container, yes?
[snip]

Of course, you cannot completely fill this container by pouring into it,
since the speed of light constrains how quickly the paint can expand outward
at the bottom/side of the container.  The container can hold enough paint to
paint the entire surface (given the standard abstract paint), but the act of
painting it would take forever.

Jim
-----
-- 
Jim Oly <jamesoly@uiuc.edu>
"The universe doesn't give you any points for doing things that are easy."
     -- Captain John Sheridan, Babylon 5

William R. Penrose (wpenrose@interaccess.com) wrote:
: In article <01bc0db2$216f18c0$b8ad11cf@lucasea-home> "Eric Lucas" <lucas@superlink.net> writes:
: >From: "Eric Lucas" <lucas@superlink.net>
: >Subject: Re: DAMN, THAT'S COOL! (Help a fellow geek)
: >Date: 29 Jan 1997 07:01:09 GMT

: >Louis Hom <lhom@nature.berkeley.edu> wrote in article
: ><5cmpbo$nds@agate.berkeley.edu>...
: >>       For example, one of my favorite things is the solid that you get
: >> when you rotate y=1/(log x) about the x-axis -- it has finite volume but
: >an
: >> infinite surface.  So if you had a container that shape, you could fill
: >it
: >> up with some finite quantity of paint, but it would never be enough to
: >> actually paint the surface of the thing.

: >Pretty cool.  Simple (obvious) explanation:  as the two branches approach
: >the asymtote, the distance between them gets infinitely small, leaving
: >infinitely little space for the layer of paint.  Now, if you could come up
: >with a paint that covers in an infinitely thin layer, you'd really have
: >something.

: But this result doesn't depend on the dimensions of atoms or anything,  It is 
: one of those weird things that calculus does, when you calculate surfaces and 
: volumes of revolution.  Even your infinitely thin paint wouldn't cover it.

	Sure it would.  'Infinitely thin paint' can cover an infinite 
surface area.  Whenever you run out, just scrape what you've painted so 
it's half as thin, thus doubling your supply.
	The inside of the shape described would _already_ be painted if 
full of paint.. by definition.

--
Nathaniel Tagg		Physics grad student		University of Guelph
	"The chances of a neutrino actually hitting something as it 
travels through all this howling emptiness are roughly comparable to that 
of dropping a ball bearing at random from a cruising 747 and hitting, 
say, an egg sandwich."		-- Douglas Adams, _Mostly_Harmless_

In article <5cp349$poq$1@news1.sympatico.ca>, gbuchanan@sympatico.ca wrote:
> >>So, I'm trying to collect posters' favorite Things in Math & Science.
> 
> Bucky Balls.  Cool.
> 
crown ethers too.

John L. Park
ChemTeam@clubnet.net
http://dbhs.wvusd.k12.ca.us/ChemTeamIndex.html
----------------------------------------------------------------------
Selected "Top 500 of the Web" by HomePC 
Magazine, December 1996
----------------------------------------------------------------------
Currently recommending: "Beethoven Violin Sonatas: 
Nos. 5 (Spring), 8, 9 (Kreutzer). Artur Rubinstein and 
Henryk Szeryng, RCA Victor Gold Seal 09026-61861-2

Gardner Buchanan (gbuchanan@localhost.on.sympatico.ca) wrote:

: Bucky Balls.  Cool.

No, trite.

ChemTeam@clubnet.net expostulated:

>In article <1997Jan29.215107.11542@news.wrc.xerox.com>, fleck@dirk (Pat
>Fleckenstein (x58885)) wrote:



>the loop of Henle, Purkinje fibers and the far-flung isles of Langerhans

Went there for my holidays once. Beautiful spot.
Not to be confused with the "Eyelets of Langerhans" which is the technical
name for the holes your shoelaces pass through.

--R.

lhom@nature.berkeley.edu (Louis Hom) writes:
>
>So, I'm trying to collect posters' favorite Things in Math & Science.

 My favorite bit of elementary math is the relationship between the 
 trig functions and exp(ix).  The special case of e^{i\pi} gets all 
 the fame and fortune, but students are much more intrigued by the 
 more general result when they first find out how to use it to 
 solve differential equations easily. 
 
 My favorite physics factoid is the volume I call the "Hubble-barn". 
 This has a cross-sectional area of 1 barn (used to quantify reaction 
 rates in nuclear and particle physics, SI value = 100 fm^2) and a 
 length of the inverse hubble constant (c/H; with H = 75 km/s/Mpc 
 this is  about 120 Ym).  The volume is about 12 liters for that 
 value of H, which is around the middle of the range spanned by 
 various controversial measurements, IIRC.  For that value, a straw 
 with the cross-sectional area of the Ba-135 nucleus that extends 
 to the edge of the universe has the volume of about pi US gallons, 
 the universere in about 10 six packs of beer.  This provides an 
 interesting way to compare different values of Hubbles constant, 
 since 55 ==> about 17 liters, while 100 ==> about 9 liters. 

 The more obscure "barn-Megaparsec" = 3 cm^3 = about 2/3 teaspoon 
 but has the advantage of being a well-defined constant.  

-- 
 James A. Carr   <jac@scri.fsu.edu>     |  "They invented band!"  
    http://www.scri.fsu.edu/~jac/       |  Spectator at 1997 inaugural parade
 Supercomputer Computations Res. Inst.  |  commenting on the Macarena done by
 Florida State, Tallahassee FL 32306    |  the Florida A&M Marching 100  

Right.  You have to define what you mean by "painting".  As I see it, it's
covering a surface with a uniform layer of paint a certain thickness. 
(Somehow you have to convert surface area to volume, and the only way to do
so is to define how thick the layer of paint is.)  If the paint is
infinitely thin, then the volume of paint used is SA*T (surface area *
thickness).  If SA is infinte, then the thickness must be 1/infinity, where
this infinity is of a greater degree than the infinity of the surface area.
 In other words, if the paint is sufficiently infinitely thin, you *can*
paint the surface.

There is a problem with the argument "if you can fill it, you can paint the
inner surface, and therefore you can paint the outer surface":  As you
approach the asymptote at infinity, the layer of paint becomes thinner and
thinner to an arbitrary degree, since it is being squeezed between to
curves that become arbitrarily close together.  Yes, if you are allowed to
paint the outer surface with an infintely thin layer, you can cover it with
a finite volume.  However, this is contrary to the definition of painting,
which generally refers to coverage with a finitely thin layer.  Therefore,
in the same sense that you can't paint the solid, you also can't really
fill it.

	Eric Lucas

Erik Max Francis <max@alcyone.com> wrote in article
<32F043B4.3CC3DD9D@alcyone.com>...
> Jeffery Young wrote:
> 
> > I know the math, but the physical explanation doesn't sit with me.
> 
> The physical explanation is the distinction between mathematical surface
> area and volume and realistic surface area and volume, when it comes to
> painting a surface or filling a volume with paint.
> 
> -- 
>         Erik Max Francis, &tSftDotIotE / email:  max@alcyone.com
>                       Alcyone Systems /   web: 
http://www.alcyone.com/max/
>  San Jose, California, United States /  icbm:  37 20 07 N  121 53 38 W
>                                     \
>                "I am become death, / destroyer of worlds."
>                                   / J. Robert Oppenheimer (quoting
legend)
> 

fleck@dirk (Pat Fleckenstein (x58885)) writes:

>     I always thought "DAMN, THAT'S COOL!" about things that feel abitrarily
>     assigned or limited:
> 	* the four-color map theorem

> 
>     For all of these, both the *fact that they are true* and the *reason
>     that they are true* are in the "DAMN, THAT'S COOL!" category.

I hope you don't include the proof for the four-colour map theorem
in your DTC category. It isn't.

 - Sten Thaning (who soon is going to figure out how to get this program
to handle .sigs)

Louis Hom wrote:
> 
> So, I'm trying to collect posters' favorite Things in Math & Science.
>         For example, one of my favorite things is the solid that you get
> when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
> infinite surface.  

Yep.

> So if you had a container that shape, you could fill it
> up with some finite quantity of paint, but it would never be enough to
> actually paint the surface of the thing.

Whoa!  If I have a finite quantity of "mathematical paint" which
has zero thickness, you're saying it will only cover a FINITE area??
Please provide a proof of this rather remarkable statement.

If, on the other hand, you're talking about "real" paint, then
show me a PHYSICAL surface with the properties you mention which
cannot be covered with Sears Latex (such an argument gets messy, 
not only from all the spilled paint, but because you have to make
certain assumptions about the molecular dynamics of paint, the 
notion of packing fraction of dissimilar molecules and aggregates,
etc. etc. ).  

I'd love to see a proof of this assertion using either "mathematical
paint" or (hehehe) Sears Latex. I can't wait to see it.

> My interest in starting this collection grew out of a desire to
> write something about the Math/Science Aesthetic and realizing that I
> didn't have a lot of interesting examples readily at hand.

Of course, there's a  infinite number of surfaces with the "property"
you mention.  Rotate y = -exp(-ax*x) (x over all space) about the 
y-axis and fill it with paint (Sears or Mathematical).  Vary the 
"a" from (0,inf) to get your infinite number of surfaces.

> I don't know if  I will ever write such an essay, but I think 
> it would be fun to gather some examples at least.  Maybe I'll archive 
>them on my web page or something. 
>         Anyway, thanks for reading, and thanks for helping out.
> --
> _______________________________________________________________________________
> Lou Hom >K '93
> lhom@nature.berkeley.edu
> http://www.ocf.berkeley.edu/~lhom

Larks

William R. Penrose wrote:
> 
> In article <01bc0db2$216f18c0$b8ad11cf@lucasea-home> "Eric Lucas" <lucas@superlink.net> writes:
> >From: "Eric Lucas" <lucas@superlink.net>
> >Subject: Re: DAMN, THAT'S COOL! (Help a fellow geek)
> >Date: 29 Jan 1997 07:01:09 GMT
> 
> >Louis Hom <lhom@nature.berkeley.edu> wrote in article
> ><5cmpbo$nds@agate.berkeley.edu>...
> >>       For example, one of my favorite things is the solid that you get
> >> when you rotate y=1/(log x) about the x-axis -- it has finite volume but
> >an
> >> infinite surface.  So if you had a container that shape, you could fill
> >it
> >> up with some finite quantity of paint, but it would never be enough to
> >> actually paint the surface of the thing.
> 
> >Pretty cool.  Simple (obvious) explanation:  as the two branches approach
> >the asymtote, the distance between them gets infinitely small, leaving
> >infinitely little space for the layer of paint.  Now, if you could come up
> >with a paint that covers in an infinitely thin layer, you'd really have
> >something.
> 
> But this result doesn't depend on the dimensions of atoms or anything,

Odd that, according to this, the thickness of a layer of paint doesn't 
depend upon the dimensions of the "atoms" (molecules) forming the paint. 
Pretty clearly shows that there's something wrong with your assumptions.

> It is one of those weird things that calculus does, when you calculate
> surfaces and volumes of revolution.  

In my experience, calculus does no weird things.  However, there are
plenty of weird explanations offered for things people don't fathom
about calculus.

> Even your infinitely thin paint wouldn't cover it.

I can't wait to see the proof of this.  When can I expect it?
 
> The lesson is:  If you are a painter, be careful of what you bid on.

Very sage advice.  Anyone with painting experience can attest to this.

> 
> Bill
> 
> ********************************************************
> Bill Penrose, President, Custom Sensor Solutions, Inc.
> 526 West Franklin Avenue, Naperville, IL 60540
> 630-548-3548, fax: 630-369-9618
> email wpenrose@interaccess.com
> ********************************************************
> Applications of gas sensors:  Contract R&D, product
> development, and consultation.
> ********************************************************


Ron Welch
President of Irrelevant Things
Lincoln Park, NJ

Xcott Craver wrote:
> 
> In article <5cmpbo$nds@agate.berkeley.edu>,
> Louis Hom <lhom@nature.berkeley.edu> wrote:
> >So, I'm trying to collect posters' favorite Things in Math & Science.
> >       For example, one of my favorite things is the solid that you get
> >when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
> >infinite surface.  So if you had a container that shape, you could fill it
> >up with some finite quantity of paint, but it would never be enough to
> >actually paint the surface of the thing.
> 
>         Or, a cylindrical can whose base is the Koch snowflake would
> do the trick, and would actually fit on a shelf.
> 
>         I think it would be more mind-boggling to point out that
> the can holds a finite amount of paint, but it takes an infinite amount
> of paint to paint THE INSIDE.  Aren't you technically painting the
> inside when you fill it up with that finite amount of paint???
> 
>         Of course, this problem stems from the abstraction that the
> layer of paint you lay down on a surface is infinitely thin, which
> isn't true in reality, and serves as an amusing reminder that areas
> and volumes are not comparable mathematically the way they are in
> the real world.

Exactly.  The problem stems from extending mathematics to the physical
world while retaining multiple assumptions which apply only in 
the abstract mathematical sense.  If I may get philosophical for a 
sentence or two, that's where mathematics and physics part company.  
In physics, a result must make sense; in mathematics, it must follow
from mathematical manipulation and, if it makes no sense, it is 
added to the list of examples of things deserving of a Nobel prize,
if only math had one.

> 
>  ,oooooooo8     o     ooooo@math.niu.edu  --  http://www.math.niu.edu/~caj/
> o888'   `88   ,888.    888
> 888          ,8'`88.   888   "What he can do with numbers would make a
> 888o.   ,oo ,8oooo88.  888    thousand-dollar-a-night hooker blush like
> `888oooo88 o88o  o888o 888    a nun."
> ____________________8o888'______________-Hank Jennings, _Twin_Peaks._ _____


Larks  -- Wondering what a night with a $1000/night hooker would
           entail.

In article <t1d8um33fr.fsf@lugn.ling.uu.se>, Sten Thaning wrote:
>
> - Sten Thaning (who soon is going to figure out how to get this program
>to handle .sigs)

alias emacs=slrn

-- 

Joe Bramblett
bofh@joeb.psyberlink.net
airnews.manager@airmail.net

lhom@nature.berkeley.edu (Louis Hom) wrote:
> So, I'm trying to collect posters' favorite Things in Math & Science.
>         For example, one of my favorite things is the solid that you get
> when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
> infinite surface.  So if you had a container that shape, you could fill it
> up with some finite quantity of paint, but it would never be enough to
> actually paint the surface of the thing.
>         My interest in starting this collection grew out of a desire to
> write something about the Math/Science Aesthetic and realizing that I
> didn't have a lot of interesting examples readily at hand.  I don't know if
> I will ever write such an essay, but I think it would be fun to gather some
> examples at least.  Maybe I'll archive them on my web page or something.
>         Anyway, thanks for reading, and thanks for helping out. 
> -- 
Well, after reading the posts in this thread I think I might be thinking
on different lines.

Cesium.  The wavelength associated with one of its isotopes is divisable
by 60, making it ideal for timing accurate clocks.  I think that's cool,
also, it has the largest stable electron cloud, making it the largest
atom, this is cool when you think the heaviest atoms are nearly twice its
weight.  And of course, the clincher is that it can be made into a really
bad chemistry "joke" such as: What do you say when someone steals your
gold?  Ay you, cesium that robber, he stole my gold.

Keep your stick on the ice,
Mike Schneider     http://www.tc.umn.edu/nlhome/g396/schne042/

In article <32F190F4.8D2@worldnet.att.net> Larks <Larks@worldnet.att.net> writes:

>> 
>> So, I'm trying to collect posters' favorite Things in Math & Science.
>>         For example, one of my favorite things is the solid that you get
>> when you rotate y=1/(log x) about the x-axis -- it has finite volume but an
>> infinite surface.  

>Yep.

>> So if you had a container that shape, you could fill it
>> up with some finite quantity of paint, but it would never be enough to
>> actually paint the surface of the thing.

>Whoa!  If I have a finite quantity of "mathematical paint" which
>has zero thickness, you're saying it will only cover a FINITE area??
>Please provide a proof of this rather remarkable statement.

>If, on the other hand, you're talking about "real" paint, then
>show me a PHYSICAL surface with the properties you mention which
>cannot be covered with Sears Latex (such an argument gets messy, 
>not only from all the spilled paint, but because you have to make
>certain assumptions about the molecular dynamics of paint, the 
>notion of packing fraction of dissimilar molecules and aggregates,
>etc. etc. ).  

Well, I finally did try it, and all of my Sears Weatherbeater ran into the 
singularity at the bottom and created a black hole.  Now I have a place to 
dump my garbage that doesn't take up much space.  I just don't dare drop it.

Bill




********************************************************
Bill Penrose, President, Custom Sensor Solutions, Inc.
526 West Franklin Avenue, Naperville, IL 60540
630-548-3548, fax: 630-369-9618
email wpenrose@interaccess.com
********************************************************
Applications of gas sensors:  Contract R&D, product
development, and consultation.
********************************************************

In article <schne042-3101970914410001@x209-65.bae.umn.edu> schne042@gold.tc.umn.edu (Mike Schneider) writes:
>weight.  And of course, the clincher is that it can be made into a really
>bad chemistry "joke" such as: What do you say when someone steals your
>gold?  Ay you, cesium that robber, he stole my gold.

So how many hafniums does it take to make a holmium?

Bill


********************************************************
Bill Penrose, President, Custom Sensor Solutions, Inc.
526 West Franklin Avenue, Naperville, IL 60540
630-548-3548, fax: 630-369-9618
email wpenrose@interaccess.com
********************************************************
Applications of gas sensors:  Contract R&D, product
development, and consultation.
********************************************************

> 
> So who said it had to be painted in finite time?
> 
>         Eric Lucas


I belive most painters would appriciate if you gave them
a job that has a time limit :)

//Luka
-- 
------------------------------------	
mailto:devalmont@geocities.com
http://www.geocities.com/CapeCanaveral/1630/
------------------------------------

Roger.Douglas@remove_this.tafensw.edu.au (Roger Douglas) wrote:
>ChemTeam@clubnet.net expostulated:
>
>>In article <1997Jan29.215107.11542@news.wrc.xerox.com>, fleck@dirk (Pat
>>Fleckenstein (x58885)) wrote:
>
>
>
>>the loop of Henle, Purkinje fibers and the far-flung isles of Langerhans
>
>Went there for my holidays once. Beautiful spot.
>Not to be confused with the "Eyelets of Langerhans" which is the technical
>name for the holes your shoelaces pass through.
>
>--R.
>
May be I should visit such exotic places too!

Eric Lucas wrote:
> 
> William R. Penrose <wpenrose@interaccess.com> wrote in article
> <wpenrose.370.0025650A@interaccess.com>...
> > In article <schne042-3101970914410001@x209-65.bae.umn.edu>
> schne042@gold.tc.umn.edu (Mike Schneider) writes:
> > >weight.  And of course, the clincher is that it can be made into a
> really
> > >bad chemistry "joke" such as: What do you say when someone steals your
> > >gold?  Ay you, cesium that robber, he stole my gold.
> >
> > So how many hafniums does it take to make a holmium?
> 
> None of yer bismuth.

Ah, go to helium.
-- 
Susan 

http://www4.ncsu.edu/unity/users/s/sjhogart/public/home.html
Suzuki GS page: http://homepage.cistron.nl/~peterh/gsresources/

Luka Crnkovic wrote:
> 
> >
> > So who said it had to be painted in finite time?
> >
> >         Eric Lucas
> 
> I belive most painters would appriciate if you gave them
> a job that has a time limit :)
> 
> //Luka
> --
> ------------------------------------
> mailto:devalmont@geocities.com
> http://www.geocities.com/CapeCanaveral/1630/
> ------------------------------------

Not if they're being paid by the hour

In article <5ctlit$o3d@gannett.math.niu.edu>,
Xcott Craver <caj@baker.math.niu.edu> wrote:
>William R. Penrose <wpenrose@interaccess.com> wrote:
>>
>>So how many hafniums does it take to make a holmium?
>>
>>Bill
>
>	Dyslexic particle physicists do it with hadrons.

Mathematicians take it to the limit.

(Physicists do it with charm :)



Norah Esty
norah@math.montana.edu
http://math.montana.edu/~norah
"Mathematics is the sex urge sublimated"
-- 
/*  Norah Esty
/*  norah@math.montana.edu
/*  http://math.montana.edu/~norah
/*    "Mathematics is the sex urge sublimated."

>Xcott Craver wrote:
>> 
>> In article <5cmpbo$nds@agate.berkeley.edu>,
>> Louis Hom <lhom@nature.berkeley.edu> wrote:
>> >So, I'm trying to collect posters' favorite Things in Math & Science.

Counter-current exchange -it really tripped me out when I first learned
how kidneys work and how the blood in some birds feet is cold,and on the
way back it picks up heat from the incoming blood

ATP

thermoacidophiles

the Belousov-Zhabotinsy reaction 

Supernovae

NMR-and, of course,the Fourier transform 

Felix

I think its VERY DAMN COOL,
how large the known universe is,
how small neutrinos are,
how really in between we are.
John

In article <5d2tp4$i00@netra.montana.edu>, norah@math.montana.edu (Norah
Esty) wrote:

= In article <5ctlit$o3d@gannett.math.niu.edu>,
= Xcott Craver <caj@baker.math.niu.edu> wrote:
= >William R. Penrose <wpenrose@interaccess.com> wrote:
= >>
= >>So how many hafniums does it take to make a holmium?
= >>
= >>Bill
= >
= >       Dyslexic particle physicists do it with hadrons.
= 
= Mathematicians take it to the limit.
= 
= (Physicists do it with charm :)

Biologists need a calculator to add, subtract or divide......but at least
they know how to multiply.

"What makes you think I'd do as you say?  Ani l'dodi."

Visit San Diego Mensa:  http://members.aol.com/sdmensa/

In alt.folklore.science,  <taguebw@wfu.edu> wrote: 
> William R. Penrose wrote:
> > 
> > Nobel prizes are reserved for really useful and rigorous
> > disciplines, like economics.
> 
> Useful and rigorous like economics?!?!?
> This must be a troll....

Warner's Paraphrase of Somebody's Precept: Never
ascribe to trolling that which can adequately be
explained by sarcasm ... except sometimes.
             ^^^^^^^
(or maybe irony ... I can never keep 'em straight)

-- 
Michael Warner 
<warner@wsu.edu>

In alt.folklore.science,
Roger Douglas <Roger.Douglas@remove_this.tafensw.edu.au> wrote: 
> ChemTeam@clubnet.net expostulated:
> 
> >In article <1997Jan29.215107.11542@news.wrc.xerox.com>, fleck@dirk (Pat
> >Fleckenstein (x58885)) wrote:
> >the loop of Henle, Purkinje fibers and the far-flung isles of Langerhans
> 
> Went there for my holidays once. Beautiful spot.

I suppose ... if you're the introspective sort.

> Not to be confused with the "Eyelets of Langerhans"
> which is the technical name for the holes your
> shoelaces pass through.

That doesn't confuse me, but when the "Aglets of Langerhans"
come into it, I sometimes forget which goes where.

-- 
Michael Warner 
<warner@wsu.edu>

wpenrose@interaccess.com (William R. Penrose) wrote:

>In article <schne042-3101970914410001@x209-65.bae.umn.edu> schne042@gold.tc.umn.edu (Mike Schneider) writes:
>>weight.  And of course, the clincher is that it can be made into a really
>>bad chemistry "joke" such as: What do you say when someone steals your
>>gold?  Ay you, cesium that robber, he stole my gold.

>So how many hafniums does it take to make a holmium?

>Bill


>********************************************************
>Bill Penrose, President, Custom Sensor Solutions, Inc.
>526 West Franklin Avenue, Naperville, IL 60540
>630-548-3548, fax: 630-369-9618
>email wpenrose@interaccess.com
>********************************************************
>Applications of gas sensors:  Contract R&D, product
>development, and consultation.
>********************************************************
Or as the New England Geologist, with a large supply of Halite said:
"I don't basalt, I just take it for granite !'
Mark Zawadzki, late of Waynesboro, Va.
"Philadelphia is ours and fairly won"

norah@math.montana.edu (Norah Esty) writes:

>In article <5ctlit$o3d@gannett.math.niu.edu>,
>Xcott Craver <caj@baker.math.niu.edu> wrote:
>>William R. Penrose <wpenrose@interaccess.com> wrote:
>>>
>>>So how many hafniums does it take to make a holmium?
>>>
>>>Bill
>>
>>	Dyslexic particle physicists do it with hadrons.

>Mathematicians take it to the limit.

>(Physicists do it with charm :)

Physicists do it with rigid bodies and simple harmonic motion :)

PEte
-- 
	Name: Petri Lipponen	        email: plippone@cc.hut.fi	
	Addr: J�mer�ntaival 11 k 222 	Espoo 02150 FINLAND
        "PGP PUBLIC KEY" available:     finger plippone@pii.hut.fi

More to the point:

What if the so called cylindrical can was made of gold, and you unrolled
the thing?

Austin Roche

In article <5coo52$ifm@agate.berkeley.edu>,
Mark Rajesh Das <rmarkd@uclink2.berkeley.edu> wrote:
>Xcott Craver (caj@holmes.math.niu.edu) wrote:
>
>: 	Or, a cylindrical can whose base is the Koch snowflake would
>: do the trick, and would actually fit on a shelf.
> Just curious, but by _one_of_ the definitions I have for a fractal
>(i.e. fractional dimension, it has infinite length, but zero area)
>wouldn't any fractal base accomplish the same (a,e)ffect?  
>
>: 	I think it would be more mind-boggling to point out that
>: the can holds a finite amount of paint, but it takes an infinite amount
>: of paint to paint THE INSIDE.  Aren't you technically painting the
>: inside when you fill it up with that finite amount of paint???
>
>: 	Of course, this problem stems from the abstraction that the 
>: layer of paint you lay down on a surface is infinitely thin, which 
>: isn't true in reality, and serves as an amusing reminder that areas
>: and volumes are not comparable mathematically the way they are in 
>: the real world.
> Yeah, I was just about to bring up the point you brought up above.
> But alas, you beat me to the point.
>
>:  ,oooooooo8     o     ooooo@math.niu.edu  --  http://www.math.niu.edu/~caj/
>: o888'   `88   ,888.    888                                                 
>: 888          ,8'`88.   888   "What he can do with numbers would make a
>: 888o.   ,oo ,8oooo88.  888    thousand-dollar-a-night hooker blush like
>: `888oooo88 o88o  o888o 888    a nun."             
>: ____________________8o888'______________-Hank Jennings, _Twin_Peaks._ _____
>
>

plippone@alpha.hut.fi (Petri Lipponen) wrote:

>norah@math.montana.edu (Norah Esty) writes:

>>In article <5ctlit$o3d@gannett.math.niu.edu>,
>>Xcott Craver <caj@baker.math.niu.edu> wrote:
>>>William R. Penrose <wpenrose@interaccess.com> wrote:
>>>>
>>>>So how many hafniums does it take to make a holmium?
>>>>
>>>>Bill
>>>
>>>	Dyslexic particle physicists do it with hadrons.

>>Mathematicians take it to the limit.

>>(Physicists do it with charm :)

>Physicists do it with rigid bodies and simple harmonic motion :)

>PEte
>-- 
Geologists do it deeper !
>	Name: Petri Lipponen	        email: plippone@cc.hut.fi	
>	Addr: J�mer�ntaival 11 k 222 	Espoo 02150 FINLAND
>        "PGP PUBLIC KEY" available:     finger plippone@pii.hut.fi

Mark Zawadzki, late of Waynesboro, Va.
"Philadelphia is ours and fairly won"

In article <32F60788.7110@lancnews.infi.net> John <jbgrosh@lancnews.infi.net> writes:

>I think its VERY DAMN COOL,
>how large the known universe is,
>how small neutrinos are,
>how really in between we are.
>John

Good point.  If we were the size of neutrinos, we'd never meet anyone else, at 
least not for eons.  If we were bigger than a few solar masses, we'd collapse 
into black holes.

It is also neat, and very mysterious, how the existence of the Universe 
depends on the precise values of a few fundamental constants.  Diddle 
some of these constants even in the nth decimal place (where n is a number 
I've forgotten), and all our atoms fall apart.  

Accident or design?  (Bait for the metaphysics types that haunt this group.)

Bill


********************************************************
Bill Penrose, President, Custom Sensor Solutions, Inc.
526 West Franklin Avenue, Naperville, IL 60540
630-548-3548, fax: 630-369-9618
email wpenrose@interaccess.com
********************************************************
Applications of gas sensors:  Contract R&D, product
development, and consultation.
********************************************************

Austin Duncan Roche wrote:
 > 
 > More to the point:
 > 
 > What if the so called cylindrical can was made of gold, and you
unrolled
 > the thing?

you'd be rich...

electronic monk

 > 
 > Austin Roche
 > 
 > In article <5coo52$ifm@agate.berkeley.edu>,
 > Mark Rajesh Das <rmarkd@uclink2.berkeley.edu> wrote:
 > >Xcott Craver (caj@holmes.math.niu.edu) wrote:
 > >
 > >:      Or, a cylindrical can whose base is the Koch snowflake would
 > >: do the trick, and would actually fit on a shelf.
 > > Just curious, but by _one_of_ the definitions I have for a fractal
 > >(i.e. fractional dimension, it has infinite length, but zero area)
 > >wouldn't any fractal base accomplish the same (a,e)ffect?
 > >

Alas! sucked in by the bait.  I saw it, just couldn't resist.

Of course, if those universal constants were even slightly different, we
wouldn't be around to debate them.