Re: [Phys-l] elliptical thinking

[John Denker <jsd@av8n.com>]

On 07/22/2008 01:17 PM, LaMontagne, Bob wrote:

> Ellipses and circles are all part of what are called "conic
> sections". If one places a basic right circular cone base first on a
> table, any horizontal cross-sectional cut through the cone produces a
> circle. If one imagines a tiny light bulb at the apex of the cone,
> the shadow of any one of these horizontal cuts is the circle that
> forms the base of the cone. Likewise, a non-horizontal cut defines
> the conic section called the ellipse.

Small point:  I was hoping somebody would point that out.  This has 
been considered the canonical definition of ellipse for most of the 
last 2000 years.

Let me add that Bob's earlier "tilt the circle" construction must
have used an /orthogonal/ projection (otherwise the square wouldn't
have turned into a rectangle).  This corresponds to the limiting
case of a very narrow cone, so that the apex is very far away from
the circle and the ellipse.

All in all, it seems that Bob's construction entirely upholds the 
letter and the spirit of of the canonical definitions.

The main reason I find this small point to be interesting is that 
there were some claims to the contrary.  


Much larger point:  it is almost(*) always a Bad Idea to worry about 
what is "the" definition.  Who cares whether it is axiomatic, so long 
as it is true?

The choice of axioms is almost never unique.  For example, if we
have a theoretical system where A implies B, B implies C, C implies 
D, and D implies A ... then any one of those four propositions will
serve equally well as "the" axiom.  As a more geometric example,
Playfair's version of the infamous 5th postulate is not the same
as Euclid's version, but it has the same consequences.

Feynman said that knowledge is like a grand tapestry.  Something 
that you've forgotten is like a hole in the tapestry.  You should
be able to repair the damage in many different ways, be reweaving
down from the top, or up from the bottom, or in from the sides.

My point is that this is true even if the missing fact had been at 
some stage treated as axiomatic.

  *) Note: Sometimes you might want to play a highly formal game
   where you set down some minimalist set of axioms and try to
   deduce the whole tapestry from there.  That's OK, but remember
   it is just a game ... and that others may choose to play with
   other axioms.   A professional who has mastered the subject
   will be able to see things from many different viewpoints, and
   will not be infatuated with any one set of axioms.