From: "Clive Tooth"
Subject: Re: Please help me with my project
Date: Sat, 6 Mar 1999 16:48:00 -0000
Newsgroups: [missing]
To: "Dave Rusin"
Keywords: What is a porism? (Poncelet's, Steiner's)
Hi Dave...
>OK, I'll confess: I don't know what a porism is, although I worked for
>a while on something I knew to be called "Poncelet's porism". How would
>you describe this class of objects to a mathematician?
>dave
A porism is a geometrical construction which is (depending on the starting
configuration) either impossible, or possible in an infinite number of ways.
Examples...
Poncelet's porism: If an n-gon can be inscribed in a given conic while
circumscribing another given conic then there are an infinite number of such
n-gons.
Steiner circles: given one circle, P, inside another circle, Q, construct a
circle, C1, which touches P and Q. Now construct a circle C2 which touches
P, Q and C1. Now construct a circle C3 which touches P, Q and C2. Continue
like this until a circle, Cn, either overlaps or touches C1. If it touches
C1 an infinite number of such chains can be constructed, and no overlapping
chain can be constructed. If it overlaps C1 then an infinite number of such
chains can be constructed, and no touching chain can be constructed.
(The word "porism" also seems to have some other meanings, but I didn't mean
those. :) )
Clive
==============================================================================
From: "Clive Tooth"
Subject: Re: Please help me with my project
Date: Sat, 6 Mar 1999 20:14:03 -0000
Newsgroups: [missing]
To: "Dave Rusin"
Hi...
>Ah, I see the pattern. It's interesting when those kinds of problems occur.
>It's perhaps more interesting that someone should have thought to give them
>a name!
>
>Thanks for your response.
>
>dave
Yes... and so far as I know it is _only_ used in a geometrical setting. If
we have a Diophantine equation which only has solutions (in fact, an
infinite number) if, for instance, k is of the form 4n+1 we would hardly
comment that the equation was in the form of a porism :)
Clive