From: eppstein@euclid.ics.uci.edu (David Eppstein)
Newsgroups: sci.math
Subject: Re: Computing polyhedron's volume from its edges
Date: 15 Jun 1998 17:37:06 -0700
In <6m49ov$qsl$1@vixen.cso.uiuc.edu> astephan@students.uiuc.edu (adam louis stephanides) writes:
: In the "Mathematical Recreations" column of July 1998, Ian Stewart
: says it has been proved that for any polyhedron, there is a
: polynomial equation relating its volume to the lengths of its edges.
: Stewart later glosses this result as "the volume of a polyhedron
: would depend solely on the lengths of its edges." But in the form
: Stewart states this, it is clearly wrong. Consider prisms with
: height 1 whose bases are rhombuses with edge lengths 1. Such prisms
: all have the same edge lengths, but can have any volume between 0 and 1.
As you guessed, the result is for triangulated polyhedra only.
The reference is:
"The Bellows Conjecture", R. Connelly, I. Sabitov, and A. Walz,
Beitr\"age zur Algebra und Geometrie (Contributions to Algebra and Geometry)
Volume 38 (1997), No. 1, pp. 1-10.
Apparently, I. Sabitov proved the same result earlier, for the case of
triangulated surfaces homeomorphic to the sphere. I can't transliterate
the Russian title and journal name, but here it is in English:
"On the problem of invariance of a flexible polyhedron", I. Sabitov,
Fundamental and Applied Mathematics, 1996.
Connelly's paper doesn't list the volume or page numbers of the Sabitov paper.
--
David Eppstein UC Irvine Dept. of Information & Computer Science
eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/
==============================================================================
From: ptwahl@aol.com (PTWahl)
Newsgroups: sci.math
Subject: Re: Computing polyhedron's volume from its edges
Date: 16 Jun 1998 00:39:19 GMT
On 15 Jun 1998 23:12:31 GMT, astephan@students.uiuc.edu
(adam louis stephanides) wrote:
>In the "Mathematical Recreations" column of July 1998, Ian Stewart
>says it has been proved that for any polyhedron, there is a
>polynomial equation relating its volume to the lengths of its edges.
>Steart later glosses this result as "the volume of a polyhedron
>would depend solely on the lengths of its edges." But in the form
>Stewart states this, it is clearly wrong. Consider prisms with
>height 1 whose bases are rhombuses with edge lengths 1. Such prisms
>all have the same edge lengths, but can have any volume between 0 and
>1. So what's the correct result? (My guess is that the polyhedra must
>have triangular faces, where this includes the "degenerate" case
>where two or more faces form a larger, triangulated face.) Stewart
>credits the result to ______ Connelly (I omitted to xerox the first
>name), Idzhad Sabitov and Anke Walz.
>
>--Adam
You want "Robert Connelly" for the blank above. (See URL:)
http://math.cornell.edu/~connelly/index.html
As for Stewart's article, re-read (second paragraph, middle)
the definition of "flexible polyhedron, " and (third paragraph)
"when a polyhedron flexes, the only things that change are the
angles at which the faces meet. Imagine that the faces are
hinged along their edges. Everything else is perfectly rigid."
The way I read it, a square face cannot turn into a rhombus.
These are not edges in thin air, but the edges of a steel sheet.
(Meant of course as a concept only: no matter is involved.)
I agree that Stewart at first appears to claim too much, and
might have been more clear. And I'm not precisely sure what
he does mean, in the passage you quote. (Several conditionals
modify that sentence: "Nevertheless, suppose ...")
I look forward to a clarification, in some future issue.
Regards,
Patrick T. Wahl,
Greeley, Colorado, USA
==============================================================================
From: astephan@students.uiuc.edu (adam louis stephanides)
Newsgroups: sci.math
Subject: Re: Computing polyhedron's volume from its edges
Date: 16 Jun 1998 15:33:55 GMT
ptwahl@aol.com (PTWahl) writes:
>As for Stewart's article, re-read (second paragraph, middle)
>the definition of "flexible polyhedron, " and (third paragraph)
>"when a polyhedron flexes, the only things that change are the
>angles at which the faces meet. Imagine that the faces are
>hinged along their edges. Everything else is perfectly rigid."
>The way I read it, a square face cannot turn into a rhombus.
>These are not edges in thin air, but the edges of a steel sheet.
>(Meant of course as a concept only: no matter is involved.)
Yes, that is true for the Bellows Conjecture, which is the main
subject of Stewart's article. And my rhombic prisms are not, of course,
counterexamples to the Bellows Conjecture. But in describing
Connelly et al's result, Stewart says, as I quoted in part, that "the
volume of a polyhedron would depend solely on the lengths of its
edges. As the polyhedron flexes, the lengths of its edges do not
change--so the volume of the polyhedron must not change either."
He doesn't say that "the lengths of its edges and the shapes of its
faces do not change." While, as you say, Stewart is not entirely
clear, I think my reading is the most natural. Of course, your version
of the result would be sufficient for proving the Bellows conjecture.
And the correct result as stated by David Eppstein is somewhat stronger
than your version.
--Adam