From: flor@balu.kfunigraz.ac.at (Peter Flor)
Newsgroups: sci.math.research
Subject: Re: flexible polyhedra
Date: Tue, 20 Jun 1995 15:11:45 LOCAL
Originator: greg@symcom.math.uiuc.edu
In article <3s64jo$fmd@news-sop.inria.fr> jpm@tonic.cma.fr (Jean-Paul Marmorat) writes:
>Can somebody help me to find references to the so-called "flexible
>polyhedra" (rigid faces but non rigid positions of the vertices).
>It seems that flexible polyhedra
>1) exist
>2) are necessarily non-convex
...
>[For starters look up papers by Robert Connelly - Greg]
I believe this topic was treated by Walter Wunderlich. - Peter Flor.
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From: ruxton@agcux.bio.ns.ca (Mike Ruxton (CHS))
Newsgroups: sci.math.research
Subject: Re: flexible polyhedra
Date: Sat, 1 Jul 1995 11:31:37 -0400
...
As Greg remarked, look for papers by R. Connelly, e.g.
The Rigidity of polyhedral surfaces, Math. Mag. 52(1979) 275-283
See also
B.Roth - Rigid and flexible frameworks AMM 88 (1981) 6-21
where it is proved e.g.
The framework of a convex polyhedron C in R3 is rigid iff every face of C is a triangle.
Cauchy proved that any convex polyhedron in R3 with rigid faces,
hinged at the edges, is completely rigid.
This information was gleaned from
Croft, Falconer & Guy - Unsolved Problems in Geometry - problems B13 & B14.
--
Michael Ruxton - Canadian Hydrographic Service - ruxton@agcux.bio.ns.ca
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
So many men, so many opinions. - Terence
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From: jpm@tonic.cma.fr (Jean-Paul Marmorat)
Newsgroups: sci.math
Subject: flexible polyhedra
Date: 10 Jul 1995 13:37:21 GMT
Many thanks to all of you who sent replies about the "flexible polyhedra"
topic:
Daniel A. Asimov, Chris Thompson, D. Ruberman, George Hart, P. Penning,
Gary O'Brien, Ross Geoghegan, Klaus Johannson, John Mitchell, Douglas J. Zare,
Larry Edwards, Chris Hillman, Jean-Daniel Boissonnat, Henry Crapo,
David Petry, A. Florian
Sorry if I forgot someone.
Many of you mention Robert Connelly (connelly@math.cornell.edu)
as the discoverer of flexible polyhedra and the result of Cauchy
"Second Memoire, J.Ecole Polytechnique 9 (1813) 87"
that convex polyhedra are rigid.
Here is a summary of your replies:
......
R.Connelly: How to built a flexible polyhedral surface
in: Geometric Topology (James C. Cantrell, ed.), Academic Press, 1979
......
See an article by (I think) Robert Connelly, who is the discoverer of
flexible polyhedra, in the Mathematical Intelligencer, circa 1978 or so.
......
I believe after Connelly's initial success a contest was held (by P.
Delign?) to find the fewest number of vertices for such an object. That
number is very small, like 7 if memory serves.
......
I believe the original paper by Robert Connelly was published
in the IHES journal (whose exact name I no longer remember;
"Publications de l'IHES" or something like that
......
I believe Robert Connelly at Cornell was the first to construct such a
polyhedron. One article of his on this appeared in Mathematics Magazine
(an AMS publication) in the late 70's, perhaps 1977.
......
Vous n'avez pas la collection de Topologie Structurale
dans le centre de Docs a Sophia? Je pense qu'il sera toujours
possible de commander la serie complete du bureau a Montreal.
Il y a pleins d'articles sur le sujet, avec references
au travail du 19e, debut 20e: Bennett, Bricard -- et plus
recemment, Michael Goldberg.
......
Hallard T. Croft & Kenneth J. Falconer & Richard K. Guy
"Unsolved Problems in Geometry"
Springer-Verlag (1991)
ISBN 0-387-97506-3
......
There are many that are made of edge-connected sub polyhedra, e.g., the
"Yoshimoto cubes" sold in toy stores, which are a cyclic chain of pieces
hinged at their edge connections. I don't know much about ones which are not
divisible into sub polyhedra; there is a picture of one on p. 68 of
Miyazaki's book (but it isn't creditied).
......
I can give you some older references.
The 'most' flexible polyhedron I know of was found by the american R
Connelly, while being in France a L'Institut des Hautes Etudes Scientifiques
35 Route de Chartres 91440 Bures sur Yvette
It was published in a paper by its director Nicolaas H. KUIPER in the
Seminaire Bourbaki 30e annee. 1977/78, nr 514.
See also an article in Le Monde of May 10 1978.
R. Connelly has also published in the Journal of Differential Geometry
around 1980, when he was at Cornell University, Ithaca, New York 14853,
Department of Math
On other flexible polyhedra:
Herman Gluck, Almost all simply connected closed surfaces,are rigid.
Lecture notes in Mathematics no 438 Geometric Topology,
Springer Verlag(1975), pp 225-239
R. Connelly 'How to build a flexible polyhedral surface.
In Geometrical Tology ed by J.C.Cantrell Acad Press 1979
pp 675-683
M. Goldberg Unstable Polyhedral Structures
Mathematics Magazine Vol 51, No 3, May 1978 pp 165-170
......
Wunderlich, W., Neue Wackelikosaeder [= New Flexible Icosahedra].
Sitzungsberichte der Oesterreichischen Akademie der Wissenschaften,
mathematisch-naturwissenschaftliche Klasse 1980, #2, 28-33.
W. W., Wackelikosaeder [= Flexible Icosahedra]. Geometriae Dedicata 11
(1981), #2, 137-146.
W. W., Kipp-Ikosaeder [= Tilted Icosahedra]. Elemente der Mathematik 36
(1981), #6, 153-158.
W. W., Kipp-Ikosaeder II: Ikosaeder mit dreizaehliger Symmetrieachse [=
Flex. Icosahedra II: Icosahedra with three-fold symmetry axis]. Elemente
der Mathematik 37 (1982), #3, 84-89.
W. W., Wackeldodekaeder [= Flexible Dodecahedra]. Elemente der Mathematik
37 (1982), #6, 153-163.
W. W., Wackelige Doppelpyramiden [= Flexible Double Pyramids].
Sitzungsberichte der Oesterreichischen Akademie der Wissenschaften,
math.-naturwiss. Klasse 1980, #5, 82-87.
W. W., Zur projektiven Invarianz von Wackelstrukturen [= On the Projective
Invariance of Shaky Structures]. Zeitschrift fuer Angewandte Mathematik und
Mechanik 60 (1980), #12, 703-708.
W. W., "Projective Invariance of Shaky Structures," Acta Mechanica 42
(1982), #3-4, 171-181.
......
________________________________________________________________
JP Marmorat Centre de Mathematiques Appliquees
Ecole des Mines de Paris, BP 207 06904 SOPHIA ANTIPOLIS Cedex
Tel: +33 93 95 74 71 --- Fax: +33 93 95 74 88
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Note: the reference to Connelly's article in the Mathematical Intelligencer
is to vol. 1 #3, pp 13-14 (!)