|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[geometry]], a '''flexible polyhedron''' is a [[Polyhedron|polyhedral]] [[surface]] that allows continuous non-[[rigid (structural)|rigid]] [[Deformation (engineering)|deformation]]s such that all faces remain rigid. The [[Cauchy's theorem (geometry)|Cauchy rigidity theorem]] shows that in dimension 3 such a polyhedron cannot be [[convex set|convex]] (this is also true in higher dimensions).
| | Her name is Felicidad Ahmad. What she loves performing is taking part in croquet and she is attempting to make it a profession. Her family members life in Delaware but she needs to transfer because of her family members. Managing people is how I make cash and it's some thing I truly enjoy.<br><br>Feel free to surf to my weblog - extended auto warranty; [http://sun-lan.de/index.php?mod=users&action=view&id=14346 More suggestions], |
| | |
| The first examples of flexible polyhedra, now called '''Bricard's octahedra''', were discovered by {{harvs|txt|authorlink=Raoul Bricard|first=Raoul |last=Bricard |year= 1897}}. They are self-intersecting surfaces [[isometry|isometric]] to an [[octahedron]]. The first example of a non-self-intersecting surface in '''R'''<sup>3</sup>, the '''Connelly sphere''', was discovered by {{harvs|txt|authorlink=Robert Connelly|first=Robert |last=Connelly |year= 1977}}.
| |
| | |
| == Bellows conjecture ==
| |
| | |
| In the late 1970s Connelly and [[D. Sullivan]] formulated the '''Bellows conjecture''' stating that the [[volume]] of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra [[homeomorphic]] to a [[sphere]] by {{harvs|txt | last1=[[:ru:Сабитов, Иджад Хакович|Sabitov]] | first1=I. Kh. | title=On the problem of the invariance of the volume of a deformable polyhedron | mr=1339277 | year=1995 | journal=Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=50 | issue=2 | pages=223–224}}
| |
| using [[elimination theory]], and then proved for general [[Orientability|orientable]] 2-dimensional polyhedral surfaces by {{harvs|txt | last1=Connelly | first1=Robert | last2=Sabitov | first2=I. | last3=Walz | first3=Anke | title=The bellows conjecture | mr=1447981 | year=1997 | journal=Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry | issn=0138-4821 | volume=38 | issue=1 | pages=1–10|url=http://www.mat.ub.es/EMIS/journals/BAG/vol.38/no.1/1.html}}.
| |
| | |
| == Scissor congruence ==
| |
| | |
| Connelly conjectured that the [[Hilbert's third problem|Dehn invariant]] of a flexible polyhedron is invariant under flexing. This is known as the '''strong bellows conjecture'''. Preservation of the Dehn invariant is known to be equivalent to [[scissors congruence]] of the enclosed region under flexing. The special case of [[mean curvature]] has been proved by Ralph Alexander.
| |
| | |
| == Generalizations ==
| |
| | |
| Flexible [[polychora]] in 4-dimensional Euclidean space and 3-dimensional [[Hyperbolic geometry|hyperbolic space]] were studied by [[Hellmuth Stachel]]. In November 2009 it was not known whether flexible polytopes exist in Euclidean space of dimension <math>n\geq 5</math>.
| |
| | |
| == See also ==
| |
| * [[Rigid origami]]
| |
| | |
| == References ==
| |
| *{{citation|first=R.|last= Bricard|title= Mémoire sur la théorie de l'octaèdre articulé|journal= J. Math. Pures Appl. |volume=5|issue= 3 |year=1897|pages= 113–148|url=http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1897_5_3_A5_0}}
| |
| *{{Citation | last1=Connelly | first1=Robert | title=A counterexample to the rigidity conjecture for polyhedra | url=http://www.numdam.org/item?id=PMIHES_1977__47__333_0 | mr=0488071 | year=1977 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=47 | pages=333–338 | doi=10.1007/BF02684342 | volume=47}}
| |
| * R. Connelly, "The Rigidity of Polyhedral Surfaces", ''Mathematics Magazine'' '''52''' (1979), 275–283
| |
| * R. Connelly, "Rigidity", in ''Handbook of Convex Geometry'', vol. A, 223–271, North-Holland, Amsterdam, 1993.
| |
| * {{MathWorld |urlname = BellowsConjecture |title = Bellows conjecture}}
| |
| * {{MathWorld |urlname = FlexiblePolyhedron |title = Flexible polyhedron}}
| |
| *{{Citation | last1=Connelly | first1=Robert | last2=Sabitov | first2=I. | last3=Walz | first3=Anke | title=The bellows conjecture | mr=1447981 | year=1997 | journal=Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry | issn=0138-4821 | volume=38 | issue=1 | pages=1–10|url=http://www.mat.ub.es/EMIS/journals/BAG/vol.38/no.1/1.html}}
| |
| * Ralph Alexander, ''Lipschitzian Mappings and Total Mean Curvature of Polyhedral Surfaces'', [[Transactions of the American Mathematical Society|Transactions of the AMS]] '''288''' (1985), 661–678
| |
| *{{Citation | last1=[[:ru:Сабитов, Иджад Хакович|Sabitov]] | first1=I. Kh. | title=On the problem of the invariance of the volume of a deformable polyhedron | mr=1339277 | year=1995 | journal=Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=50 | issue=2 | pages=223–224}}
| |
| * [[Hellmuth Stachel|H. Stachel]], ''Flexible octahedra in the hyperbolic space'', in ''Non-Euclidean geometries. János Bolyai memorial volume''(Eds. A. Prékopa et al.). New York: Springer. Mathematics and its Applications (Springer)'''581''', 209–225 (2006). ISBN 0-387-29554-2.
| |
| * [[Hellmuth Stachel|H. Stachel]], ''Flexible cross-polytopes in the Euclidean 4-space'', J. Geom. Graph. '''4''', No.2 (2000), 159–167.
| |
| | |
| ===Popular level===
| |
| *D. Fuchs, S. Tabachnikov, ''Mathematical Omnibus: Thirty Lectures on Classic Mathematics''
| |
| | |
| [[Category:Nonconvex polyhedra]]
| |
| [[Category:Mathematics of rigidity]]
| |
| [[Category:Discrete geometry]]
| |
| [[Category:Polyhedra]]
| |
Her name is Felicidad Ahmad. What she loves performing is taking part in croquet and she is attempting to make it a profession. Her family members life in Delaware but she needs to transfer because of her family members. Managing people is how I make cash and it's some thing I truly enjoy.
Feel free to surf to my weblog - extended auto warranty; More suggestions,