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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).
 
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&ndash;283
* R. Connelly, "Rigidity", in ''Handbook of Convex Geometry'', vol. A, 223&ndash;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&ndash;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]]

Revision as of 20:30, 24 January 2014

In geometry, a flexible polyhedron is a polyhedral surface that allows continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this is also true in higher dimensions).

The first examples of flexible polyhedra, now called Bricard's octahedra, were discovered by Template:Harvs. They are self-intersecting surfaces isometric to an octahedron. The first example of a non-self-intersecting surface in R3, the Connelly sphere, was discovered by Template:Harvs.

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 Template:Harvs using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by Template:Harvs.

Scissor congruence

Connelly conjectured that the 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 space were studied by Hellmuth Stachel. In November 2009 it was not known whether flexible polytopes exist in Euclidean space of dimension n5.

See also

References

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • 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.


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  • Ralph Alexander, Lipschitzian Mappings and Total Mean Curvature of Polyhedral Surfaces, Transactions of the AMS 288 (1985), 661–678
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • 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.
  • 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