Flexible polyhedron

File:Ste-anim.gif, the simplest possible non-self-crossing flexible polyhedron]]

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| footer = Bricard octahedron with a rectangle and antiparallelogram respectively as its equator

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The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by {{harvs|txt|authorlink=Raoul Bricard|first=Raoul |last=Bricard |year= 1897}}. They are self-intersecting surfaces isometric to an octahedron. The first example of a flexible non-self-intersecting surface in $\backslash mathbb\{R\}^3$, the Connelly sphere, was discovered by {{harvs|txt|authorlink=Robert Connelly|first=Robert |last=Connelly |year= 1977}}. Steffen's polyhedron is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra.{{sfnp|Alexandrov|2010}}

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=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 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 | issn=0138-4821 | volume=38 | issue=1 | pages=1–10|url=https://www.emis.de/journals/BAG/vol.38/no.1/1.html}}.{{sfnp|Sabitov|1995}}{{sfnp|Connelly|Sabitov|Walz|1997}} The proof extends Piero della Francesca's formula for the volume of a tetrahedron to a formula for the volume of any polyhedron. The extended formula shows that the volume must be a root of a polynomial whose coefficients depend only on the lengths of the polyhedron's edges. Since the edge lengths cannot change as the polyhedron flexes, the volume must remain at one of the finitely many roots of the polynomial, rather than changing continuously.{{sfnp|Demaine|O'Rourke|2007}}

Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. This was known as the strong bellows conjecture or (after it was proven in 2018) the strong bellows theorem.{{sfnp|Gaĭfullin|Ignashchenko|2018}} Because all configurations of a flexible polyhedron have both the same volume and the same Dehn invariant, they are scissors congruent to each other, meaning that for any two of these configurations it is possible to dissect one of them into polyhedral pieces that can be reassembled to form the other. The total mean curvature of a flexible polyhedron, defined as the sum of the products of edge lengths with exterior dihedral angles, is a function of the Dehn invariant that is also known to stay constant while a polyhedron flexes.{{sfnp|Alexander|1985}}

Flexible 4-polytopes in 4-dimensional Euclidean space and 3-dimensional hyperbolic space were studied by {{harvs|first=Hellmuth|last=Stachel|authorlink=Hellmuth Stachel|year=2000|txt}}.{{sfnp|Stachel|2000}} In dimensions $n\backslash geq\; 5$, flexible polytopes were constructed by {{harvtxt|Gaifullin|2014}}.{{sfnp|Gaifullin|2014}}

• Flexagon
• Kaleidocycle
• Kokotsakis polyhedron
• Rigid origami

{{reflist|30em}}

{{refbegin|30em}}

• {{citation

| last = Alexander | first = Ralph

| doi = 10.2307/1999957

| issue = 2

| journal = Transactions of the American Mathematical Society

| mr = 776397

| pages = 661–678

| title = Lipschitzian mappings and total mean curvature of polyhedral surfaces. I

| volume = 288

| year = 1985| jstor = 1999957

| doi-access = free

}}.

• {{citation

| last = Alexandrov | first = Victor

| arxiv = 0901.2989

| doi = 10.1007/s00022-011-0061-7

| issue = 1–2

| journal = Journal of Geometry

| mr = 2823098

| pages = 1–13

| title = The Dehn invariants of the Bricard octahedra

| volume = 99

| year = 2010}}.

• {{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|access-date=2008-07-27|archive-url=https://web.archive.org/web/20120216011406/http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1897_5_3_A5_0|archive-date=2012-02-16|url-status=dead}}
• {{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}}
• {{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 | issn=0138-4821 | volume=38 | issue=1 | pages=1–10 | url=https://emis.de/journals/BAG/vol.38/no.1/1.html }}
• {{citation

| last = Gaifullin | first = Alexander A.

| doi = 10.1134/S0081543814060066

| issue = 1

| journal = Proceedings of the Steklov Institute of Mathematics

| mr = 3482593

| pages = 77–113

| title = Flexible cross-polytopes in spaces of constant curvature

| volume = 286

| year = 2014| arxiv = 1312.7608

}}.

• {{citation

| last1 = Gaĭfullin | first1 = A. A.

| last2 = Ignashchenko | first2 = L. S.

| doi = 10.1134/S0371968518030068

| isbn = 5-7846-0147-4

| issue = Topologiya i Fizika

| journal = Trudy Matematicheskogo Instituta Imeni V. A. Steklova

| mr = 3894642

| pages = 143–160

| title = Dehn invariant and scissors congruence of flexible polyhedra

| volume = 302

| year = 2018}}.

• {{Citation | last1=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}}
• {{citation

| last = Stachel | first = Hellmuth | authorlink = Hellmuth Stachel

| contribution = Flexible octahedra in the hyperbolic space

| doi = 10.1007/0-387-29555-0_11

| editor = A. Prékopa|display-editors=etal

| isbn = 978-0-387-29554-1

| mr = 2191249

| pages = 209–225

| publisher = Springer | location = New York

| series = Mathematics and its Applications

| title = Non-Euclidean geometries (János Bolyai memorial volume)

| volume = 581

| year = 2006| citeseerx = 10.1.1.5.8283 }}.

• {{citation

| last = Stachel | first = Hellmuth | authorlink = Hellmuth Stachel

| issue = 2

| journal = Journal for Geometry and Graphics

| mr = 1829540

| pages = 159–167

| title = Flexible cross-polytopes in the Euclidean 4-space

| url = http://www.geometrie.tuwien.ac.at/stachel/cross.pdf

| volume = 4

| year = 2000}}.

{{refend}}

{{refbegin|30em}}

• {{citation

| last = Connelly | first = Robert | authorlink = Robert Connelly

| doi = 10.2307/2689778

| issue = 5

| journal = Mathematics Magazine

| mr = 551682

| pages = 275–283

| title = The rigidity of polyhedral surfaces

| volume = 52

| year = 1979| jstor = 2689778 }}.

• {{citation

| last = Connelly | first = Robert | authorlink = Robert Connelly

| editor-last = Klarner, David A. | editor-first =

| contribution = Flexing surfaces

| doi = 10.1007/978-1-4684-6686-7_10

| isbn = 978-1-4684-6688-1

| pages = 79–89

| publisher = Springer

| title = The Mathematical Gardner

| year = 1981}}.

• {{citation

| last = Connelly | first = Robert | authorlink = Robert Connelly

| contribution = Rigidity

| contribution-url = http://www.math.cornell.edu/~web7620/Rigidity-Connelly.pdf

| location = Amsterdam

| mr = 1242981

| pages = 223–271

| publisher = North-Holland

| title = Handbook of convex geometry, Vol. A, B

| year = 1993}}.

• {{citation

| last1 = Demaine | first1 = Erik D. | author1-link = Erik Demaine

| last2 = O'Rourke | first2 = Joseph | author2-link = Joseph O'Rourke (professor)

| contribution = 23.2 Flexible polyhedra

| doi = 10.1017/CBO9780511735172

| isbn = 978-0-521-85757-4

| mr = 2354878

| pages = 345–348

| publisher = Cambridge University Press, Cambridge

| title = Geometric Folding Algorithms: Linkages, origami, polyhedra

| title-link=Geometric Folding Algorithms

| year = 2007}}.

• {{citation

| last1 = Fuchs | first1 = Dmitry

| last2 = Tabachnikov | first2 = Serge

| doi = 10.1090/mbk/046

| isbn = 978-0-8218-4316-1

| location = Providence, RI

| mr = 2350979

| publisher = American Mathematical Society

| title = Mathematical Omnibus: Thirty lectures on classic mathematics

| year = 2007

| contribution = Lecture 25. Flexible polyhedra

| pages = 345–360}}

{{refend}}

• {{MathWorld2 |urlname = FlexiblePolyhedron |title = Flexible polyhedron |urlname2 = BellowsConjecture |title2 = Bellows conjecture}}

{{Mathematics of paper folding}}

Category:Nonconvex polyhedra

Category:Mathematics of rigidity

Category:Flexible polyhedra