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Cauchy’s rigidity theorem 83 Now let n ≥ 4. If for any i ∈ {2, . . ., n − 1} we have α i = α ′ i , then the q corresponding vertex can be cut off by introducing the diagonal from q i−1 to q i+1 resp. from q i−1 ′ to q′ i+1 , with q i−1 q i+1 = q′ i−1 q′ i+1 , so we are done by induction. Thus we may assume α i < α ′ i for 2 ≤ i ≤ n − 1. Now we produce a new polygon Q ∗ from Q by replacing α n−1 by the largest possible angle α ∗ n−1 ≤ α′ n−1 that keeps Q∗ convex. For this we replace q n by qn, ∗ keeping all the other q i , edge lengths, and angles from Q. i α i Q: α n−1 If indeed we can choose α ∗ n−1 = α ′ n−1 keeping Q ∗ convex, then we get q 1 q n < q 1 qn ∗ ≤ q′ 1 q′ n , using the case n = 3 for the first step and induction as above for the second. Otherwise after a nontrivial move that yields Q ∗ : q 1 α ∗ n−1 q n q 1 q ∗ n > q 1 q n (1) we “get stuck” in a situation where q 2 , q 1 and qn ∗ are collinear, with Q ∗ : q 1 α ∗ n−1 q ∗ n q 2 q 1 + q 1 qn ∗ = q 2 q∗ n . (2) q 2 Now we compare this Q ∗ with Q ′ and find q 2 qn ∗ ≤ q 2 ′ q′ n (3) by induction on n (ignoring the vertex q 1 resp. q 1 ′ ). Thus we obtain q 1 q ∗ n q ′ 1 q′ n (∗) ≥ q ′ 2 q′ n − q ′ 1 q′ 2 (3) ≥ q 2 qn ∗ (2) − q 1 q 2 = q 1 qn ∗ (1) > q 1 q n , where (∗) is just the triangle inequality, and all other relations have already been derived. □ We have seen an example which shows that Cauchy’s theorem is not true for non-convex polyhedra. The special feature of this example is, of course, that a non-continuous “flip” takes one polyhedron to the other, keeping the facets congruent while the dihedral angles “jump.” One can ask for more: Could there be, for some non-convex polyhedron, a continuous deformation that would keep the facets flat and congruent? It was conjectured that no triangulated surface, convex or not, admits such a motion. So, it was quite a surprise when in 1977 — more than 160 years after Cauchy’s work — Robert Connelly presented counterexamples: closed triangulated spheres embedded in R 3 (without self-intersections) that are flexible, with a continuous motion that keeps all the edge lengths constant, and thus keeps the triangular faces congruent.
84 Cauchy’s rigidity theorem A beautiful example of a flexible surface constructed by Klaus Steffen: The dashed lines represent the non-convex edges in this “cut-out” paper model. Fold the normal lines as “mountains” and the dashed lines as “valleys.” The edges in the model have lengths 5, 10, 11, 12 and 17 units. The rigidity theory of surfaces has even more surprises in store: only very recently Idjad Sabitov managed to prove that when any such flexing surface moves, the volume it encloses must be constant. His proof is beautiful also in its use of the algebraic machinery of polynomials and determinants (outside the scope of this book). References [1] A. CAUCHY: Sur les polygones et les polyèdres, seconde mémoire, J. École Polytechnique XVIe Cahier, Tome IX (1813), 87-98; Œuvres Complètes, IIe Série, Vol. 1, Paris 1905, 26-38. [2] R. CONNELLY: A counterexample to the rigidity conjecture for polyhedra, Inst. Haut. Etud. Sci., Publ. Math. 47 (1978), 333-338. [3] R. CONNELLY: The rigidity of polyhedral surfaces, Mathematics Magazine 52 (1979), 275-283. [4] I. KH. SABITOV: The volume as a metric invariant of polyhedra, Discrete Comput. Geometry 20 (1998), 405-425. [5] J. SCHOENBERG & S.K. ZAREMBA: On Cauchy’s lemma concerning convex polygons, Canadian J. Math. 19 (1967), 1062-1071.
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Martin Aigner Günter M. Ziegler Pr
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Prof. Dr. Martin Aigner FB Mathemat
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VI Preface to the Fourth Edition Wh
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VIII Table of Contents 21. A theore
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Six proofs of the infinity of prime
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Six proofs of the infinity of prime
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Bertrand’s postulate Chapter 2 We
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Bertrand’s postulate 9 times. Her
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Bertrand’s postulate 11 by compar
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Binomial coefficients are (almost)
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Binomial coefficients are (almost)
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Representing numbers as sums of two
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The law of quadratic reciprocity Ch
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The law of quadratic reciprocity 25
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Every finite division ring is a fie
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Page 55 and 56: 46 Three times π 2 /6 For m = 1, 2
Page 57 and 58: 48 Three times π 2 /6 1 f(t) = 1 t
Page 59 and 60: 50 Three times π 2 /6 (3) It has b
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Page 85 and 86: 76 Three applications of Euler’s
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Page 97 and 98: 88 Touching simplices For our examp
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Page 109 and 110: 100 Borsuk’s conjecture To obtain
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A theorem of Pólya on polynomials
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On a lemma of Littlewood and Offord
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On a lemma of Littlewood and Offord
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Cotangent and the Herglotz trick Ch
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Cotangent and the Herglotz trick 15
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Buffon’s needle problem Chapter 2
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Buffon’s needle problem 157 The c
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Combinatorics 25 Pigeon-hole and do
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162 Pigeon-hole and double counting
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Tiling rectangles Chapter 26 Some m
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Tiling rectangles 175 Third proof.
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Tiling rectangles 177 References [1
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180 Three famous theorems on finite
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182 Three famous theorems on finite
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Shuffling cards Chapter 28 How ofte
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Shuffling cards 187 Indeed, if A i
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Shuffling cards 189 Let T be the nu
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Shuffling cards 191 Theorem 1. Let
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Shuffling cards 193 Proof. (1) We
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Lattice paths and determinants Chap
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Lattice paths and determinants 197
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Lattice paths and determinants 199
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Cayley’s formula for the number o
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Identities versus bijections Chapte
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Identities versus bijections 209 Pr
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Identities versus bijections 211 As
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Completing Latin squares Chapter 32
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Completing Latin squares 215 Proof
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Completing Latin squares 217 be dis
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Graph Theory 33 The Dinitz problem
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222 The Dinitz problem In 1976 Vizi
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224 The Dinitz problem X A bipartit
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226 The Dinitz problem G : L(G) : a
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228 Five-coloring plane graphs From
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230 Five-coloring plane graphs is s
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232 How to guard a museum The follo
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234 How to guard a museum “Museum
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236 Turán’s graph theorem Let us
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238 Turán’s graph theorem Thus b
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Communicating without errors Chapte
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Communicating without errors 243 cl
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Communicating without errors 245 Le
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Communicating without errors 247 On
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Communicating without errors 249 No
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The chromatic number of Kneser grap
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258 Of friends and politicians w 1
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Probability makes counting (sometim
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Proofs from THE BOOK 269
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Index acyclic directed graph, 196 a
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Index 273 matrix-tree theorem, 203
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