Volumes of polyhedra in spaces of constant curvature

Transcription

1 Volumes of polyhedra in spaces of constant curvature Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk State University Russia Branched Coverings, Degenerations, and Related Topis 010 Hiroshima, Japan 08-1 March, 010 Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

2 Introduction The calculation of the volume of a polyhedron in 3-dimensional space E 3, H 3,orS 3 is a very old and difficult problem. The first known result in this direction belongs to Tartaglia ( ) who found a formula for the volume of Euclidean tetrahedron. Now this formula is known as Cayley-Menger determinant. More precisely, let be an Euclidean tetrahedron with edge lengths d ij, 1 i < j 4. Then V = Vol(T ) is given by V 1 0 d1 d13 d 14 = 1 d1 0 d3 d4. 1 d31 d3 0 d34 1 d41 d4 d43 0 Note that V is a root of quadratic equation whose coefficients are integer polynomials in d ij, 1 i < j 4. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, 010 / 34

3 Introduction Cauchy theorem (1813) states that if the faces of a convex polyhedron are made of metal plates and the polyhedron edges are replaced by hinges, the polyhedron would be rigid. In spite of this there are non-convex polyhedra which are flexible. Bricard, 1897 (self-interesting flexible octahedron) Connelly, 1978 (the first example of true flexible polyhedron) The smallest example is given by Steffen (14 triangular faces and 9 edges). 7 3 a c 8 c e 10 f 6 a g 1 5 g 1 b 1 b d 5 11 d h 4 5 h e f 6 Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

4 Introduction Surprisely, but the result can be generalized on any Euclidean polyhedron in the following way. Theorem 1 (I. Kh. Sabitov, 1996) Let P be an Euclidean polyhedron. Then V = Vol(P) is a root of an even degree algebraic equation whose coefficients are integer polynomials in edge lengths of P depending on combinatorial type of P only. Example P 1 P (All edge lengths are taken to be 1) Polyhedra P 1 and P are of the same combinatorial type. Hence, V 1 = Vol(P 1 ) and V = Vol(P ) are roots of the same algebraic equation a 0 V n + a 1 V n a n V 0 =0. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

5 Introduction Bellows Conjecture Very important consequence of Sabitov s theorem is a positive solution of the Bellows Conjecture proposed by Dennis Sullivan. Theorem (R. Connelly, I. Sabitov and A. Walz, 1997) All flexible polyhedra keep a constant volume as they are flexed. It was shown by Victor Alexandrov (Novosibirsk, 1997) that Bellows Conjecture fails in the spherical space S 3. In the hyperbolic space H 3 the problem is still open. Any analog of Sabitov s theorem is unknown in both spaces S 3 and H 3. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

6 Spherical orthoscheme Theorem 3 (L. Schläfli) The volume of a spherical orthoscheme with essensial dihedral angles A, B and C A C B S 3 is given by the formula V = 1 4S(A, B, C), where S( π x, y, π z) =Ŝ(x, y, z) = ( ) D sin x sin z m cos mx cos my +cosmz 1 D +sinx sin z m x + y z m=1 and D cos x cos z cos y. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

7 Hyperbolic orthoscheme The volume of a biorthogonal tetrahedron (orthoscheme) was calculated by Lobachevsky and Bolyai in H 3 and by Schläfli in S 3. Theorem 4 (J. Bolyai) The volume of hyperbolic orthoscheme T is given by the formula T D z A β C CD AB T T CBA BCD B α Vol (T )= tan γ z tanβ 0 z sinh zdz ( ). cosh z cos α 1 cosh z cos γ 1 Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

8 Hyperbolic orthoscheme The following theorem is the Coxeter s version of the Lobachevsky result. Theorem 5 (Lobachevsky, Coxeter) The volume of a hyperbolic orthoscheme with essential dihedral angles A, B and C A C B H 3 is given by the formula V = i S(A, B, C), 4 where S(A, B, C) is the Schläfli function. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

9 Ideal polyhedra Consider an ideal hyperbolic tetrahedron T with all vertices on the infinity Opposite dihedral angles of ideal tetrahedron are equal to each other and A + B + C = π. Theorem 6 (J. Milnor, 198) x Vol(T )=Λ(A)+Λ(B)+Λ(C), where Λ(x) = log sint dt is the Lobachevsky function. More complicated case with only one vertex on the infinity was investigated by E. B. Vinberg (1993). Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34 0

10 Ideal polyhedra Let O be an ideal symmetric octahedron with all vertices on the infinity. O: Then C = π A, D = π B, F = π E and the volume of O is given by Theorem 7 (Yana Mohanty, 00) ( Vol (O) = Λ ( π + A + B + E ( π + A B E +Λ ) +Λ ) ( π A B + E +Λ ( π A + B E )). ) Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

11 Volume of tetrahedron Despite of the above mentioned partial results, a formula the volume of an arbitrary hyperbolic tetrahedron has been unknown until very recently. The general algorithm for obtaining such a formula was indicated by W. Y. Hsiang (1988) and the complete solution of the problem was given by Yu. Cho and H. Kim (1999), J. Murakami, M. Yano (001) and A. Ushijima (00). In these papers the volume of tetrahedron is expressed as an analytic formula involving 16 Dilogarithm of Lobachevsky functions whose arguments depend on the dihedral angles of the tetrahedron and on some additional parameter which is a root of some complicated quadratic equation with complex coefficients. A geometrical meaning of the obtained formula was recognized by G. Leibon from the point of view of the Regge symmetry. An excellent exposition of these ideas and a complete geometric proof of the volume formula was given by Y. Mohanty (003). Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

12 Volume of tetrahedron We suggest the following version of the integral formula for the volume. Let T = T (A, B, C, D, E, F ) be a hyperbolic tetrahedron with dihedral angles A, B, C, D, E, F.Weset V 1 = A + B + C, V = A + E + F, V 3 = B + D + F, V 4 = C + D + E (for vertices) H 1 = A + B + D + E, H = A + C + D + F, H 3 = B + C + E + F, H 4 =0 (for Hamiltonian cycles). Theorem 8 (D. Derevnin and M., 005) The volume of a hyperbolic tetrahedron is given by the formula Vol (T )= 1 4 z z 1 log 4 i=1 cos V i +z sin H dz, i +z where z 1 and z are appropriate roots of the integrand. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

13 Volume of tetrahedron More precisely, the roots in the previous theorem are given by the formulas z 1 = arctan K K 1 arctan K 4 K 3, z = arctan K K 1 + arctan K 4 K 3 and K 1 = K = 4 (cos(s H i )+cos(s V i )), i=1 4 (sin(s H i )+sin(s V i )), i=1 K 3 =(sinasin D +sinb sin E +sinc sin F ), K 4 = K1 + K K 3, S = A + B + C + D + E + F. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

14 Volume of tetrahedron Recall that the Dilogarithm function is defined by Li (x) = x 0 log(1 t) dt. t We set l(z) =Li (e iz ) and note that I(l(z)) = Λ( z ). The following result is a consequence of the above theorem. Theorem 9 (J. Murakami, M. Yano, 001) Vol(T )= 1 I(U(z 1, T ) U(z, T )), where U(z, T )= 1 (l(z)+l(a + B + D + E + z) +l(a + C + D + F + z)+l(b + C + E + F + z) l(π + A + B + C + z) l(π + A + E + F + z) l(π + B + D + F + z) l(π + C + D + E + z)). Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

15 More deep history It is surprising that, more than a century ago, in 1906, the Italian mathematician G. Sforza found the formula for the volume of a non-euclidean tetrahedron. This fact became known during a discussion of the author with J. M. Montesinos at the conference in El Burgo d Osma (Spain) in August 006. Let G be Gram matrix for hyperbolic tetrahedron T.Weset c ij =( 1) i+j G ij, where G ij is ij-th minor of matrix G. Theorem 10 (G. Sforza, 1906) The volume of a hyperbolic tetrahedron T is given by the following formula Vol (T )= 1 4 A A 0 log where A 0 is a root of the equation det G =0. c 34 det G sin A c 34 + det G sin A da, Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

16 More deep history ProofofSforzaformula We start with the the following theorem. Theorem 11 (Jacobi) Let G =(a ij ) i,j=1,...,n be an n n matrix with det G =Δ.Denoteby C =(c ij ) i,j=1,...,n the matrix formed by elements c ij =( 1) i+j G ij, where G ij is ij-th minor of matrix G.Then det (c ij ) i, j=1,...,k =Δ k 1 det (a ij ) i, j=k+1,...,n. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

17 Sforza formula Apply the theorem to Gram matrix G for n =4and k = 1 cos A x x x x x x G = cos A 1 x x x x x x, C = x x x x x x c 33 c 34. x x x x x x c 43 c 44 We have By Cosine Rule c 33 c 44 c 34 =Δ(1 cos A). sinh l A = cosh l A = c 34 c33 c 44, hence c 34 c 33c 44 c 33 c 44 = sin A c33 c 44 Δ. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

18 Sforza formula Since exp(±l A )=coshl A ± sinh l A we have exp(l A )= c 34 +sina Δ c33 c 44, exp( l A )= c 34 +sina Δ c33 c 44. Hence, exp(l A )= c 34 +sina Δ c 34 sin A Δ, and l A = 1 log c 34 +sina Δ c 34 sin A Δ. By the Schläfli formula dv = 1 l α dα, α {A, B, C, D, E, F } A α l A V = ( )da = 1 4 A 0 A A 0 log c 34 ΔsinA c 34 + ΔsinA. The integration is taken over path from (A, B, C, D, E, F ) to (A 0, B, C, D, E, F ) where A 0 is a root of Δ=0. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

19 Symmetric polyhedra A tetrahedron T = T (A, B, C, D, E, F ) is called to be symmetric if A = D, B = E, C = F. Theorem 1 (Derevnin-Mednykh-Pashkevich, 004) Let T be a symmetric hyperbolic tetrahedron. Then Vol(T ) is given by π/ (arcsin(a cos t)+arcsin(b cos t)+arcsin(c cos t) arcsin(cos t)) dt sin t, Θ where a =cosa, b =cosb, c =cosc, Θ (0,π/) is defined by sin A sinh l A = sin B sinh l B = sin C sinh l C =tanθ, and l A,l B,l C are the lengths of the edges of T with dihedral angles A, B, C, respectively. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

20 Sine and cosine rules Sine and cosine rules for hyperbolic tetrahedron Let T = T (A, B, C, D, E, F ) be a hyperbolic tetrahedron with dihedral angles A, B, C, D, E, F and edge lengths l A,l B,l C,l D,l E,l F respectively. Consider two Gram matrices 1 cos A cos B cos F G = cos A 1 cos C cos E cos B cos C 1 cos D cos F cos E cos D 1 and 1 coshl D cosh l E cosh l C G = cosh l D 1 coshl F cosh l B cosh l E cosh l F 1 coshl A. cosh l C cosh l B cosh l A 1 Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

21 Sine and cosine rules Starting volume calculation for tetrahedra we rediscover the following classical result: Theorem 13 (Sine Rule, E. d Ovidio, 1877, J. L. Coolidge, 1909, W. Fenchel, 1989) sin A sin D sin B sin E sin C sin F det G = = = sinh l A sinh l D sinh l B sinh l E sinh l C sinh l F det G. The following result seems to be new or at least well-forgotten. Theorem 14 (Cosine Rule, M. Pashkevich and M., 005) cos A cos D cos B cos E det G = cosh l B cosh l E cosh l A cosh l D det G. Both results are obtained as a consequence of Theorem 11 relating complimentary minors of matrices G and G. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

22 Symmetric octahedra Octahedron O = O(a, b, c, A, B, C) having mmm symmetry Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, 010 / 34

23 Symmetric octahedra Theorem 15 (Sine-Tangent Rule, N. Abrosimov, M. Godoy and M., 008) Let O(a, b, c, A, B, C) be a spherical octahedra having mmm-symmetry. Then the following identities hold sin A tan a = sin B tan b = sin C tan c = T =K C, where K and C are positive numbers defined by the equations K =(z xy)(x yz)(y xz), C =xyz x y z +1, and x =cosa, y =cosb, z =cosc. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

24 Symmetric polyhedra: volume of mmm octahedron Theorem 16 (N. Abrosimov, M. Godoy and M., 008) Let O = O(A, B, C) be a spherical octahedron having mmm symmetry. Then volume V = V (O) is given θ ( ) dτ arth(x cos τ)+arth(y cos τ)+arth(z cos τ)+arth(cos τ) π cos τ, where X =cosa, Y =cosb, Z =cosc and 0 θ π/ is a root of the equation tan (1 + X )(1 + Y )(1 + Z ) θ + =0. 1+X + Y + Z Moreover, θ is given by the Sine Tangent rule sin A tan a = sin B tan b = sin C tan c =tanθ. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

25 Symmetric polyhedra: Euclidean mmm octahedron For the Euclidean case the following result holds. Theorem 17 (R. V. Galiulin, S. N. Mikhalev, I. Kh. Sabitov, 004) Let V be the volume of an Euclidean octahedron O(a, b, c, A, B, C) with mmm symmetry. Then V is a positive root of equation 9V =(a + b c )(a + c b )(b + c a ). Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

26 Symmetric polyhedra: octahedron with m symmetry D,d D,d m m Octahedron O = O(a, b, c, d, A, B, C, D), having m symmetry. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

27 Symmetric polyhedra: volume of m octahedron Theorem 18 (N. Abrosimov, M. Godoy and M., 008) Let O = O(A, B, C, D) be a spherical octahedron having m symmetry. Then the volume V = V (O) is given by θ ( arth(x cos τ)+arth(y cos τ)+arth(z cos τ)+arth(w cos τ)) dτ cos τ, π where X =cosa, Y =cosb, Z =cos C+D, W =cos C D and θ, 0 θ π/ is given by Sine Tangent rule sin A tan a = sin B tan b C+D sin = tan c+d = sin C D tan c d =tanθ. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

28 Symmetric polyhedra: Euclidean m octahedron For the Euclidean case the following result holds. Theorem 19 (R. V. Galiulin, S. N. Mikhalev, I. Kh. Sabitov, 004) Let V be the volume of an Euclidean octahedron O(a, b, c, d, A, B, C, D) with m symmetry. Then V is a positive root of equation 9V =(a +b c d )(a b + cd)(b a + cd). Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

29 Symmetric polyhedra: volume of spherical hexahedron C B B A C A C A B A B C Hexahedron combinatorial cube H(A, B, C) Theorem 0 (N. Abrosimov, M. Godoy and M., 008) Volume of a spherical hexahedron H(A, B, C) with mmm symmetry is equal Θ Re π (arcth ( X cos t )+arcth ( Y cos t )+arcth ( Z cos t )+arcth ( 1 dt cos t )) sin t, where Θ, 0 Θ π is defined by tan Θ+ (XYZ + X + Y + Z 1) 4(X + YZ)(Y + XZ)(Z + XY ) =0, X =cosa, Y =cosb and Z =cosc. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

30 Lambert cube The Lambert cube Q(α, β, γ) is one of the simplest polyhedra. By definition, this is a combinatorial cube with dihedral angles α, β and γ at three noncoplanar edges and with right angles at all other edges. The volume of the Lambert cube in hyperbolic space was obtained by R. Kellerhals (1989) in terms of the Lobachevsky function. We give the volume formula of the Lambert cube in spherical space. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

31 Lambert cube: spherical volume Theorem 1 (D. A. Derevnin and M., 009) π The volume of a spherical Lambert cube Q(α, β, γ), given by the formula <α,β,γ <π is where V (α, β, γ) = 1 4 (δ(α, Θ) + δ(β,θ) + δ(γ,θ) δ(π, Θ) δ(0, Θ)), and Θ, π δ(α, Θ) = π Θ < Θ <πis defined by tan Θ= K + L =tanα, M =tanβ, N =tanγ. dτ log(1 cos α cos τ) cos τ K + L M N, K =(L + M + N +1)/, Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

32 Lambert cube: hyperbolic volume Remark. The function δ(α, Θ) can be considered as a spherical analog of the function Δ(α, Θ) = Λ(α +Θ) Λ(α Θ). Then the main result of R.Kellerhals (1989) for hyperbolic volume can be obtained from the above theorem by replacing δ(α, Θ) to Δ(α, Θ) and K to K. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

33 Lambert cube: hyperbolic volume As a consequence of the above mentioned volume formula for Lambert cube we obtain Proposition 1 (D. A. Derevnin and M., 009) Let L (α, β, γ) be a spherical Lambert cube such that cos α +cos β +cos γ =1. Then Vol L (α, β, γ) = 1 4 ( π (π α) (π β) (π γ) ). Before a similar statement for spherical orthoscheme was done by Coxeter. Proposition (H. S. M. Coxeter, 1935) Let T (α, β, γ) be a spherical orthoscheme such that cos α +cos β +cos γ =1. Then Vol T (α, β, γ) = 1 4 (β ( π α) ( π γ) ). Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34

34 Rational Volume Problem The following problem is widely known and still open. Rational Volume Problem. Let P be a spherical polyhedron whose dihedral angles are in πq. Then Vol (P) π Q. Examples 1. Since cos π 3 +cos π 3 +cos 3π 4 =1, by Proposition 1 we have Vol L ( π 3, π 3, 3π 4 )= π.. Let P be a Coxeter polyhedron in S 3 (that is all dihedral angles of P are π n for some n N). Then the Coxeter group Δ(P) generated by reflections in faces of P is finite and Vol (P) = Vol (S 3 ) Δ(P) = π Δ(P) π Q. Alexander Mednykh (NSU) Volumes of polyhedra 08-1 March, / 34