Algebra versus analysis in the theory of flexible polyhedra

ALGEBRA VERSUS ANAL YSIS IN THE THEOR Y OF FLEXIBLE POL YHEDRA VICTOR ALEXANDRO V Abstract. Two basic theorems of the theory of flexible p olyhedra were prov en b y completely differen t methods: R. Alexander used analysis, namely , t he Stok es theorem, to prov e th at the total mean curv ature r emains constan t dur- ing the flex, while I.Kh. Sabitov used algebra, namely , the theory of r esultan ts, to prov e that the orien ted volume remains con stan t dur ing th e flex. W e show that n one o f t hese met hods can be used to prov e the b oth theorems. As a b y-pro duct, we pro v e that the total mean curv ature of an y p olyhedron in the Euclidean 3-space is not an algebraic function of its edge length s. A po lyhedron (mor e pr e c isely , a p olyhedr a l surface) is said to be flexible if its spatial shap e can b e changed contin uo usly due to changes o f its dihedr al angle s only , i. e., if every face remains congrue nt to itself during the flex. I n other words, a p olyhedro n P 0 is flexible if it is included in a contin uous family { P t } , 0 6 t 6 1, of p olyhedr a P t such tha t, for every t , the cor resp onding faces of P 0 and P t are congruent while the p olyhedr a P 0 and P t are not congruent. The family { P t } , 0 6 t 6 1, is called the flex o f P 0 . Self-int ersectio ns a re p ossible bo th for P 0 and P t provided the con verse is not for m ulated explicitly . Without loss of gener ality we assume that the faces of the p olyhedra a re triangular. Flexible self-intersection free sphere-homeomor phic poly hedra in Euclidea n 3- space were constructed by R. Connelly thirty years ago [4], [6]. Since that time, v arious non-trivial properties of flex ible polyhedra were discov ered in the Euclidean 3-space [10] and 4-space [12] (for r esults in the h ype r bo lic 3-space, see [13]). Let us formulate tw o o f them in a form suitable for our purp oses . Let P b e a closed orient ed p olyhedr on in R 3 , let E b e the set of its edg es, le t | ℓ | be the length of the edge ℓ , and let α ( ℓ ) b e the dihedral ang le o f P at the edge ℓ measured from inside of P . The sum M ( P ) = 1 2 X ℓ ∈ E | ℓ |  π − α ( ℓ )  is called the total me an curvatur e of P . Theorem 1. L et P 0 b e a flexible p olyhe dr on in R 3 and let { P t } , 0 6 t 6 1 , b e its flex . The total me an curvatur e M ( P t ) is indep endent of t .  Theorem 1 w as proved by R. Alexander [2] for all E uclidean n -spaces , n > 3, though no example of a flexible p olyhedr o n is kno wn for n > 5. Date : July 6, 2010. 1991 Mathematics Subje ct Classific ation. Pr imary 52 C25; Secondary 51M20. Key wor ds and phr ases. Flexible polyhedron, volume, infinitesimal bending, total mean cur- v ature, algebraic function. The author was supp orted by the F ederal Program ‘Research and educational resourses of inno v ative Russia in 2009–2013’ (con tract 02.740.11.0457 ) and the Russian F oundation for Basic Researc h (gran t 10–01–9100 0–ANF). 1 2 VICTOR ALEXANDR OV Let P 0 be a flexible p olyhedro n and let { P t } , 0 6 t 6 1, b e its flex. Let r t be the p oint of the p olyhedron P t which corresp onds to the p oint r 0 ∈ P 0 . It follows from the definition of a flexible p olyhedro n that, for a ny curve γ 0 ⊂ P 0 , the length of the cur ve γ t = { r t | r 0 ∈ γ 0 } ⊂ P t is indep endent of t . The rea der can easily verify the well-known fact that, for a n y curve γ 0 ⊂ P 0 , the length of the curve e γ t = { r 0 + t v | r 0 ∈ γ 0 } is stationa ry a t t = 0 , where v = d dt | t =0 r t is the velocity vector of the p oint r t at t = 0. Obviously , the vector fie ld v is linear on every face of P 0 . This leads to the follo wing well-kno wn definition: a vector field w on a po lyhedron P , which is linear on ev ery face o f P , is said to b e its infinitesimal flex if, for any curve γ ⊂ P , the length of the curve e γ ( t ) = { r + t w | r ∈ γ } is stationary at t = 0, see [7] for mor e detail. Of course, the velocity vector field of a flexible po lyhedron is its infinitesimal flex (but the conv er se is no t necessarily true). In [2 ] Theo r em 1 w a s obtained as an obvious corollary of the following Theorem 2 prov ed for R n , n > 3 . Theorem 2. L et P b e a close d oriente d p olyhe dr on in R 3 , let w b e its infinites- imal flex , and let P ( t ) = { r + t w | r ∈ P } . Then d dt | t =0 M  P ( t )  = 0 .  In [2] Theorem 2 was pr ov ed with the help of the Stok es theorem. Later several authors observed (see, e. g., [3], [1 1]) that Theorem 2 follows immediately from the Schl¨ afli differential formula, whic h, in turn, is based on the Stokes theorem. In any case, all known pr o ofs of Theorems 1 and 2 belo ng to Analysis. In [8] I.Kh. Sabitov prov ed ano ther highly non-trivial prop erty of the flexible po lyhedra that may b e form ulated as follows and who se many-dimensional analog is not known yet. Theorem 3. If { P t } is a fl ex of an orientable p olyhe dr on in R 3 , then the oriente d volume of P t is c onstant in t .  In [8], [9], and later in [5] Theore m 3 was obtained as an obvious co rollary of the following Theore m 4 v alid since every co n tinuous mapping, whose image is a finite set, is constant. Theorem 4. F or t he set P K of al l (not ne c essarily flex ible) close d p olyhe dr a in R 3 with triangular fac es and with a pr escrib e d c ombinatorial struct ur e K ther e exists a universal p olynomial p K of a single variabl e whose c o efficients ar e universal p olynomials in the e dge lengths of a p olyhe dr on P ∈ P K and su ch that the oriente d the volume of any P ∈ P K is a r o ot of p K .  In [8] and [9] The o rem 4 was proved with the help o f the theory o f resultants, while in [5] it was proved with the help of the theory o f places. In any ca se, all known pro o fs o f Theorems 3 and 4 belo ng to Algebra. Now reca ll that the deriv ativ e of the volume o f a def ormable doma in in R 3 equals o ne third of the flux acr oss the b oundary of the velo city vector o f a p oint of the b oundary of the domain, i. e., equals one third o f the in tegral of the normal comp onent o f the velo cit y ov er the b oundary . Hence, an ‘infinitesimal version’ of Theorem 3 should rea d that, for every orientable p olyhe dr on P , the flux R P ( w , n ) dP of any infinitesimal flex w e quals zer o. Theorem 5 s hows that this is not the cas e and, th us, that Theorem 3 can not be prov ed by means of Analysis lik e Theorem 1. Theorem 5 . Ther e is a close d oriente d p olyhe dr on P in R 3 with the fol lowing pr op erties : (i) the flux acr oss P of some infinitesimal fl ex w of P is non-zer o ; (ii) P c ontains no vertex V whose star lies in a plane ; FLEXIBLE POL YHEDRA: ALGEBRA VS ANAL YSIS 3 A B C D V A B C D V B 1 Figure 1. Polyhedron T 1 Figure 2. Polyhedron T 2 (iii) P c ontains no vertex V such that some thr e e e dges of P incident to V lie in a plane. Pr o of. Let T 0 = AB C D b e an arbitrar y tetrahedron in R 3 . L e t a p olyhedron T 1 be obtained from T 0 by triang ulation of the fa ce AB C that uses one additional vertex V (see Fig . 1). Let a vector field w 1 be linea r o n each face of T 1 , be equa l zero at the v ertices A , B , C , and D , and be equal a non-zero vector p erp endicular to the fa c e AB C at the vertex V . It is eas y to chec k that w 1 is a n infinitesimal flex o f T 1 and its flux a cross T 1 is non-zero (b ecause the scalar pro duct ( w 1 , n ) is either everywhere non-negative or everywhere non-p ositive; here n stands for the unit normal v ector field on T 1 ). Hence, T 1 satisfies the condition (i), though it do es not satisfy the conditions (ii) a nd (iii). Let B 1 be an a rbitrary p oint in R 3 which do es not lie in the plane AB C . Replace the face AC V o f the po lyhedron T 1 with the lateral surface of the triangula r pyramid AC V B 1 with the ba s e AC V . D enote the resulting po ly hedron b y T 2 (see Fig . 2 ). It is easy to check that the infinitesimal flex w 1 of T 1 can be extended in a unique wa y to an infinitesimal flex of T 2 . Denote it by w 2 and obser ve that the flux of w 2 across T 2 equals the flux of w 1 across T 1 (beca use the flux of w 2 across the triangular p yramid AC V B 1 equals zero). Hence, T 2 satisfies the conditions (i) and (ii), though it do es not sa tisfy the condition (iii). Recall the constr uction of the Brica rd flexible o cta he dr on o f t yp e 1. Consider a dis k -homeomorphic piece-wise linear surface S in R 3 comp osed of fo ur triangles AB V , B A 1 V , A 1 B 1 V , and B 1 AV such that | AB | = | A 1 B 1 | and | B 1 A | = | B A 1 | (see Fig. 3 ). It is known [6] that such a spatial quadrilateral AB A 1 B 1 is symmetric with resp ect to the line L pa ssing thro ugh the middle p oints of its diagonals AA 1 and B B 1 . Glue tog e ther S and its symmetric image with resp ect to L (see Fig. 4). The resulting p olyhe dr al surfa ce with self-in tersections is flexible (b ecause S is flex ible) and is combinatorially equiv alent to the surface of the regular o ctahedron. This is known as the Brica rd o ctahedron of type 1. Construct the B ricard o ctahedron of type 1 suc h that its triangles AB V and AV B 1 are co ngruent with the co r resp onding triang les AB V and AV B 1 of the p oly- hedron T 2 . Remove tho se tria ngles from T 2 and fro m the Brica r d oc ta hedron and glue together the remaining pa rts of those p olyhedr a. Obse r ve that the r esulting po lyhedron P satisfies the conditions (i)–(iii). In fact, the infinitesimal flex w 2 of T 2 can b e extended in a unique w ay to an infinitesimal flex of P . Denote it by w and observe that the flux of w across P equals the flux o f w 2 across T 2 (beca use 4 VICTOR ALEXANDR OV A B V A 1 B 1 L A B V A 1 B 1 Figure 3. Disk S Figure 4. Bric ard octahedr on the restriction o f the flux w o n the Bricard octahedr on is generated b y some its flex and, th us equals zero, s ince, e. g., the oriented v olume of the Bricard o cta hedron is known to b e consta n t for every flex [5], [7 ]–[10]).  Observe that an ‘alg ebraic version’ o f Theor em 2 should read that the total me an curvatur e of any oriente d p olyhe dr on is a r o ot of some universal p olynomial p in a single variable whose c o efficients ar e universal p olynomials in the e dge lengths of the p olyhe dr on. This statement would, obviously , imply Theor em 1. But this statement a lso implies that the tota l mean curv a ture of a polyhedr on is a n algebr a ic function of its edge lengths. Theorem 6 shows that this is not the case and, thus, that Theorem 1 can not b e prov ed b y means of Algebra like Theorem 3. Theorem 6. The total me an curvatur e of any close d oriente d p olyhe dr on in R 3 is not an algebr aic function of its e dge lengths. Pr o of. F o r ev ery K , the class of all poly hedra of combinatorial type K contains a family ∆ K of p olyhedr a depending on a sing le indep enden t v ariable l > 0 which may b e describ ed as follows (see Fig. 5): there are t wo tria ngles AB C and AC D in R 3 such tha t ( α ) | B C | = | C D | = | B D | = 1 and | AB | = | AC | = | AD | = l ; ( β ) every P ∈ ∆ K has the triangles AB C and AC D as their face s ; ( γ ) every vertex of P ∈ ∆ K either coincides with A , B , C , or D or is an int erior po in t of the segment B D or is an interior point of the triangle B C D ; ( δ ) the set o f the interior p oints of every edge of P ∈ ∆ K either c o incides with AB , AC , or AD o r lies in the open segment B D or lies in one of the open triangles AB D or B C D . A dir ect calcula tion shows that, for every P ∈ ∆ K , the total mea n curv a ture M ( P ) of P equals M ( P ) = 3 2  π − ϕ ( l )  + 3 2 l  π − ψ ( l )  = 3 2  π − arccos 1 2 √ 3 √ 4 l 2 − 1  + 3 2 l  π − arccos 2 l 2 − 1 4 l 2 − 1  , where ϕ ( l ) is the inner dihedral angle o f P at the edge B C and ψ ( l ) is the inner dihedral angle of P at the edge AC . Consider the right-hand side of the last for mu la as a function o f complex v ar iable l . Obviously , this function has a non-algebra ic singularity (known also a s a lo garithmic branch p oint) ov er l = 0 (as well as over l = ± 1 / √ 3 and l = ± i p 48 / 13). Recall that a function w = f ( z ) of a single complex v ariable z is called algebraic, if there is a po lynomial p ( w , z ) in tw o v a riables which do es not v anish ident ically and such that p ( f ( z ) , z ) ≡ 0. It is known that an a nalytic function of a single FLEXIBLE POL YHEDRA: ALGEBRA VS ANAL YSIS 5 A B C D Figure 5. Polyhedron of the class ∆ K complex v ar iable is an algebraic function if and only if it has a finite num ber of branches and a t most algebra ic singularities [1, p. 306]. Hence, the total mean curv ature M ( P ) is not an algebraic function of the v ariable l for the p olynomials P of the cla ss ∆ K . Th us, M ( P ) is not a n alg ebraic function of the edge lengths of the p olyhedron P .  References [1] L. Ahlfors, Comp lex analysis , 3rd ed. McGra w-H i ll, 1979. [2] R. Alexander, Lipschitzian mappings and total me an curvatur e of p olyhe dr al surfac es, I, T r ans. Amer. M ath. So c. 288 (1985), 661–67 8. [3] F. Almgren and I. Rivin, The me an curvatur e inte gr al is inv ariant under b ending, in: The Epstein Birthda y Sc hrift, Unive rsity of W arwick, 1998, pp. 1–21. [4] R. Connelly , Conje ctur e s and o p en questions in rigidity, in: Pro c. Int. Congr. Math., Hels inki 1978, V ol. 1, 1980, pp. 407 –414. [5] R. Connelly , I. Sabitov, and A. 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Appl. 20 (2004), 31–45. [12] H. Stac hel, Flexible cross-p olytop es in the Euclidean 4- space, J. Geom. Graph. 4 (2000), 159–167. [13] H. Stac hel, Fl exible octahedra in the h yper bolic space, in: A. Pr´ ek opa et al. Non-Euclidean geometries. J´ anos Bolyai memorial volume, Springer, New Y ork, 2006, pp. 209–225 . Sobolev Institute of Ma thema tics, Kopt yug a ve., 4, Novo sibirsk, 63 0090, Russia and Dep ar tment of Ph ysics, Novosibirsk St a te University, Pirogov str., 2, Novosibirsk, 630090, Russia E-mail addr ess : alex@math.nsc. ru