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Figure 1: Convex oblique triangular bipyramid p(t) theory of flexible polyhedra. For more details about this theory the reader is referred to [8], [5], and the references given there. 2. Basic definitions. In this section, we specify the terminology used. We study compact boundary-free polyhedral surfaces in Euclidean 3-space. For short, we call them polyhedral surfaces or polyhedra. A polyhedral surface is called convex if it is the boundary of a convex set. Bipyramid is any polyhedral surface combinatorially equivalent to the boundary of the convex hull of a regular tetrahedron and its image under reflection in one of its faces. Two polyhedral surfaces, p and q, are called intrinsically isometric if there exists a one-to-one correspondence f : p q, which preserves the length of any curve. Such f : p q is called an intrinsic deformation→ of a polyhedral surface p. An intrinsic deformation f : p → q is called volume increasing if p is orientable and the 3-dimensional volume enclosed by p is→ less than the 3-dimensional volume enclosed by q. 3. Auxiliary polyhedron p(t). In this section, we build and study a continuous family of auxiliary convex oblique triangular bipyramids p(t). The parameter t is a real number, which takes values from an interval 0 vertex of p(t). C is the middle point of the edge EF . It is shown in Fig. 1 for convenience of further explanations. The bipyramid p(t) is assumed to be symmetric with respect to the planes of the triangles ABD and AEF . By definition, the lengths of some edges of p(t) are as follows: AB = AD = 10, BE = BF = DE = DF = 13, and EF = 24. The lengths of the edges| | AE| and| AF are| supposed| | to| be| equal| to| each| other and depend| | on the parameter t, which has the following geometric meaning: t is equal to the nonoriented angle between the straight line segments AB and AC. According to this definition, t may take any value from 0 to t0 = π/6. The value t0 = π/6 corresponds to the case, when the triangles BEF and DEF lie in a common plane and, thus, the quadrilateral ABCD is in fact the regular triangle, whose sides have the lengths 10. In order to express AE as a function of t, we proceed as follows. Applying the Pythagorean theorem to the right triangle| | BCE, we find BC = 5. The cosine law applied to the triangle ABC yields BC 2 = AB 2 + AC 2 2 AB AC cos| t.| Thus, AC = 10cos t √100 cos2 t 75. In order to obtain| a| convex| | bipyramid| | −p(|t), we|| must| choose the| maximal| value± of AC in− the previous formula. So, we get AC = 10cos t + √100 cos2 t 75. Now, applying the Pythagorean| | theorem | | −

2 E′

D′

A′ C′

B′

F′

Figure 2: Nonconvex oblique triangular bipyramid q(t) to the right triangle ACE, we find

AE = q69 + 200cos2 t + 20cos t 100 cos2 t 75. (1) | | p − In order to provide the reader with a better opportunity to visualize the spatial form of the bipyramid p(t) for small values of t, we expand the right-hand side of the formula (1) in the Maclaurin series 117 AE =3√41 t2 + O(t4), (2) | | − 2√41 and note that 3√41 19.209 .... We conclude this≈ section with the computation of the volume of p(t), vol p(t), as a function of 1 t. Obviously, area(ABC)= 2 AB AC sin t, where area(ABC) stands for the area of the triangle ABC. On the other hand, vol|p(t)=|| 2|EF area(ABC). After simplifications, this yields 3 | | vol p(t) = 80(10cos t + 100 cos2 t 75) sin t. (3) p − Expanding the right-hand side of the formula (3) in the Maclaurin series, we get

vol p(t) = 1200t 1400t3 + O(t5). − In particular, it follows from the latter formula that vol p(t) 0 as t 0. 4. Auxiliary polyhedron q(t). In this section, we build→ and study→ a continuous family of auxiliary nonconvex oblique triangular bipyramids q(t). The parameter t is a real number, which ′ ′ takes values from an interval 0

3 (ii) suppose the straight line segment XY entirely lies in p(t) (i. e., XY is either an edge of p(t) or one of the following straight line segments: BC, CD, CE, CF ); (iii) suppose also that the straight line segment X′Y ′ entirely lies in q(t); (iv) then we put by definition X′Y ′ = XY . It follows immediately from this| definition| | that| q(t) is intrinsically isometric to p(t) as soon as p(t) and q(t) exist. From Sect. 3 we know that p(t) does exist for 0 t

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ C E A C < A E < C E + A C | | − | | | | | | | | or, equivalently,

′ ′ 3.3397 ... 12 5√3 < A E < 12+5√3 20.6603 .... (4) ≈ − | | ≈ Using the formula (2), we find A′E′ = 3√41 19.209 ... for t = 0 and conclude that (4) holds true for t = 0. It follows from (1)| that| A′E′ ≈is a continuous function in t in a neighborhood of | | ′ the point t = 0. Thus, there exists a constant t0 > 0 such that the inequalities (4) hold true for ′ ′ all 0

A′E′ 2 69 438 A′E′ 2 A′E′ 4 692 cos α = | | − . Thus sin α = p | | − | | − . 10√3 A′E′ 10√3 A′E′ | | | | Substituting the expressions for area(A′B′D′) and sin α to (5), we get 5 vol q(t)= 438 A′E′ 2 A′E′ 4 692. (6) 3p | | − | | − Recall that A′E′ = AE and AE is the function in t defined by the formula (1). Substitute (1) in (6) and expand| | the| right-hand| | side| of the formula obtained in the Maclaurin series (the result of substitution itself is too complicated and we prefer not to wright it here). The first two members of the Maclaurin series are as follows 3750 vol q(t)=50√23 + t2 + O(t4). √23

In particular, we conclude from the latter formula that vol q(t) 50√23 239.79 as t 0. ∗ → ≈ ∗ → ′ 6. Proof of Theorem 1. Given a constant c> 0, find t such that 0 c. (7) vol p(t∗)

We can satisfy (7) because we know from Sect. 4 that vol q(t) 50√23 as t 0 and we know from Sect. 3 that vol p(t) 0 as t 0. From Sect. 4 we also know→ that p(t∗) and→q(t∗) are intrinsically isometric to each other.→ By→ definition, put P = p(t∗) and Q = q(t∗). The bipyramids P and Q satisfy all the conditions of Theorem 1. 

4 References

[1] Alexandrov, A.D.: Convex polyhedra. Springer, Berlin (2005). Zbl 1067.52011 [2] Alexandrov, V.A.: How one can crush a milk carton in such a way as to enlarge its volume. Soros. Obraz. Zh. (2), 121–127 (2000) (in Russian). Zbl 0947.52012 [3] Bleecker, D.D.: Volume increasing isometric deformations of convex polyhedra. J. Differ. Geom. 43(3), 505–526 (1996). Zbl 0864.52003 [4] Burago, Yu.D., Zalgaller, V.A.: Isometric piecewise-linear immersions of two-dimensional man- ifolds with polyhedral metric into R3. St. Petersbg. Math. J. 7(3), 369–385 (1996). Russian original published in Algebra Anal. 7(3), 76–95 (1995). Zbl 0851.52018 [5] Gaifullin, A.A.: Generalization of Sabitov’s theorem to polyhedra of arbitrary dimensions. Discrete Comput. Geom. 52(2), 195–220 (2014). Zbl 1314.52008 [6] Milka, A.D., Gorkavyy, V.A.: Volume increasing bendings of regular polyhedra. Zb. Pr. Inst. Mat. NAN Ukr. 6(2), 152–182 (2009) (in Russian). Zbl 1199.52007 [7] Pak, I.: Inflating the cube without stretching. Am. Math. Mon. 115(5), 443–445 (2008). Zbl 1145.52006 [8] Sabitov, I.Kh.: Algebraic methods for solution of polyhedra. Russ. Math. Surv. 66(3), 445–505 (2011). Zbl 1230.52031 [9] Samarin, G.A.: Volume increasing isometric deformations of polyhedra. Comput. Math., Math. Phys. 50(1), 54–64 (2010). Zbl 1224.52023

Victor Alexandrov Sobolev Institute of Mathematics Koptyug ave., 4 Novosibirsk, 630090, Russia and Department of Physics Novosibirsk State University Pirogov st., 2 Novosibirsk, 630090, Russia e-mail: [email protected]

Originally submitted to arXive: July 22, 2016

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