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Platonic Compounds of Cylinders Platonic Compounds of Cylinders Oleg Ogievetsky1;2;3 and Senya Shlosman1;4;5 1Aix Marseille Universit´e,Universit´ede Toulon, CNRS, CPT UMR 7332, 13288, Marseille, France 2I.E.Tamm Department of Theoretical Physics, Lebedev Physical Institute, Leninsky prospect 53, 119991, Moscow, Russia 3Kazan Federal University, Kremlevskaya 17, Kazan 420008, Russia 4Inst. of the Information Transmission Problems, RAS, Moscow, Russia 5Skolkovo Institute of Science and Technology, Moscow, Russia ... and the left leg merely does a forward aerial half turn every alternate step. Monty Python Abstract Generalizing the octahedral configuration of six congruent cylinders touching the unit sphere, we exhibit configurations of congruent cylin- ders associated to a pair of dual Platonic bodies. arXiv:1904.02043v1 [math.MG] 3 Apr 2019 1 Contents 1 Introduction 3 2 Interpretation of the configuration O6: A4-symmetric configurations 7 3 Generalizations of the configuration O6: δ-rotation process 10 3.1 Neighboring tangent lines . 10 3.1.1 Pair O ........................... 11 3.1.2 Pair I ........................... 13 3.2 Comment about calculations in Subsection 3.1 . 15 4 Platonic compounds: global picture 17 4.1 Octahedron/Cube . 17 4.2 Icosahedron/Dodecahedron . 19 5 Minima 25 5.1 Octahedron/Cube . 26 5.2 Icosahedron/Dodecahedron . 27 6 Concluding remarks 32 References 35 2 1 Introduction In this paper we study compounds of cylinders. By a compound { of cylin- ders we mean a configuration of infinite open nonintersecting right circular congruent cylinders in R3; touching the unit sphere S2, such that some of the cylinders in { touch each other. We denote by r ({) the common radius of cylinders forming the compound. Let M k be the manifold of k-tuples of straight lines m = fu1; :::; ukg tangent to the sphere S2. The manifold M k has dimension 3k: The space of compounds of k cylinders can be conveniently identified with the manifold M k: the straight line corresponding to a cylinder is its unique ruling touching the unit sphere. Therefore the space of compounds of k cylinders is 3k- dimensional. For a configuration m we denote by d (m) the minimal distance between the lines of the configuration. If { is a compound with tangent rulings m, then d (m) r ( ) = : (1) { 2 − d (m) By a slight abuse of notation we allow the lines ui to intersect; then r ({) = d (m) = 0: For every k ≥ 3 let us define the number Rk = max r ({) : {2M k By definition, Rk is the maximal possible radius of k congruentp nonintersect- 2 ing cylinders touching S : It is easyp to see that R3 = 3 + 2 3, and a natural (open) conjecture is that R4 = 1 + 2: The compound C6 of six unit parallel cylinders has appeared in one question of Kuperberg [K] in 1990. For quite some time the common be- lief was that R6 = r (C6) = 1; until M. Firsching [F] in 2016 has found 6 6 by numerical exploration of the manifold M a compound {F 2 M with r ({F ) ≈ 1:049659: That finding has triggered our interest in the problem, 3 and in the paper [OS] we have discovered a curve C6(t) in the compound 6 space M , C6 (0) = C6; along which thep function r (C6 (t)) increases to its 1 maximal value on this curve, rm = 8 3 + 33 ≈ 1:093070331; taken at the compound Cm: The compounds C6 and Cm are shown on Figure 1 (the green unit ball is in the center). Figure 1: Two compounds of cylinders: the compound C6 of six parallel cylinders of radius 1 (on the left) and the compound Cm of six cylinders of radius ≈1:0931 (on the right) We have shown in [OS-C6] that in fact the compound Cm is a point of local maximum (modulo global rotations) of the function r : for every compound { close to Cm we have r ({) < rm: We believe that in fact 1 p R = r = 3 + 33 : 6 m 8 For values of k larger than 3 the quantities Rk are unknown; at the end of this paper we present some lower bounds for few values of k. 6 In his paper Kuperberg has presented yet another compound, O6 2 M with the same radius r (O6) = 1. 4 The compound O6 is shown on Figure 2; the central unit ball is again shown in green. Figure 2: Compound O6 Contrary to C6, the compound O6 is indeed a point of local maximum (modulo global rotations) of the function r : for every compound { close to O6 one has r ({) < 1: This was proved in our paper [OS-O6]. The configuration O6 of tangent lines is shown on Figure 3. Figure 3: Configuration O6 of tangent lines 5 Sometimes the configuration O6 is called `octahedral' because, probably, the tangency points lie at the vertices of the regular octahedron. We present an interpretation of the configuration O6 which relates it to the configuration of rotated edges of the regular tetrahedron. Following this interpretation we introduce configurations of tangent lines for each pair of dual Platonic bodies, that is, for the tetrahedron/tetrahedron, octahedron/cube and icosa- hedron/dodecahedron; for the pair tetrahedron/tetrahedron this is precisely the locally maximal configuration O6. The number of lines in our configu- rations is equal to the number of edges of either of Platonic bodies in the pair, that is, twelve for the pair octahedron/cube and thirty for the pair icosahedron/dodecahedron. We recall the definition of a critical configurations from [OS-O6]. We call a cylinder configuration critical if for each small deformation t that keeps the radii of the cylinders, either (T1) some cylinders start to intersect, or else (T2) the distances between all of them increase, but by no more than ∼ ktk2 ; or stay zero, for some. Here ktk is the natural norm, see details in [OS-O6]. A critical configuration is called a locally maximal configuration if all its deformations are of the first type. Any other critical configuration is called a saddle configuration, and the deformations of type (T2) are then called the unlocking deformations. For the pair octahedron/cube we find two possible candidates for the critical points; For the pair icosahedron/dodecahedron we find four possible candidates for the critical points. The paper is organized as follows. In Section 2 we present an interpreta- tion of the configuration O6 relating it to the edges of the tetrahedron. In Sections 3, 4 and 5 we describe configurations of tangent lines associated to a pair of dual Platonic bodies. We conclude the article by several conjectures concerning local maxima and critical points for the configurations related to the pairs octahedron/cube and icosahedron/dodecahedron. 6 2 Interpretation of the configuration O6: A4-symmetric configurations We shall now give an interpretation of the configuration O6 which shows that it rather deserves to be called the tetrahedral configuration. There is a family of configurations possessing the A4 symmetry (A4 is the group of proper symmetries of the regular tetrahedron, alternating group on four letters). We start with the configuration of the tangent to the unit sphere lines which are continuations of the edges of the regular tetrahedron. The points of the sphere at which tangent lines pass are the edge middles of the regular tetrahedron. The initial position of the edges of the tetrahedron are shown in blue on Figure 4. Figure 4: Sphere tangent to tetrahedron edges Then each edge is rotated around the diameter of the unit sphere, passing through the middle of the edge, by an angle δ. On Figure 4 the point A (in green) is the middle of the edge UV . The line, passing through the point A and rotated by the angle δ, is shown in red. The lines passing through other middles of edges are rotated according to A4 symmetry. The positions of other lines are well defined because the stability subgroup of the edge UV of the regular tetrahedron is generated by the rotation by angle π around the diameter of the sphere passing through the point A and, clearly, the red line on Figure 4 is invariant under this rotation. The minimum of the squares of the distances between the lines is given 7 by the formula 16T 2 4 sin2(2δ) d2 = ≡ − : (2) (3T 2 + 1)(T 2 + 3) cos(2δ) − 2 cos(2δ) + 2 Here T = tan(δ). When δ changes from 0 to π=2, the value of d2 increases, achieves the maximal value 1 for δ0 = π=4, and afterwards decreases to 0, see Figure 5. Figure 5: Graph of d2(T ) For δ 6= 0; π=4; π=2 the symmetry of the configuration is A4. For δ = 0 the symmetry group is S4, the full symmetry group of the regular tetrahedron. For δ = π=4 the configuration is centrally symmetric, its symmetry group is A4 × C2, where the cyclic group C2 is generated by the central reflection I. For δ = π=4 this is exactly the configuration O6. For δ = π=2 the tangent lines become again the continuations of the edges of the regular tetrahedron. The convex span of the edge middles of the regular tetrahedron is the regular octahedron. Our above observation boils down to the fact that ro- tating the edges in the plane tangent to the mid-sphere (the sphere which touches the middle of each edge of P) we obtain the configuration O6 when the angle of rotation is equal to π=4. The following pictures of models made with the help of a tensegrity kit (Figure 6) and its Martian version (Figure 7) might be helpful for under- standing.
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