PARTIAL COVERING OF A CIRCLE BY EQUAL CIRCLES. PART II: THE CASE OF 5 CIRCLES

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1 PARTIAL COVERING OF A CIRCLE BY EQUAL CIRCLES. PART II: THE CASE OF 5 CIRCLES Zsolt Gáspár, Tibor Tarnai, Krisztián Hincz Abstract. How must n equal circles of given radius r be placed so that they cover as great a part of the area of the unit circle as possible? In this Part II of a two-part paper, a conjectured solution of this problem for n = 5 is given for r varying from the maximum packing radius to the minimum covering radius. Results are obtained by applying a mechanical model described in Part I. A generalized tensegrity structure is associated with a maximum area configuration of the 5 circles, and by using catastrophe theory, it is pointed out that the equilibrium paths have bifurcations, that is, at certain values of r, the type of the tensegrity structure and so the type of the circle configuration changes. 1 Introduction Zahn [18] raised the following mathematical problem: How must n equal circles of given radius r be arranged so that the area of the part of the unit circle covered by the n circles will be as large as possible? In this problem r max (n) r R (n) min, where r(n) max is the maximum radius of n equal circles that can be packed in the unit circle without overlapping, and R (n) min is the minimum radius of n equal circles which can cover the unit circle without interstices. So, Zahn s intermediate problem lies between the densest circle packing problem and the thinnest circle covering problem with respect to the unit circle. It is apparent that if r varies continuously from r max (n) to R (n) min, a transition from the maximum packing to the minimum covering is obtained. The present paper deals with the intermediate problem for n = 5 circles. A general survey of the best results obtained so far for arrangements of n equal circles on the unit circle for the packing, covering and intermediate problems has been given in Part I [6]. To our knowledge, the first published result concerning the best packing configuration of 5 circles was provided by Pacioli [12] more than five hundred years ago. In the contemporary mathematics, without knowing Pacioli s result, Kravitz [8] came up with the same arrangement. The fact that Pacioli s and Kravitz s conjectured solution is the real solution of the packing problem was proved independently of each other by Graham [7] and Pirl [13]. Budapest University of Technology and Economics, gaspar@ep-mech.me.bme.hu Budapest University of Technology and Economics, tarnai@ep-mech.me.bme.hu Budapest University of Technology and Economics, hincz@eik.bme.hu JoCG 5(1), ,

2 The problem of the minimum covering of a circle with 5 equal circles was raised in connection with a fairground game that was popular around the turn of the nineteenth to the twentieth century. As Rouse Ball and Coxeter [15] write: The problem of completely covering a fixed red circular space by placing over it, one at a time, five smaller equal circular tin discs is familiar to frequenters of English fairs. Its effectiveness depends on making the tin discs as small as possible, and therefore leads to the interesting geometrical question of finding the size of the smallest tin discs which can be used for the purpose. Neville [10] provided a numerical solution to this problem, but optimality of his covering configuration was proved only almost seventy years later by Bezdek [1]. Curiously enough, however, as pointed out by Melissen [9], the numerical values of the minimum radius given by Neville, and by Bezdek were different from each other, but both were in error. The correct numerical value of the minimum radius of the 5 equal circles was given by Melissen [9], and Nurmela [11]. As to the intermediate problem, very few results are available. For 5 equal circles, the only known results are due to Zahn [18] who provided numerical solutions for four different values of r(r = k/16, k = 6, 7, 8, 9). The three-digit accuracy he used in his calculations was acceptable to give good approximation of the numerical value of the maximum covered area, but in some cases it was not sufficient to determine the correct circle configuration and to identify its symmetry. For n = 2 equal circles of radius r such that 0.5 < r < 1, Zahn proved that, in the maximum area configuration, the centres of the two equal circles and the centre of the unit circle are collinear, and in the three double overlaps that appear (one formed by the two equal circles, and two formed by one of the equal circles and the exterior of the unit circle) the chord lengths are equal. For n = 3 and 4, the centres of the equal circles form an equilateral triangle and a square, respectively, in both the maximum packing and the minimum covering configurations. Therefore one may think that, in the transition from packing to covering, the structure of the intermediate circle arrangements remains unchanged, that is, the centres of the circles form equilateral triangles and squares for all maximum area configurations. For n = 3 this is indeed the case. By using the mechanical model, and the method outlined in Part I [6] and applied in the present work, we could show numerically that the threefold symmetry of the configurations is maintained. For n = 4, however, contrary to expectations, the fourfold symmetry of the packing and covering configurations is reduced to twofold in between (the square is deformed to rhombi which then regain the shape of a square). The details of these investigations for n = 3 and 4 (together with those for n = 6 and 7) will be published elsewhere. The next value of n for consideration is 5, but a more substantial reason can be advanced why the number 5 in circle arrangements on the unit circle is of particular interest. The main point is that 5 is the smallest value of n for which the structure of the maximum packing configuration and that of the minimum covering configuration are substantially different. In the maximum packing and in the minimum covering the centres of the circles form a regular pentagon and an elongated pentagon having bilateral symmetry, respectively. It is obvious that, in the transition from packing to covering, the structure of the intermediate circle arrangements should change. Thus, it is an important question asked also by Connelly [2]: How does the intermediate arrangement change for n = 5 when JoCG 5(1), ,

3 r varies from the maximum packing radius to the minimum covering radius? The aim of this paper is to answer this question. By using the mechanical models and general procedures shown in Part I [6], in the case n = 5, the solution to the intermediate problem for different values of r will be presented. According to the method in [6], for a given value of r, a locally optimal arrangement is looked for by proceeding along stable equilibrium paths. In the transition from packing to covering, it can occur, however, and for n = 6 it does occur (this will be published in the mentioned forthcoming paper), that there is no stable equilibrium path leading to a locally optimal configuration, that is, this configuration can be reached only by proceeding along an unstable equilibrium path. Since the existence of such a locally optimal configuration is not known in advance, unstable equilibrium configurations should also be investigated. Although for n = 5 we have found that the case above does not occur, to make the study complete we will show configurations along unstable equilibrium paths we investigated in search for a locally optimal arrangement. Although the solutions to the packing and covering problems for n = 5 are already known it will also be demonstrated how to find the (locally) optimal arrangements by using mechanical models in these cases. An extensive computational study will be reported in the present paper. The outline of the rest of the paper is the following. In Section 2, we demonstrate how the optimality of a known packing can be shown by using a mechanical model. In Section 3, we show the mechanical models of more covering constructions. Partial coverings are discussed in Section 4. First, the applied generalized tensegrity is presented (Section 4.1), then we introduce the method of dynamic relaxation by which, in many cases, the optimal configurations are obtained (Section 4.2). In Section 4.3, with a procedure using the stiffness matrix, we investigate the equilibrium path, all branches one by one. We determine their singular points (we classify them), at each segment of the equilibrium path we show which elements of the tensegrity are active and whether the respective equilibrium configurations are stable. We present the data of the characteristic points of the equilibrium paths in the form of tables. The obtained results are summarized in Section 5. 2 The maximum packing The optimal (maximum) packing of 5 equal circles in the unit circle can be found with uniform heating of a planar bar-and-joint assembly (tensegrity structure composed of struts and joints) associated with the circle system. The joints of the assembly are the centres of the equal circles, and the bars connect the centres of two circles touching each other. Thus the length of each bar is 2r. The joints at a distance of 1 r from the centre of the unit circle are supported against displacement in the radial direction. (One of these joints is also supported against tangential displacement in order to prevent rotation of the whole assembly about the centre of the unit circle.) If a state of self-stress can appear in an arrangement, then the arrangement is (locally) optimal provided no tensile force appears in any of the bars (struts), and consequently no reaction force directed to the exterior of the circle of radius 1 r arises at the supports. In all bars of the bar-and-joint assembly shown in Fig. 1, the forces are equal and form a state of self-stress, and thus the arrangement of the circles is (at least locally) optimal JoCG 5(1), ,

4 and has D 5 symmetry in the plane. (In this paper, C n and D n, n = 1, 2, 3,..., denote the cyclic and dihedral symmetry groups, respectively [3], [19]. In particular, C 1 means that there is no symmetry at all, while D 1 expresses simple bilateral symmetry.) Figure 1: The maximum packing of 5 equal circles and its mechanical model The radius of the equal circles is r max = sin(π/5) 1 + sin(π/5) = To simplify the notation here and throughout the paper, r max stands for r (5) max. 3 The minimum covering The reader who does not know the solution to the covering problem might expect that the optimal (minimum) covering with 5 equal circles also has D 5 symmetry, where the radius of the circles is R D5 = 1 2 cos(π/5) = This covering is shown in Fig. 2(a) where, for ease of checking, the associated bar-andjoint assembly (tensegrity structure composed of cables and joints) is also presented. The assembly has two kinds of joints. The joints of the first kind are the centres of the equal circles. The joints of the second kind are those points where the boundaries of three or more equal circles (or the boundary of the unit circle and the boundaries of two or more equal circles) intersect, and which are not interior points of any of the equal circles. The bars (cables) of length r always connect a first kind joint and a second kind joint. The joints on the boundary of the unit circle are supported against displacement in radial direction. One of these joints is also supported against tangential displacement. JoCG 5(1), ,

5 Figure 2: (a) Covering having D 5 symmetry, and its mechanical model, (b) the model with a triple joint, (c) the minimum covering Figure 3: (a) Covering having an egg shape, and its mechanical model, (b) the model with a double joint, (c) the minimum covering To the second kind joint at the centre of the unit circle 5 bars (cables) join, thus, according to [6], this joint should be considered as a triple joint (Fig. 2(b)). At uniform cooling of the model, 4 bars are in compression (cables slacken), so the newly appearing two joints together with the joining bars can be removed. The temperature of the remaining part of the structure can be decreased further until it reaches a state of self-stress in the form shown in Fig. 2(c) where the radius of the equal circles is R min = To simplify the notation here and throughout the paper, R min stands for R (5) min. (Numerical data for particular arrangements are given in Table 1 at the end of Section 4.4.) It should be noted that a state of self-stress can also appear in the bar-and-joint assembly associated with another covering as well (if one of the central bars of the structure in Fig. 2(a) was removed). Figure 3(a) shows a circle arrangement with a so called egg shape where the radius of the equal circles is JoCG 5(1), ,

6 R egg = Here, 4 bars (cables) meet at the joint closest to the centre of the unit circle, and therefore this joint and two of the bars connected to it should be doubled (Fig. 3(b)). Under uniform cooling, with removal of 3 bars in compression (slackened cables), the optimal covering is again obtained. This is shown in Fig. 3(c) which is the same as Fig. 2(c) but rotated by 72 degrees. (If a different choice of the two bars to be doubled is made, the same optimal configuration is obtained, but with a rotation in the opposite direction.) 4 Transition from packing to covering 4.1 The model If r is such that r max r R min then the equal circles can intersect the unit circle and also can intersect each other. Additionally, three equal circles can have an area in common, and so a generalized tensegrity structure is used for finding the maximum-area partial covering. This model consists of elements which are of three types: (a) a cable element connecting the centre of the unit circle and the centre of one of the equal circles, (b) a strut element connecting the centres of two equal circles, (c) a triangle element determined by the centres of three equal circles. (In the figures, cables, struts and triangle elements are represented with dashed lines, solid lines and shaded triangles, respectively.) Proper supports are introduced to prevent rigidbody displacements of the structure. An element is called active if a non-zero force arises in it. For the investigation of the whole range of r, it is not known in advance which elements will be active, and so all possible elements (5 cables, 10 struts, and 5 triangle elements) are taken into account as shown in Fig. 4. Figure 4: Elements and supports of the model of partial covering Forces (S) arising in the elements can be calculated as functions of the lengths (L) JoCG 5(1), ,

7 of the elements [6]. For a cable: and for a strut: { S = 0 if L 1 r 4L 2 (1 + L 2 r 2 ) 2 /L if L > 1 r, (1) S = { 4r 2 L 2 if L < 2r 0 if L 2r. (2) The angles of a triangle element as functions of the edge lengths (L i ) are (numbering the edges in cyclic order): β i = arccos L2 i 1 + L2 i+1 L2 i 2L i 1 L i+1, i = 1, 2, 3. (3) The radius of the circumcircle of the triangle (for any i) is R = L i 2 sin β i. (4) If none of the edge lengths of the triangle element is greater than 2r, then by introducing the parameters a i = L2 i L2 i 1 L2 i+1 4L i 1, b i = c i sinβ i+1, c i = the edge forces are obtained as r 2 L 2 i /4, d i = R 2 L 2 i /4, 2c i if 0 < a i b i c Q i = i + d i if a i b i < 0 < a i c i d i if a i < 0 < a i + b i 0 if a i + b i < 0. (5) 4.2 Solution with the method of dynamic relaxation Starting from r max, r is increased in small steps of r = (using at each step, the previous position to begin the iteration), and the method of dynamic relaxation [4, 17] is used to obtain equilibrium positions. To characterize a circle arrangement the coverage of the unit circle is introduced. Coverage (C) is defined as the ratio of the area (A c ) of the part of the unit circle covered by the equal circles and the area of the unit circle: C = A c π JoCG 5(1), ,

8 where the covered area can be written as Here A stands for the surplus area such that A = 4 5 i=0 j=i+1 A c = 5r 2 π A. A ij i=1 j=i+1 k=j+1 where A ij and A ijk denote the area of the intersection of two circles (i and j) and of three circles (i, j and k), respectively (that is, the area of a double and a triple overlap). The zero subscript, i = 0, refers to the exterior of the unit circle, while the other values of the subscripts, e.g. j = 1, 2,..., 5, refer to the interior of the jth one of the equal circles. A ijk C 0.94 D 5 egg C 1 pumpkin r Figure 5: Variation of the coverage of the unit circle with the radius r of the equal circles Figure 5 shows the variation of the coverage of the unit circle with the radius r for the region 0.5 r and gives also characteristics of the arrangement. These form and symmetry characteristics (defining 4 types of configuration) are as follows: (a) all 5 cables have equal length (D 5 symmetry), (b) there are 3 different cable lengths, and the lengths of the two shortest cables are equal (egg shape), (c) the lengths of the cables are all different (there is no symmetry, i.e., C 1 symmetry), (d) there are 3 different cable lengths, and the lengths of the two longest cables are equal (pumpkin shape). An arrangement with egg or pumpkin shape has bilateral symmetry (D 1 symmetry). Figure 6 presents the number of iteration steps, needed to find an equilibrium position at dynamic relaxation, as a function of the step-wise increased radius r. In the iteration, JoCG 5(1), ,

9 i the fictitious masses at the nodes were of unit magnitude, and the iteration was repeated until the unbalanced nodal forces dropped below It is striking that the number of iteration steps changes dramatically at a change of type and close to the complete covering iteration steps D 5 egg C 1 pumpkin r Figure 6: Number of iteration steps versus radius r It should be noted that if, beginning with R min, we go backwards along the same steps of r, then, in the vicinity of the changes of type, we do not always obtain the same arrangement: the change of type is delayed, or in other words, a certain kind of hysteresis occurs. This fact also shows that the exact locus of the changes of type cannot be determined easily with the method of dynamic relaxation, therefore, to follow the change of state and to characterize the stability of the equilibrium positions, the stiffness matrix of the structure should be determined. 4.3 Use of the stiness matrix The tangent stiffness matrices of the elements can be determined with the relationships presented in Part I [6], and from these the tangent stiffness matrix of the structure can be constructed. Increasing the radius of the equal circles in small steps and using the data of the previous equilibrium state as initial values in each step we can travel along an equilibrium path. From the point of view of stability, every equilibrium position can be characterized on the basis of the number of the negative eigenvalues of the stiffness matrix. If the number of the negative eigenvalues changes at a step then the locus of the change of sign can also be determined with arbitrary degree of accuracy (e.g. with the secant method). (We note here that a change of sign can occur where the matrix becomes singular, or where the topology of the active elements changes.) JoCG 5(1), ,

10 4.3.1 Equilibrium path of arrangements having D 5 symmetry Figure 1 has shown that, in the case of radius r max, we have the optimal arrangement with D 5 symmetry. Figure 7 illustrates the changes occurring in the equilibrium states while D 5 symmetry is preserved. Here and in all forthcoming figures of equilibrium paths, solid, dashed and dotted lines show that the stiffness matrix has zero, one and two negative eigenvalues, respectively. At the radius r 1 = the stiffness matrix becomes singular, and here the equilibrium path can bifurcate. After r 7 = every element will be active. It is true, however, that only one edge force of each triangle element is different from zero, because the common part of three equal circles is bounded by only two circles, and this edge force is just the opposite of the force in the respective new strut. As mentioned in Section 3, for radius R D5 (non-optimal) covering of the unit circle is obtained. Figure 7: Characteristics of the equilibrium path of the arrangements having D 5 symmetry Let us investigate the type of the bifurcation (catastrophe point, see e.g. [5, 16]) occurring at radius r 1. The stiffness matrix has two zero eigenvalues, and therefore this cannot be a cuspoid catastrophe. The structure in the plane has 5 axes of symmetry, so nor can it be an umbilic catastrophe, since in such a case the expression consisting of third-degree terms of the function of the surplus covered area would have 1 or 3 root lines. Equally it cannot belong to any of the typical classes of the double cusp catastrophe either, as in such a case the expression consisting of fourth-degree terms would have at most 4 root lines. Figure 8 presents one of eigenvectors (displacements of the circle centres) associated with the zero eigenvalues where the vertical axis remains an axis of symmetry. The active part of the 5-jet (see e.g. [14]) of the surplus area function given in a suitable cylindrical coordinate system (A, u, φ) can be written in the form j 5 A = 1 2 λu u u5 cos 5φ (6) JoCG 5(1), ,

11 Figure 8: Displacements leading to an egg shape at the point of bifurcation at r 1 where λ = r r 1 and u means the displacements given in Fig. 8. Function (6) indicates that this point of bifurcation belongs to the very degenerate class 12 of the double cusp catastrophe [5]. The primary equilibrium path (u = 0) bifurcates into 10 secondary paths (Fig. 9). The paths corresponding to the angles r ϕ u Figure 9: Bifurcation of the equilibrium path at r 1 φ = 2i π, i = 0, 1,..., 4 5 are stable, but those corresponding to the angles φ = (2i + 1) π, i = 0, 1,..., 4 5 JoCG 5(1), ,

12 are unstable (the stiffness matrix has one negative eigenvalue). The former paths correspond to equilibrium configurations having an egg shape, and the latter paths correspond to equilibrium configurations having a pumpkin shape. If r > r 1 then the stiffness matrix corresponding to the path u = 0 has two negative eigenvalues Equilibrium paths corresponding to congurations having an egg shape At the point r 1, the D 5 path (Fig. 7) bifurcates into 5 identical equilibrium paths corresponding to configurations having an egg shape. Characteristics of segments of these paths are illustrated in Fig. 10. Figure 10: Characteristics of the equilibrium path of the arrangements having an egg shape At the point r 2 = a new strut becomes active, as a consequence of this the eigenvalues of the stiffness matrix jump, but all remain positive, so the next segment of the path is also stable. At r 3 = the stiffness matrix is singular, here the equilibrium path can bifurcate. At r 4 = also a triangle element becomes active, but the stiffness matrix still has one negative eigenvalue. As mentioned earlier, at R egg a (non-optimal) covering of the unit circle is obtained. At r 3, where there is a bifurcation, the stiffness matrix has a zero eigenvalue, and thus, according to the catastrophe theory, a cuspoid catastrophe occurs. Along the tertiary equilibrium paths, the structure has no axis of symmetry in the plane, but the equilibrium path is symmetric, that is, at the point of bifurcation there is a standard cusp catastrophe, or according to the terminology of the theory of structural stability, there is a stable-symmetric point of bifurcation [5, 16]. JoCG 5(1), ,

13 4.3.3 Equilibrium paths corresponding to congurations having a pumpkin shape At the point r 1, the D 5 path (Fig. 7) bifurcates also into 5 additional identical equilibrium paths corresponding to configurations having a pumpkin shape. Characteristics of segments of these paths are illustrated in Fig. 11. Figure 11: Characteristics of the equilibrium path of the arrangements having a pumpkin shape At the point r 5 = two new struts become active, as a consequence of this the eigenvalues jump, and all become positive, so the next segment of the path is stable. At r 6 = two triangle elements also become active, but all the eigenvalues of the stiffness matrix remain positive. As seen earlier, at R min the optimal covering of the unit circle is obtained Equilibrium paths corresponding to asymmetric congurations As shown in Fig. 10, the equilibrium path bifurcates at radius r 3, but in a small enough neighbourhood of the point of bifurcation the topology of the active elements is the same; and the structure corresponding to the tertiary paths has no axis of symmetry. According to the stable symmetric bifurcation the equilibrium configurations are stable, the circle arrangements are locally optimal. Equilibrium paths, branching from the path corresponding to one of the egg-shaped configurations, join to the r 5 point of the neighbouring branches corresponding to pumpkin shapes. Therefore, at these points, four equilibrium paths meet. At the meeting point of the four paths, the centres of the five equal circles are in the same position, but the length of two struts is equal to 2r. At this point the graph of the constitutive function of the strut element has a kink, and so the element can be considered both active and JoCG 5(1), ,

14 passive. On the lower segment of the pumpkin branch both elements are passive, but on its upper segment both ones are active. On each of the two joining asymmetric branches one element (different on the two branches) is active (Fig. 11). 4.4 Summary of the results so far In Fig. 12, we show the top view of the equilibrium paths given in Fig. 9, supplemented with the asymmetric branches, and indicate the stable and unstable equilibrium configurations. Figure 12: Top view of the equilibrium paths The stable configurations correspond to the (locally) optimal arrangements of the circles. The value of the maximum coverage as a function of the radius r of the circles is given in Fig. 13. An enlarged detail of this coverage diagram above the segment (r 1, r 6 ) is presented in Fig. 14 where the arrangement of the active elements is given on each subsegment. The numerical data of circle arrangements at particular radii are compiled in Table 1 where the radius and the coordinates of the centres of the equal circles are given. Additionally, the type of the arrangements in the vicinity of these radii is presented, and for each type the numbers of active cable elements (c), strut elements (s) and triangle elements (t) are indicated. (If more than one possibility can occur, then each of them is given.) JoCG 5(1), ,

15 1 0.9 C r max r 1,2,3,5 r 6 R min r Figure 13: Variation of the maximum coverage of the unit circle with radius r of 5 equal circles. Radius r varies between the packing (r max ) and covering (R min ) limits. Special radii at which transitions take place are marked along the bottom of the figure. Their values are given in Table 1 C 1 double cusp cusp nonsmooth bifurcation D egg egg pumpkin pumpkin C 1 C 1 D 5 egg C 1 pumpkin r 1 r 1 r 2 r 3 r 5 r 6 r Figure 14: Enlarged detail of the maximum coverage diagram. Representative generalized tensegrity structures within distinctive ranges of r allow the activation of new elements and changes in arrangement types to be followed as the circle radius is increased JoCG 5(1), ,

16 Table 1: Data for particular points of the equilibrium paths (Radius r, coverage C, coordinates x i, y i of the centre of the ith circle, form and symmetry type of the branches meeting at that point, and numbers of active elements (cable, strut, triangle) c, s, t are listed.) r C i x i y i type c/s/t r max = ;5 ± D 5 5/5/0 3;4 ± D 5 5/5/0 r 1 = ;5 ± pumpkin 5/5/0 3;4 ± egg 5/5/ egg 5/5/0 r 2 = ;5 ± egg 5/6/0 3;4 ± egg 5/6/0 r 3 = ;5 ± C 1 5/6/0 3;4 ± egg 5/6/0 r 4 = ;5 ± egg 5/6/1 3;4 ± C 1 5/6/0 r 5 = ;5 ± pumpkin 5/5/0 3;4 ± pumpkin 5/7/ pumpkin 5/7/0 r 6 = ;5 ± pumpkin 5/7/2 3;4 ± D 5 5/5/0 r 7 = ;5 ± D 5 5/10/5 3;4 ± R min = ;5 ± pumpkin 5/7/2 3;4 ± R egg = ;5 ± egg 5/6/1 3;4 ± R D5 = ;5 ± D 5 5/10/5 3;4 ± JoCG 5(1), ,

17 4.5 Additional (unstable) equilibrium paths For finding optimal arrangements, it is sufficient to determine stable equilibrium paths in the interval (r max, R min ). However, in order to understand the changes better, in Section 4.3, we have dealt also with unstable paths, which have segments corresponding to radii larger than R min. So, the question arises: Do there exist additional unstable equilibrium paths? The answer is positive. Of course if 0 < r < r max then the equal circles can be packed in the unit circle such that the equal circles are disjoint. In this way all elements are passive, and the generalized tensegrity structure is in equilibrium with zero element forces. The coverage is C = 5r 2. However, there also exist equilibrium paths with active elements Arrangement of circles along a straight line If we have n equal circles such that 1/n r 1 and the n equal circles are arranged along a diameter of the unit circle so that the lengths of the chords connecting the points of intersection of the adjacent equal circles are equal to the lengths of the chords connecting the points of intersection of the first and last circles with the unit circle, then a model in equilibrium is obtained. The absolute value of the force arising in the elements of the model is n S n = 2 2 r 2 1 n 2 1, (7) and the central visual half-angle of a chord in one of the equal circles and in the unit circle are α n = arcsin ( ) Sn 2r β n = arcsin ( Sn 2 ), respectively. In the case of n = 5, the circle arrangement and the (somewhat distorted) structure composed of 2 cables and 4 struts in alignment with a straight line are shown in Fig. 15, and the coverage of the unit circle is written C = 1 π (5r2 π 10r 2 α 5 + 2β 5 ). (8) It should be noted that, in the case of 5 equal circles, if the circle centres are collinear and r is increased, then r 8 = 0.2 is the first value of radius r where the active elements can be in equilibrium. At radius r = three new strut elements and also three degenerate triangle elements appear (and with further increase in radius r, even quadruple and quintuple overlaps will also be present), but as the forces arising in these elements cancel each other out, the coverage given by (8) is valid up to complete covering, that is, up to r = R str = 1. JoCG 5(1), ,

18 Figure 15: Arrangement of n = 5 equal circles along a straight line and its mechanical model Cross-like arrangement If r 10 = 1/3 r 1/ 5 = r 12 then the cross shape shown in Fig. 16(a) provides an equilibrium configuration for which the model consists of 4 active cable elements and 4 active strut elements. The two straight lines need not necessarily be mutually perpendicular. Here, expression (7) should be applied with n = 3, and the coverage will be similar to (8): C = 1 π (5r2 π 12r 2 α 3 + 4β 3 ). (9) If r 12 r 1/ 2 = R + then, according to Fig. 16(b), the model is supplemented with four new active strut elements and triangle elements, and since the two straight lines are perpendicular to each other, the configuration has D 4 symmetry. Here, there is equilibrium if AB = CD in Fig. 16(b), and these distances are exactly equal to the absolute values of the forces in the elements: S = 1 80r r 4 + 6r 2 1. (10) (We note that the central circle is able to move freely in a certain domain without disturbing the equilibrium.) The coverage is C = 1 π (4r2 π 8r 2 α 3 + 4β 3 ). (11) We note here that, in the case of R +, four of the equal circles just cover the unit circle, so the fifth circle is redundant A type arrangement If r 9 = 1/4 r 8/23 = r 11 then the shape in Fig. 17(a) also provides an equilibrium configuration, where the centres of four equal circles are collinear and the fifth is independent JoCG 5(1), ,

19 Figure 16: Cross-like circle arrangement and its model for (a) r 10 r r 12, (b) r 12 r R + of them. The mechanical model consists of two active cable elements and three active strut elements. The cable force can be calculated with the expression (7) by substituting n = 4. After modifying (8) accordingly we obtain the coverage in the form At radius r 11 the coverage is C = C = 1 π (5r2 π 8r 2 α 4 + 2β 4 ). (12) If r 11 r = r 13, then the shape shown in Fig. 17(b) provides an equilibrium configuration, where a new active cable and two new struts appear in the model. 2 At r 13 = = , circles i = 1, 2, 5 will have a point in common. 187 In this arrangement the coverage is C = (Note that here x 3 = 2x 2, y 3 = 2y 2, whereas at r 11 x 3 = 3x 2, y 3 = y 2 = 0.) If r 13 r 3/2 = R 4+1 then, according to Fig. 17(c), an active triangle element also appears. Furthermore, at larger values of r, additional elements appear, but the forces arising in them cancel, and thus this model remains sufficient. At radius R 4+1, the circles JoCG 5(1), ,

20 Figure 17: The 4+1 type circle arrangement and its model for (a) r 9 r r 11, (b) r 11 r r 13, (c) r 13 r R 4+1 i = 1, 3, 4 just cover the unit circle (identically to the optimal covering with 3 equal circles), and the positions of circles i = 2 and 5 coincide with positions of circles i = 3 and 4. JoCG 5(1), ,

21 4.5.4 Summary of the results for unstable congurations Coverage corresponding to different types of equilibrium configurations is summarized in Fig. 18. There, the solid curve shows the best coverage, dashed lines indicate the coverage corresponding to the other equilibrium paths. On this scale it is not possible to show the difference between the curve corresponding to the optimal arrangements and the curves corresponding to the unstable branches of D 5, egg, pumpkin arrangements. C R min R + R 4+1 R str 1 (e) (a) (b) (c) (d) 0 0 r 8 r 9 r 10 r max 1 r Figure 18: Coverage in the cases of different types of equilibrium configurations: (a) disjoint, (b) straight line, (c) 4+1 type, (d) cross-like, (e) optimal Radii and coordinates of the centres of equal circles in particular configurations are given in Table 2. In the final two columns, it is also indicated in the vicinity of a particular radius what types of equilibrium arrangement can occur, and how many active elements of each kind are present in the model. (The strut elements and the triangle elements are not considered active when the strut force and the edge force acting along the same line are in equilibrium.) JoCG 5(1), ,

22 Table 2: Data for particular points of the other equilibrium paths (Notation as in Table 1.) r C i x i y i type c/s/t disjoint 0/0/0 r 8 = ;5 0 ± 0.4 straight 2/4/0 3;4 0 ± disjoint 0/0/0 r 9 = ;5 ± /3/0 3;4 ± disjoint 0/0/0 r 10 = ;5 ± cross 4/4/0 3;4 0 ± /3/0 r 11 = ;5 ± /5/0 3;4 ± cross 4/4/0 r 12 = ;5 ± cross 4/8/4 3;4 0 ± /5/0 r 13 = ;5 ± /5/1 3;4 ± R + = ;5 ± cross 4/8/4 3;4 0 ± R 4+1 = ;5 ± /5/1 3;4 ± R str = ;5 0 0 straight 2/4/0 3;4 0 0 JoCG 5(1), ,

23 5 Conclusions (a) We have determined numerically the optimal partial coverings of the unit circle with five circles for all values of radius r in the interval r max (5) r R (5) min and have shown the coverage of the optimum circle arrangements (maximum-area partial coverings) in Fig. 13. (b) A stable equilibrium position, where the stiffness matrix is positive definite, is usually guarantees only a local optimum of the circle configuration. However, in the case of the partial covering of the unit circle with five equal circles, for any value of r, the stuctures corresponding to the stable equilibrium states are equivalent, so the local optima at the same time are also global optima. (c) It is known from the literature that the optimum packing has D 5 symmetry, and the optimum covering has D 1 symmetry. By using the generalized tensegrity model, we pointed out that, starting from the optimum packing and increasing the radius of the circles, arrangements with D 5 symmetry are optimal in a region. This ceases to exist at a degenerate type of the double cusp catastrophe, where the D 5 branch is intersected by five secondary paths. Although these paths have horizontal tangents at the common point of bifurcation, they are not symmetrical: initially the egg shape is stable and the pumpkin shape is unstable, then both shapes are unstable (and in this case the optimum configuration has C 1 symmetry), and finally the pumpkin shape is stable and the egg shape is unstable. (d) For a fixed value of r, the number of stable equilibrium configurations depends on the sort of symmetry of the circle arrangement. For D 5 1, for D 1 5, and for C 1 10 equilibrium configurations exist (while rotation of the whole arrangement is prevented). (e) The constitutive function of the elements is not smooth, and thus the eigenvalues of the stiffness matrix change discontinuously. In most cases, the sign of the eigenvalues did not change at the jumps, so the equilibrium path had only a kink there. At the meeting point of the pumpkin branch and the asymmetric branches, however, a non-smooth bifurcation occurred. (f) We have shown that there also exist many unstable equilibrium paths, but not leading to locally optimal configurations. 6 Acknowledgements The research reported here was partially supported by OTKA grant no. K81146 awarded by the Hungarian Scientific Research Fund. This paper was also supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The authors are grateful to Professor P. W. Fowler for advice and comments. References [1] Bezdek, K.: Über einige Kreisüberdeckungen, Beiträge Algebra Geom. 14, 7-13 (1983). [2] Connelly, R.: Maximizing the area of unions and intersections of discs. Lecture at the Discrete and Convex Geometry Workshop, Alfréd Rényi Institute of Mathematics, Budapest, July 4-6 (2008). JoCG 5(1), ,

24 [3] Coxeter, H.S.M.: Introduction to Geometry, Wiley, New York (1961). [4] Day, A.S.: An introduction to dynamic relaxation, The Engineer 219, (1965). [5] Gáspár, Zs.: Mechanical models for the subclasses of catastrophes, in Phenomenological and Mathematical Modelling of Structural Instabilities (eds: M. Pignataro and V. Gioncu), CISM Courses and Lectures no. 470, Springer, Wien, pp (2005). [6] Gáspár, Zs., Tarnai, T. and Hincz, K.: Partial covering of a circle by equal circles, Part I: The mechanical models, J. Comput. Geom. 5 (1), (2014). [7] Graham, R.L.: Sets of points with given minimum separations (Solution to Problem E 1921), Amer. Math. Monthly 75, (1968). [8] Kravitz, S.: Packing cylinders into cylindrical containers, Math. Mag. 40, (1967). [9] Melissen, H.: Packing and covering with circles, PhD thesis, University of Utrecht (1997). [10] Neville, E.H.: On the solutions of numerical functional equations, illustrated by an account of a popular puzzle and of its solution, Proc. London Math. Soc. 2nd Ser. 14, (1915). [11] Nurmela, K.J.: Covering a circle by congruent circular discs, Preprint, Department of Computer Science and Engineering, Helsinki University of Technology (1998). [12] Pacioli, L.: Summa de Arithmetica, Geometria, Proportioni et Proportionalita, Venice (1494). [13] Pirl, U.: Der Mindestabstand von n in der Einheitskreisscheibe gelagenen Punkten, Math. Nachr. 40, (1969). [14] Poston, T., Stewart, I.: Catastrophe Theory and its Applications, Pitman, London (1978). [15] Rouse Ball, W.W., Coxeter, H.S.M.: Mathematical Recreations and Essays, thirteenth edition, Dover, New York (1987), (first edition: 1892). [16] Thompson, J.M.T., Hunt, G.W.: Elastic Instability Phenomena, Wiley, Chichester, (1984). [17] Topping, B.H.V. and Iványi, P.: Computer Aided Design of Cable Membrane Structures, Saxe-Coburg Pub., Kippen, Scotland (2007). [18] Zahn, C.T.: Black box maximization of circular coverage, J. Res. Nat. Bur. Standards Sect. B 66, (1962). [19] Weil, H.: Symmetry, Princeton University Press (1952). JoCG 5(1), ,