Recon guring Convex Polygons Oswin Aichholzer  Erik D. Demaine y Je Erickson z Ferran Hurtado x Mark Overmars { Michael Soss k Godfried T. Toussaint k Abstract We prove that there is a motion from any convex polygon to any convex polygon with the same counterclockwise sequence of edge lengths, that preserves the lengths of the edges, and keeps the polygon convex at all times. Furthermore, the motion is \direct" (avoiding any intermediate canonical con guration like a subdivided triangle) in the sense that each angle changes monotonically throughout the motion. In contrast, we show that it is impossible to achieve such a result with each vertex-to-vertex distance changing monotonically. We also demonstrate that there is a motion between any two such polygons using three-dimensional moves known as pivots, although the complexity of the motion cannot be bounded as a function of the number of vertices in the polygon. 1 Introduction This paper is concerned with linkages modeled by polygons (primarily in the plane), whose vertices represent hinges and whose edges represent rigid bars. A fundamental question about such linkages is whether it is possible to reach every polygon with the same sequence of edge lengths by motions that preserve the edge lengths. Several papers have shown that the answer to this question is yes for various types of polygons; we call this a universality  Institut fur Grundlagen der Informationsverarbeitung, Technische Universitat Graz, Schiestattgasse 4, A-8010 Graz, Austria, email: oaich@igi.tu-graz.ac.at. Supported by the Austrian Programme for Advanced Research and Technology (APART). y Department of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, email: eddemaine@uwaterloo.ca. z Department of Computer Science, University of Illinois, Urbana, IL 61801, USA, email: jeffe@cs.uiuc.edu. Supported by a Sloan Fellowship. x Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya, Pau Gargallo 5, 08028- Barcelona, Espa~na, email: hurtado@ma2.upc.es. Supported by Proyectos MEC-DGES-SEUID PB98-0933 and CUR Gen. Cat. SGR1999-00356 { Department of Computer Science, Utrecht University, Padualaan 14, De Uithof, 3584 CH Utrecht, The Netherlands, email: markov@cs.uu.nl. k School of Computer Science, McGill University, 3480 University Street, Montreal, Quebec H3A 2A7, Canada, email: fsoss, godfriedg@cs.mcgill.ca. 1 result. If edges are allowed to cross each other, then this is true in every dimension [12, 16]. If edges are not allowed to cross, universality does not hold in general for polygons in 3-D [2, 5], but has been shown for polygons in the plane and motions in 3-D [1, 2], for polygons and motions in the plane [9], for polygons in 3-D with simple projections [4], and for all polygons in 4-D and higher dimensions [8]. All of these papers show universality by proving that every polygon can be convexi ed, that is, moved to a convex (planar) polygon while preserving edge lengths. Convex polygons are used as an intermediate state; because motions can be reversed and concatenated, all that remains is to show that a convex polygon can be moved to every other convex polygon with the same counterclockwise sequence of edge lengths. This fact is established in [12] when edges are allowed to cross. No proof has been published for the case in which edges cannot cross. The basic idea in the proof in [12] of universality of convex polygons is to show how to recon gure every convex polygon into another intermediate state, a \canonical triangle." In the rst half of this paper, we show that this intermediate state can be avoided. Speci cally, a convex polygon can be moved into any other convex polygon with the same counterclockwise sequence of edge lengths in such a way that each vertex angle varies monotonically with time (either never increasing or never decreasing). In this sense, the motion goes directly from the source to the destination. Our motion is also of the simplest type possible [3]: it can be decomposed into a linear number of moves, each of which changes only four joint angles at once. In the second half of this paper, we study the same problem of recon guring convex polygons, under a more restrictive type of move. Speci cally, we study motions consisting of a sequence of pivots, which are the simplest kind of motion in three dimensions, changing only two joint angles at once. Such motions are popular in biology and physics circles; see Section 5. It may seem that the freedom to move in three dimensions is a signi cant advantage, but in fact the limited motions make it diĘcult to change angles in the plane. Nonetheless, we show that it is possible to simulate our planar motions by a sequence of pivots. Thus we obtain the result that a convex polygon can be pivoted to any other convex polygon with the same counterclockwise sequence of edge lengths. The paper is organized as follows. In Section 2 we introduce some basic notation that we will use throughout the paper. Section 3 proves the theorem about angle-monotone motions in the plane, using an old lemma of Cauchy and Steinitz. Section 4 shows an example in which a di erent type of monotonicity cannot be achieved. Finally, Section 5 proves the theorem about pivots in three dimensions. 2 Notation For a polygon P , we denote its vertices by v 1 ; : : : ; v n in counterclockwise order, its edges by e i = (v i ; v i+1 ), and its edge lengths by ` i = je i j = jv i ell-known result characterizes the edge lengths for which convex con gurations exist: 2 Lemma 1 (Lemma 3.1 of [12]) The edge lengths ` 1 ; : : : ; ` n admit a convex con guration precisely if ` i  P j 6=i ` j for all i. A motion or recon guration is a continuous function from the unit interval [0; 1] (rep- resenting time) to a con guration, where each con guration is a polygon with the same counterclockwise sequence of edge lengths. An angle-monotone motion is a motion in which each vertex angle is a monotone function in time. In the following, we split our results into two components: theorems give the existential result, and propositions give the additional computational result. 3 Recon guring between Two Convex Con gurations Consider two convex con gurations C and C 0 of the same sequence of edge lengths. We think of C as the source con guration and C 0 as the destination con guration. Label each angle of C by + if it needs to get bigger in order to match the corresponding angle in C 0 , by on the sphere. His key lemma about alternations in such +; labeling that comes from two distinct convex con gurations, there are at least four sign alternations. Proof (Sketch): Because the con gurations are distinct, not all labels are 0. By circularity, the number of alternations between + and hain speci es that the ends of the chain should get further apart, whereas the latter chain speci es the opposite. It is this last part of the argument that needs careful analysis; for details, see [20] for Steinitz's original (complicated) proof, [10] for a simpler proof due to Isaac J. Schoenberg, or [18] for another elementary proof. 2 The idea is to take vertices v i ; v j ; v k ; v l in cyclic order around the polygon, whose angles are labeled +; v 1 ; v 2 ; v 3 ; v 4 , there is a motion that decreases the angles at v 1 and v 3 , and increases the angles at v 2 and v 4 . The motion can continue until one of the angles reaches 0 or . Proof: We consider the following viewpoint: v 1 is pinned to the plane, and v 3 moves along the directed line from v 1 to v 3 (see Figure 2). The motions of v 2 and v 4 are determined by 3 v j v i v k v l Figure 1: Applying a quadrangle motion to a convex polygon by taking vertices labeled +; osition I.25 [11] to triangle v 1 ; v 2 ; v 3 , because jv 1 y-Steinitz lemma (Lemma 2), there must be at least four sign alternations when compared to any future quadrangle we will visit. This proves that the angles at v 1 and v 3 are decreasing throughout the motion. 2 v 1 v 2 v 4 v 4 v 2 v 1 v 3 v 3 Figure 2: Moving a convex quadrangle as in Lemma 3. 4 We are now in the position to prove the main theorem of this section: Theorem 1 Given two convex con gurations C; C 0 of the same edge lengths ` 1 ; : : : ; ` n , there is an angle-monotone motion from C to C 0 that involves O(n) moves each of which changes only four vertex angles at once. Proof: Consider con guration C. By Lemma 2, we can nd vertices v i ; v j ; v k ; v l in cyclic order around the polygon, whose angles are labeled +; hes the angle in C 0 . (No angle will ever reach 0 or  because of our stopping condition.) Repeat this process until all angles match. The result is a sequence of motions from C to C 0 . There are at most n moves, because each motion changes the label of an angle from + or done in O(n) time on a pointer machine with real numbers. Proof: The rst part is to maintain the vertices of the quadrangle, v i ; v j ; v k ; v l , throughout the motion. We maintain four consecutive blocks I; J; K; L of the same sign; speci cally, we maintain the rst and last vertex in each block. This can be found initially in linear time by scanning along the polygon's vertices in order. The desired vertices v i ; v j ; v k ; v l are identi ed with the rst vertex in the corresponding block. When the label of one of them switches to 0, it and the block's rst vertex advance to the next element in the block. If this was the last element (the block is empty), we make the following modi cations. If I becomes empty, we advance it to the block of +'s after L. Similarly, if L becomes empty, it retreats to the block before I. If K becomes empty, it advances to the block after L, the blocks J and L merge to produce a new J , and L advances to the block after K. The case of J becoming empty is symmetric. The second part is to apply the quadrangle motions from Lemma 3. This involves com- puting the time at which the quadrangle motion stops, and then updating the coordinates. These computations can be done analogous to Lemma 7 of [3]. Basically, we compute the times at which each angle would match the desired angle in C 0 , and take the minimum of these times. At worst, each time can be computed by solving a degree-four polynomial, which reduces to an arithmetic expression involving square and cube roots. 2 4 Distance-Monotone Motions We have shown that an angle-monotone motion between any two convex con gurations of a common sequence of edge lengths can be computed in linear time. An interesting consequence is that any polygon can be moved to a unique inscribed con guration [19], in which the vertices are cocircular, a natural generalization of regular polygons. It is interesting to note that we cannot hope for a distance-monotone motion between any two convex polygons, in which every distance between a pair of vertices varies monotonically 5 with time. (This is in direct contrast to convexi cation of a polygon [9], where all distances can be made to increase.) An example is shown in Figure 3. Because the dotted lines are the same length in both con gurations, these distance must be preserved throughout the motion; in other words, the chains v 1 ; v 2 ; v 3 and v 4 ; v 5 ; v 6 must move rigidly. The problem is thus reduced to moving a quadrangle v 1 ; v 3 ; v 4 ; v 6 , which can be moved in only two di erent ways. Only one motion decreases jv 1 con gurations. v 4 v 5 v 2 v 1 v 3 v 6 v 2 v 1 v 3 v 4 v 5 v 6 v 1 v 2 v 3 v 5 v 6 v 4 Figure 3: (Left and right) An example for which a distance-monotone motion is impossible. (Middle) The transition between jv 2 show that a convex polygon can be recon gured to any other convex polygon (with the same edge lengths) by the use of a three-dimensional motions called pivots. Let v i and v j be two vertices of a polygon. A pivot on v i v j is a motion whereby the section of the polygon between v i and v j (denoted henceforth as [v i ; v j ]) is rotated about the diagonal v i v j . Examples of pivots are illustrated in Figures 4 and 5. v i v j v j v i Figure 4: A pivot on v i v j . Pivots are of great interest to polymer physicists and molecular biologists, who are con- sider polygons as models of large molecules and are interested in the con gurations that they 6 v 4 v 3 v 1 v 2 v 4 v 3 v 1 v 2 v 4 v 1 v 3 v 2 v 2 v 1 v 4 v 3 v 4 v 2 v 3 v 1 v 1 v 4 v 3 v 2 v 1 v 4 v 3 v 2 v 1 v 4 v 2 v 3 v 4 v 1 v 3 v 2 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 4 v 1 v 3 v 2 Figure 5: The same transformation as illustrated in Figure 2 but accomplished with three pivots, shown chronologically from left to right. (Top row) Bird's eye view. (Bottom row) Oblique view. can take. This motion has been used in many contexts over the last few decades in physics as well as mathematics circles [15, 13, 14, 22]. Erd}os-Nagy [21] ips are special cases of pivots with planar polygons in 3D where the pairs of vertices that de ne the pivots are determined by lines of support of the polygon and each rotation has an angle of . Another special type of pivot which is a natural generalization of Erd}os-Nagy ips is as follows. Let P be a polygon in R d and let H be a hyperplane supporting the convex hull of P and containing at least two vertices of P . Reect one of the resulting polygonal chains across H. Let us call such motions hyperplane-ips. The rst person to propose these hyperplane- ips appears to be Gustave Choquet [7] in 1945 for applications to curve stretching. He claimed in [7] (but published no proof) that after a suitable choice of a countable number of hyperplane-ips the polygons generated converge to planar convex polygons. These results were rediscovered in 1973 by Sallee [16]. In 1994 Millett [15], in connection with exploring varieties, proposed a \walk" algorithm consisting of a sequence of pivots to take any equilateral polygon (knot) in 3D into any other. (Millett allows self-crossings during the motions.) The interest in equilateral polygons comes from molecular biology where homogeneous macromolecules or polymers such as DNA are modeled by polygons with equal length edges. Here the vertices correspond to the mers and the edges to the bonding force between them. To establish the walk Millett proposed taking an arbitrary equilateral polygon P in 3D to a planar regular polygon. His algorithm consists of three parts: (1) convert P to a planar star-shaped polygon P 0 , (2) convert P 0 to a convex polygon P 00 and (3) convert P 00 to a regular polygon. However, his algorithm for part (1) does not always work correctly. His procedure may yield non-simple planar polygons in which all turns are right turns and the winding number is high, invalidating step (2) of the algorithm. Toussaint [21] proposed an alternate walk algorithm to convexify a 3D polygon that generalizes Millett's theorem to polygons in d dimensions with no restrictions on edge lengths. Millett [15] showed in step (3) of his procedure that any convex planar polygon with equal 7 edge lengths can be taken to any other via a bounded number (as a function of n) of pivots in 3D. In this section we demonstrate that this procedure also works for non-equilateral convex polygons, although in that case an unbounded number of pivots may be required. We now prove the main theorem of this section. Theorem 2 Any planar convex polygon can be recon gured into any other planar convex polygon using pivots. Proof: We use similar logic as in the proof of Theorem 1 in that we rst locate a quadrangle v 1 v 2 v 3 v 4 whose vertices can be labelled to the second quadrangle of Figure 6. Then we pivot on v 2 v 4 to achieve a planar quadrangle, as shown in the third and fourth pictures of the gure. We will defer a discussion of the outcome of these pivots until later, and instead now demonstrate that no collisions can occur during these two pivots. During the rst pivot, the polygon is folded along one crease like a taco up to an angle of =2, so no collisions are possible there. During the second pivot, the subpolygon from v 1 to v 2 (which we will denote as [v 1 ; v 2 ]) and [v 4 ; v 1 ] cannot collide, as these move in concert; likewise for [v 2 ; v 3 ] and [v 3 ; v 4 ]. We can also readily see that the subpolygon [v 4 ; v 1 ] remains motionless, and that [v 3 ; v 4 ] rotates about v 2 v 4 by at most =2 and therefore moves strictly upward. Therefore these two polygonal chains cannot intersect; by a symmetrical argument, neither can [v 1 ; v 2 ] and [v 2 ; v 3 ]. The only remaining possibility is the collision of [v 2 ; v 3 ] and [v 4 ; v 1 ] (and symmetrically [v 3 ; v 4 ] and [v 1 ; v 2 ]). Note that after the rst pivot the subpolygon [v 2 ; v 3 ] points upward from v 2 v 3 in a vertical plane. After the second pivot, which is of at most =2, this subpolygon must lie above a horizonal plane through v 2 v 3 . Therefore [v 2 ; v 3 ] and [v 4 ; v 1 ] cannot collide, because the latter still lies below the former in the original plane of the polygon. Following these rst two pivots, the quadrangle, but not the polygon, is planar. We now perform a pivot on v 4 v 1 to bring the quadrangle back into its original plane, and then perform pivots on the remaining three edges of the quadrangle to bring the polygon into the same plane as the quadrilateral. These last motions are also illustrated in Figure 6. We have shown that no collisions occur during these pivots, but it remains to be shown that any quadrangle desired can be achieved through the repetition of these motions. Con- sider again the rst quadrangle of the top row of Figure 6. Let x be the closest point from v 2 on the line v 1 v 3 . By the law of cosines, the distance between v 2 and v 4 is expressed by (v 2 v 4 ) 2 = (v 2 x) 2 + (v 4 x) 2 (second quadrangle of the gure), \v 2 xv 4 is =2, so this term is equal to zero. Therefore after each series of pivots, v 2 and v 4 come closer together by j2(v 2 x)(v 4 x) cos \v 2 xv 4 j. Thus we always make considerable progress toward our goal con- guration, unless our goal con guration is one where either v 2 x, v 4 x, or cos \v 2 xv 4 are zero. If the goal has both v 2 x and v 4 x as zero, then our goal con guration is self-intersecting and therefore invalid, so this case needs no consideration. In the other instances, this implies that in the goal, v 1 , v 2 (or v 4 , but this case is symmetric to v 2 ), and v 3 are collinear. Thus 8 we can state that v 2 is the only vertex in between v 1 and v 3 , else we violate convexity of the goal con guration. When v 1 , v 2 , and v 3 are almost collinear, a pivot about v 1 v 3 of any angle (even as much as ) will not cause any self-intersections to arise. Therefore, if we pivot until v 2 v 4 is the same distance as v 4 x, and perform the remaining pivots to restore planarity of the polygon, v 1 v 2 v 3 will be collinear as desired. 2 v 4 v 1 v 1 v 4 v 2 v 2 x v 3 v 3 x v 4 v 1 v 1 v 4 v 2 v 2 x v 3 v 3 x v 1 v 1 v 4 v 3 v 4 v 2 v 2 v 3 v 1 v 1 v 4 v 3 v 4 v 2 v 2 v 3 v 4 v 3 v 1 v 1 v 4 v 2 v 2 v 3 x v 4 v 3 v 1 v 1 v 4 v 2 v 2 v 3 x x x + + Illustration of the pivots used in Theorem 2. (Top row) Bird's eye view. (Bottom row) Oblique View. The geometric progression of the above proof hints at the notion that there may exist some polygons for which the number of pivots required to move between any two arbitrary goal con gurations may not be bounded by a function of the number of edges in the polygon. In fact, we will soon show this to be the case. Before proving this statement in Theorem 3, we require the following lemma. We draw the reader's attention to Figure 7 which may serve as a useful visual aid during the course of the proof of Lemma 4. Lemma 4 Let v 1 v 2 v 3 v 4 be a planar convex quadrangle. After two pivots, suppose the quad- rangle is once again planar, resulting in a quadrangle v 00 1 v 00 2 v 00 3 v 00 4 . Then \v 00 2 v 00 1 v 00 4 will be at least the original value of the expression j\v 2 v 1 v 3 v 3 is the rst pivot (or both), and the case where it is preceded by a pivot on v 2 v 4 . If the pivot on v 1 v 3 occurs rst, then the pivot occurs on a planar polygon. (If both pivots are on v 1 v 3 , we can merge them into a single pivot, and thus the argument is identical.) Ignoring intersections for the time being, let v 2 rotate freely around the diagonal v 1 v 3 . The point v 2 traces out a circle in space centered on v 1 v 3 ; thus \v 2 v 1 v 3 is constant. Since \v 4 v 1 v 3 does not vary during the pivot, the resulting \v 0 2 v 0 1 v 0 4 is at least the di erence of these two angles. If the pivot on v 2 v 4 occurs rst, then the next pivot must occur on v 1 v 3 and must bring the quadrangle into a planar position. We can also visualize this as the triangle 4v 1 v 3 v 4 rotating about v 1 v 3 until it is coplanar with the triangle 4v 1 v 3 v 2 . In this case, the distance v 2 v 4 , which was constant during the previous pivot, is now increasing. By the law of sines, \v 2 v 1 v 4 must have increased. 2 The next theorem follows easily from Lemma 4. 9 Theorem 3 There exist polygons which require arbitrarily many pivots to achieve a goal con guration. Proof: Examine the leftmost parallelogram in Figure 7. This polygon has the property that because it is a parallelogram, there exist con gurations where it is as at as desired; that is, where \v 1 is arbitrarily close to zero. Furthermore, because it not a rhombus, \v 2 v 1 v 3 6= \v 4 v 1 v 3 . In fact, due to the law of sines, sin \v 2 v 1 v 3 sin \v 4 v 1 v 3 = v 2 v 3 v 3 v 4 : Furthermore, for small angles \x, sin x  x. Therefore as \v 1 gets smaller and smaller, for every two pivots \v 1 is only able to be reduced to \v 0 1 according to the expression \v 0 1  j v 2 v 3 (We note that although one cannot achieve a con guration where \v 1 = 0, this is not a valid con guration as the polygon would be at and therefore self-intersecting.) While this proves the theorem for the case where every two pivots restores the polygon to a planar con guration, we have not directly proven the theorem for arbitrary pivots. However, this is easily remedied by considering each pivot as a pair of pivots on the same diagonal, the rst to bring the quadrangle into a planar non-intersecting position and the second to produce the original pivot as desired. 2 v 1 v 0 1 v 0 1 v 00 1 v 3 v 0 3 v 0 3 v 00 3 v 00 4 v 4 v 0 4 v 0 4 v 0 2 v 00 2 v 0 2 v 2 Figure 7: Two pivots performed to recon gure a non-rhombus parallelogram. Acknowledgments The second author thanks Anna Lubiw for helpful discussions that inspired some of this research. This work was done at the Workshop on Computational Polygonal Entanglement Theory, organized by Godfried Toussaint on February 4{11, 2000 at the Bellairs Research Institute in Holetown, Barbados. We thank Vida Dujmovic for important discussions at that meeting. 10 References [1] Boris Aronov, Jacob E. Goodman, and Richard Pollack. Convexi cation of planar poly- gons in R 3 . Manuscript, October 1999. http://www.math.nyu.edu/faculty/pollack/ convexifyingapolygon10-27-99.ps. [2] T. Biedl, E. Demaine, M. Demaine, S. Lazard, A. Lubiw, J. O'Rourke, M. Overmars, S. Robbins, I. Streinu, G. Toussaint, and S. Whitesides. Locked and unlocked polyg- onal chains in 3D. Technical Report 060, Smith College, 1999. A preliminary version appeared in the Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, Maryland, January 1999, pages 866{867. [3] Therese C. Biedl, Erik D. Demaine, Sylvain Lazard, Steven M. Robbins, and Michael A. Soss. Convexifying monotone polygons. In Proceedings of the 10th Annual International Symposium on Algorithms and Computation, volume 1741 of Lecture Notes in Computer Science, pages 415{424, Chennai, India, December 1999. An expanded version is Tech- nical Report CS-99-03, Department of Computer Science, University of Waterloo, 1999. [4] Jorge Alberto Calvo, Danny Krizanc, Patrick Morin, Michael Soss, and Godfried Tou- ssaint. Convexifying polygons in 3D, 2000. [5] Jason Cantarella and Heather Johnston. Nontrivial embeddings of polygonal intervals and unknots in 3-space. Journal of Knot Theory and Its Rami cations, 7(8):1027{1039, 1998. [6] A. L. Cauchy. Sur les polygones et polyedres (second memoire). J.  Ecole Polytechnique, 9:87{98, 1813. [7] Gustave Choquet. Varietes et corps convexes. In Association Francaise pour l'Avancement des Sciences, pages 91{93, 1945. Congres de la Victoire-1945. [8] Roxana Cocan and Joseph O'Rourke. Polygonal chains cannot lock in 4D. In Proceedings of the 11th Canadian Conference on Computational Geometry, Vancouver, Canada, August 1999. http://www.cs.ubc.ca/conferences/CCCG/elec proc/c17.ps.gz. [9] Robert Connelly, Erik D. Demaine, and Gunter Rote. Straightening polygonal arcs and convexifying polygonal cycles. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, Redondo Beach, California, November 2000. [10] Peter R. Cromwell. Equality, rigidity, and exibility. In Polyhedra, chapter 6, pages 219{247. Cambridge University Press, Cambridge, 1997. [11] Sir Thomas Little Heath. The thirteen books of Euclid's Elements translated from the text of Heiberg with introduction and commentary. Dover Publications, New York, 1956. [12] W. J. Lenhart and S. H. Whitesides. Recon guring closed polygonal chains in Euclidean d-space. Discrete and Computational Geometry, 13:123{140, 1995. 11 [13] Neal Madras and Gordon Slade. The Self-Avoiding Walk. Birkhauser, Boston, 1993. [14] Neal Madras and Alan D. Sokal. The pivot algorithm: A highly eĘcient Monte Carlo method for the self-avoiding walk. Journal of Statistical Physics, 50:109{186, 1988. [15] K. Millett. Knotting of regular polygons in 3-space. Journal of Knot Theory and its Rami cations, 3:263{278, 1994. [16] G. T. Sallee. Stretching chords of space curves. Geometriae Dedicata, 2:311{315, 1973. [17] G. T. Sallee. Stretching chords of space curves. Geometriae Dedicata, 2:311{315, 1973. [18] I. J. Schoenberg and S. K. Zaremba. Cauchy's lemma concerning convex polygons. Canadian Journal of Mathematics, 19(4):1062{1071, 1967. [19] Peter Schreiber. On the existence and constructability of inscribed polygons. Beitrage zur Algebra und Geometrie (Contributions to Algebra and Geometry), 34(2):195{199, 1993. [20] Ernst Steinitz and Hans Rademacher. Vorlesungen  uber die Theorie der Polyeder. Springer-Verlag, Berlin, 1934. Reprinted 1976. [21] G. T. Toussaint. The Erd}os-Nagy theorem and its rami cations. In 11th Canadian Conference on Computational Geometry, pages 9{12, August 16-18 1999. Vancouver, Canada. [22] E. J. Janse van Rensburg, S. G. Whittington, and N. Madras. The pivot algorithm and polygons: results on the FCC lattice. Journal of Physics A: Mathematical and General Physics, 23:1589{1612, 1990. 12