Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space

Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space

Yukiko Yamauchi Corresponding author. Address: 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan. Fax: +81-92-802-3637. Email: yamauchi@inf.kyushu-u.ac.jp Kyushu University, Japan.    Taichi Uehara Kyushu University, Japan.    Shuji Kijima Kyushu University, Japan.    Kyushu University, Japan.    Masafumi Yamashita Kyushu University, Japan.
Abstract

Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, we investigate the plane formation problem that requires a swarm of robots moving in the three dimensional Euclidean space to land on a common plane. The robots are fully synchronous and endowed with visual perception. But they do not have identifiers, nor access to the global coordinate system, nor any means of explicit communication with each other. Though there are plenty of results on the agreement problem for robots in the two dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three dimensional space. This paper presents a necessary and sufficient condition for fully-synchronous robots to solve the plane formation problem that does not depend on obliviousness i.e., the availability of local memory at robots. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can form a plane from every regular polyhedron (except a regular icosahedron), whose symmetry is usually considered to be higher than any semi-regular polyhedrdon.

Keywords. symmetry breaking, mobile robots, plane formation, rotation group.

1 Introduction

Self-organization in a swarm of mobile computing entities frequently called robots, agents or sensor nodes, has gained much attention as sensing and controlling devices are developed and become cheaper. It is expected that mobile robot systems perform patrolling, sensing, and exploring in a harsh environment such as disaster area, deep sea, and space without any human intervention. Theoretical aspect of such mobile robot systems in the two dimensional Euclidean space (2D-space or plane) attracts much attention and distributed control of mobile robots with very weak capabilities has been investigated [1, 2, 4, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The robots are anonymous, oblivious (memory-less), have neither access to the global coordinate system nor explicit communication medium. For robots moving in the three dimensional Euclidean space (3D-space), we first investigate the plane formation problem, which is a fundamental self-organization problem that requires robots to occupy distinct positions on a common plane without making any multiplicity, mainly motivated by an obvious observation that robots on a plane would be easier to control than those moving in 3D-space.

In this paper, a robot is anonymous and is represented by a point in 3D-space. A robot repeats executing a “Look-Compute-Move” cycle, during which, it observes, in a Look phase, the positions of all robots by taking a snapshot, which we call a local observation in this paper, computes the next position based on a given deterministic algorithm in a Compute phase, and moves to the next position in a Move phase. This definition of Look-Compute-Move cycle implies that it has full vision, i.e., the vision is unrestricted and the move is atomic, i.e., each robot does not stop en route to the next position and we do not care which route it takes. A robot is oblivious if in a Compute phase, it uses only the snapshot just taken in the preceding Look phase, i.e., the output of the algorithm depends neither on a snapshot nor computation of the past cycles. Otherwise, a robot is non-oblivious. A robot has no access to the global -- coordinate system and all actions are done in terms of its local -- coordinate system. We assume that all local coordinate systems are right-handed. A configuration of such robot system is a set of points observed in the global coordinate system. Each robot obtains a configuration translated with its local coordinate system in a Look phase.

The robots can see each other, but do not have direct communication capabilities; communication among robots must take place solely by moving and observing robots’ positions with tolerating possible inconsistency among the local coordinate systems. The robots are anonymous; they have no unique identifiers and are indistinguishable by their looks and execute the same algorithm. Finally, they are fully synchronous (FSYNC); they all start the -th Look-Compute-Move cycle simultaneously and synchronously execute each of its Look, Compute, and Move phases.

The purpose of this paper is to show a necessary and sufficient condition for the robots to solve the plane formation problem.111Because multiplicity is not allowed, gathering at one point (i.e., point formation) is not a solution for the plane formation problem. The line formation problem in 2D-space is the counterpart of the plane formation problem in 3D-space and is unsolvable from an initial configuration if is a regular polygon (i.e., the robots occupy the vertices of a regular polygon), intuitively because anonymous robots forming a regular polygon cannot break symmetry among themselves and lines they propose are also symmetric, so that they cannot agree on one line from them [23]. Hence symmetry breaking among robots would play a crucial role in the plane formation problem.

The pattern formation problem requires robots to form a target pattern from an initial configuration and our plane formation problem is a subproblem of the pattern formation problem in 3D-space. To investigate the pattern formation problem in 2D-space, which contains the line formation problem as a subproblem, Suzuki and Yamashita [23] used the concept of symmetricity to measure the degree of symmetry of a configuration in 2D-space.222 The symmetricity was originally introduced in [24] for anonymous networks to investigate the solvability of some agreement problems. Let be a configuration. Then its symmetricity is the order of the cyclic group of , with the rotation center being the center of the smallest enclosing circle of , if . That is, is the number of angles such that rotating by () around produces itself, which intuitively means that the robots forming a regular -gon in may not be able to break symmetry among them.333We consider a point as a regular -gon with an arbitrary center and a set of two points as a regular -gon with the center at the midpoint. However, when , the symmetricity is defined to be independently of its rotational symmetry. This is the crucial difference between the cyclic group and the symmetricity that reflects the fact that the robot at can translate into another configuration with symmetricity by simply leaving . The following result has been obtained [18, 23, 25]: A target pattern is formable from an initial configuration , if and only if divides .

In this paper, based on the results in 2D-space, we measure the symmetry among robots in 3D-space by rotation groups, each of which is defined by a set of rotation axes and their arrangement. In 3D-space, such rotation groups with finite order are classified into the cyclic groups, the dihedral groups, the tetrahedral group, the octahedral group, and the icosahedral group. We call the cyclic groups and the dihedral groups two-dimensional (2D) rotation groups in the sense that the plane formation problem is obviously solvable from a configuration on which only a 2D rotation group acts, since there is a single rotation axis or a principal rotation axis and all robots can agree on a plane perpendicular to the axis and containing the center of the smallest enclosing ball of the robots. Then the oblivious (thus, non-oblivious) FSYNC robots can easily solve the plane formation problem by moving onto the agreed plane.

The other three rotation groups are recognized as the groups formed by the rotations on the corresponding regular polyhedra and they are also called polyhedral groups. A regular polyhedron consists of congruent regular polygons and all its vertices are congruent. A regular polyhedron has vertex-transitivity, that is, there are rotations that replace any two vertices with keeping the polyhedron unchanged as a whole. For example, we can rotate a cube around any axis containing two opposite vertices, any axis containing the centers of opposite faces, and any axis containing the midpoints of opposite edges. For each regular polyhedron, the rotations applicable to it form a group and, in this way, the tetrahedral group, the octahedral group, and the icosahedral group are defined.444 There are five regular polyhedra; regular tetrahedron, regular cube, regular octahedron, regular dodecahedron, and a regular icosahedron. A cube and a regular octahedron are dual each other, and so are a regular dodecahedron and a regular icosahedron. A tetrahedron is a self-dual. Since the same rotations are applicable to a regular polyhedron and its dual, there are three rotation groups. We call them three-dimensional (3D) rotation groups.

When a 3D rotation group acts on a configuration, the robots are not on one plane. In addition, the vertex-transitivity among the robots may allow some of them to have identical local observations. This may result in an infinite execution, where the robots keep symmetric movement in 3D-space and never agree on a plane. A vertex-transitive set of points is obtained by specifying a seed point and a set of symmetry operations, which consists of rotations around an axis, reflections for a mirror plane (bilateral symmetry), reflections for a point (central inversion), and rotation-reflections [9]. However, it is sufficient to consider vertex-transitive sets of points obtained by transformations that preserve the center of the smallest enclosing ball of the robots and keep Euclidean distance and handedness, in other words, direct congruent transformations, since otherwise, the robots can break the symmetry in a vertex-transitive set of points because all local coordinate systems are righthanded. Such symmetry operations consist of rotations around some axes. (See [8] for more detail.)

We define the rotation group of a configuration in 3D-space as the rotation group that acts on the configuration, i.e., a set of points. Let and be a set of points in 3D-space and its rotation group, respectively. Then the robots are partitioned into vertex-transitive subsets regarding , so that for each subset, the robots in it may have the same local observation. We call this decomposition -decomposition of . The goal of this paper is to show the following theorem:

Theorem 1.1

Let and be an initial configuration and the -decomposition of , respectively. Then irrespective of obliviousness, FSYNC robots can form a plane from if and only if (i) is a 2D rotation group, or (ii) is a 3D rotation group and there exists a subset such that .

We can rephrase this theorem as follows: FSYNC robots cannot form a plane from an initial configuration if and only if is a 3D rotation group and for each . The impossibility proof is by a construction based on the -decomposition of the robots. Obviously , and are the cardinalities of the 3D rotation groups and when a vertex-transitive set has a cardinality in , the corresponding rotation group allows “symmetric” local coordinate systems of those robots that allows identical local observations of those robots. Thus they move symmetrically regarding the rotation group that results in an infinite execution where the robots’ positions keep the 3D rotation group forever. Local memory at robots does not improve the situation since there exists an initial configuration where the positions and local coordinate systems of robots are symmetric and the contents of local memory at robots are identical, e.g., empty. Hence we have the same impossibility result for non-oblivious FSYNC robots.

Theorem 1.1 implies the following, which is somewhat counter-intuitive: The plane formation problem is solvable, even if the robots form a regular polyhedron except the regular icosahedron in an initial configuration , while it is unsolvable for the semi-regular polyhedra except an icosidodecahedron.

For the possibility proof, we present a plane formation algorithm for oblivious FSYNC robots, that non-oblivious FSYNC robots can execute by ignoring the content of their local memory. The proposed algorithm consists of a symmetry breaking algorithm and a landing algorithm. When the rotation group of an initial configuration is a 3D rotation group, the symmetry breaking algorithm translates into another configuration whose rotation group is a 2D rotation group. From the condition of Theorem 1.1, the -decomposition of contains one of the above five (semi-)regular polyhedra, i.e., a regular tetrahedron, a regular octahedron, a cube, a regular dodecahedron, and an icosidodecahedron. The symmetry breaking algorithm breaks the symmetry of these five polyhedra so that the resulting configuration as a whole has a 2D rotation group. Then the robots can agree on a plane as described before for the 2D-rotation groups and the landing algorithm assigns distinct landing points on the agreed plane. The landing algorithm is quite simple but contains some technical subtleties. We describe the entire plane formation algorithm with its correctness proofs.

Related works.  Autonomous mobile robot systems in 2D-space has been extensively investigated and the main research interest has been the computational power of robots. Many fundamental distributed tasks have been introduced, for example, gathering, pattern formation, partitioning, and covering. These problems brought us deep insights on the limit of computational power of autonomous mobile robot systems and revealed necessary assumptions of such systems to complete a given task. We survey the state of the art of autonomous mobile robot systems in 2D-space since there is few research on robots in 3D-space. The book by Flocchini et al. [14] contains almost all results on autonomous mobile robot systems up to year 2012.

Asynchrony and movement of robots are considered to be subject to the adversary. In other words, we consider the worst case scenario. Besides fully synchronous (FSYNC) robots, there are two other types of robots, semi-synchronous (SSYNC) and asynchronous (ASYNC) robots. The robots are SSYNC if some robots do not start the -th Look-Compute-Move cycle for some , but all of those who have started the cycle synchronously execute their Look, Compute and Move phases [23]. The robots are ASYNC if no assumptions are made on the execution of Look-Compute-Move cycles [17]. The movement of a robot is non-rigid if in each Move phase, the robot moves at least unknown minimum moving distance , but after moving it may stop on any arbitrary point on the track to the next position. If the length of the track to the next position is smaller than , it stops at the next position. If a robot reaches its next position in any Move phase, its movement is rigid. Most existing papers consider non-rigid movement of robots. Another important assumption is whether the robots agree on the clockwise direction, i.e., chirality. Most existing literature assumes non-rigid movement and chirality.

One of the most general form of formation tasks for autonomous mobile robot systems is the pattern formation problem that requires the robots to form a given target pattern. The pattern formation problem in 2D-space includes the line formation problem as a subproblem and Yamashita et al. investigated its solvability for each of the FSYNC, SSYNC and ASYNC models [18, 23, 25], that are summarized as follows: (1) For non-oblivious FSYNC robots, a pattern is formable from an initial configuration if and only if divides . (2) Pattern is formable from by oblivious ASYNC robots if is formable from by non-oblivious FSYNC robots, except for being a point of multiplicity 2.

This exceptional case is called the rendezvous problem. Indeed, it is trivial for two FSYNC robots, but is unsolvable for two oblivious SSYNC (and hence ASYNC) robots [23]. On the other hand, oblivious SSYNC (and ASYNC) robots can converge to a point. Therefore it is a bit surprising to observe that the point formation problem for more than two robots is solvable even for ASYNC robots. The result first appeared in [23] for SSYNC robots and then is extended for ASYNC robots in [5]. As a matter of fact, except the existence of the rendezvous problem, the point formation problem for more than two robots (which is also called as the gathering problem) is the easiest problem in that it is solvable from any initial configuration , since when is a point of multiplicity , and is always a divisor of by the definition of the symmetricity, where is the number of robots.

The other easiest case is a regular -gon (frequently called the circle formation problem), since . A circle is formable from any initial configuration, like the point formation problem for more than two robots. Recently the circle formation problem for oblivious ASYNC robots () is solved without chirality [15].

Das et al. considered formation of a sequence of patterns by oblivious SSYNC robots with rigid movement [11]. They showed that the symmetricity of each pattern of a formable sequence should be identical and a multiple of the symmetricity of an initial configuration. Such sequence of patterns is a geometric global memory formed by oblivious robots.

To circumvent the symmetricity and enable arbitrary pattern formation, Yamauchi and Yamashita proposed a randomized algorithm that allows the robots to probabilistically break the symmetricity of the initial configuration and showed that the oblivious ASYNC robots can form any target pattern with probability  [27].

The notion of compass was first introduced in [16] that assumes agreement of the direction and/or the orientation of - local coordinate systems. Flocchini et al. showed that if the oblivious ASYNC robots without chirality agree on the directions and orientations of and axes, they can form any arbitrary target pattern [17].

Flocchini et al. showed that agreement of the directions and orientation of both axes of local coordinate systems allows oblivious ASYNC robots with limited visibility to solve the point formation problem [16]. A robot has limited visibility if it can observe other robots within unknown fixed distance from itself. Agreement of the direction and the orientation of two axes can be replaced by agreement of direction and the orientation of one axis and chirality. Souissi et al. investigate the effect of the deviation of one axis from the global coordinate system at robots with chirality on the point formation problem and first introduced unreliable compasses, called eventually consistent compass, that is inaccurate for an arbitrary long time, i.e., it has an arbitrary deviation and the deviation dynamically changes, but eventually stabilizes to accurate axes [22]. Izumi et al. investigated the maximum static and dynamic deviation of compass for the point formation problem of two oblivious ASYNC robots [20].

Robustness of autonomous mobile robot systems has been discussed against error in sensing, computation, control, and several kinds of faults. A system is self-stabilizing if it accomplishes its task from an arbitrary initial configuration. A self-stabilizing system can tolerate any finite number of transient faults by considering the configuration after the final fault as an arbitrary initial configuration [12]. Suzuki and Yamashita pointed out that any oblivious mobile robot system is self-stabilizing since it does not depend on previous cycles [23]. Cohen and Peleg considered error in sensing, computation, and control, and showed acceptable range of them for oblivious ASYNC robots to converge to a point [7]. Two fundamental types of permanent faults in distributed computing are crash fault that stops the faulty entity and Byzantine fault that allows arbitrary (malicious) behavior of faulty entity. Cohen and Peleg considered the effect of crash faults at robots on the convergence problem for oblivious ASYNC robots [6]. Bouzid et al. considered the effect of Byzantine faults at robots on the convergence problem in one-dimensional space (i.e., line) for SSYNC and ASYNC robots [4]. Agmon and Peleg considered both crash faults and Byzantine faults for the point formation problem [1].

Efrima and Peleg considered the partitioning problem that requires the robots to form teams of size that divides  [13]. Without any compass, the partition problem is unsolvable from a symmetric initial configuration and they considered the availability of compass and asynchrony among robots. Izumi et al. proposed an approximation algorithm for the set cover problem of SSYNC robots that requires that for a given set of target points, there is at least one robot in a unit distance from each target point [19]. In contrast to the pattern formation problem, these problems have no (absolute) predefined final positions.

Computational power of robots with limited visibility and without any additional assumption has been also discussed. Yamauchi and Yamashita showed that oblivious FSYNC (thus SSYNC and ASYNC) robots with limited visibility have substantially weaker formation power than the robots with unlimited visibility [26]. Ando et al. proposed a convergence algorithm for oblivious SSYNC robots with limited visibility [2] while Flocchini et al. assumed consistent compass for convergence of oblivious ASYNC robots with limited visibility [16].

Peleg et al. first introduced the luminous robot model where each robot is equipped with externally and/or internally visible lights [21]. Light is an abstraction of both local memory and communication medium. Das et al. investigated the class of tasks that the luminous robots can accomplish [10]. They provided simulation algorithms for oblivious robots with constant number of externally visible bits to simulate robots without lights in stronger synchronization model.

All these papers discuss autonomous mobile robot systems in 2D-space and little is known when the robots are placed in 3D-space. This paper first investigates autonomous mobile robot systems in 3D-space and give a characterization of the plane formation problem.

Organization.   In Section 2, we first define the robot model and introduce the rotation group for points in 3D-space. Then we briefly show our main idea for the symmetry breaking algorithm. We start with some properties imposed on the robots by their rotation group in Section 3. In Section 4, we prove Theorem 1.1 by showing the impossibility of symmetry breaking and by presenting a plane formation algorithm for oblivious FSYNC robots for solvable instances. Finally, Section 5 concludes this paper by giving some concluding remarks.

2 Preliminary

2.1 Robot model

Let be a set of anonymous robots each of which is represented by a point in 3D-space. Their indices are used just for description. Without loss of generality, we assume , since all robots are already on a plane when . By we denote the global -- coordinate system. Let be the position of at time in , where is the set of real numbers. A configuration of at time is denoted by . We assume that the robots initially occupy distinct positions, i.e., for all . In general, can be a multiset, but it is always a set throughout this paper since the proposed algorithm avoids any multiplicity.555 It is impossible to break up multiple oblivious FSYNC robots (with the same local coordinate system) on a single position as long as they execute the same algorithm. Our algorithm is designed to avoid any multiplicity. However, we need to take into account any algorithm that may lead to a configuration with multiplicity when proving the impossibility result by reduction to the absurd. The robots have no access to . Instead, each robot has a local -- coordinate system , where the origin is always its current location, while the direction of each positive axis and the magnitude of the unit distance are arbitrary but never change. We assume that and all are right-handed. Thus is either a uniform scaling, transformation, rotation, or their combination of . By we denote the coordinates of a point in .

Each robot repeat a Look-Compute-Move cycle. We investigate fully synchronous (FSYNC) robots in this paper. They all start the -th Look-Compute-Move cycle simultaneously and synchronously execute each of its Look, Compute, and Move phases. We specifically assume without loss of generality that the -th Look-Compute-Move cycle starts at time and finishes before time . At time , each robot simultaneously looks and obtains a set666 Since changes whenever moves, notation is more rigid, but we omit parameter to simplify its notation.

We call the local observation of at . Next, computes its next position using an algorithm , which is common to all robots. If uses only , we say that is oblivious. Thus is a total function from to , where is the set of all configurations.777A configuration generally contains multiplicities and contains such configurations. However we do not assume multiplicity detection ability of robots. Thus the input to an algorithm is a set of points. As we will show later, the proposed pattern formation algorithm makes no multiplicity during any execution thus the input to the algorithm is always a set of points. Otherwise, we say is non-oblivious, i.e., can use past local observations and past outputs of . We say that a non-oblivious robot is equipped with local memory. Finally, moves to in before time . Thus we assume rigid movement.

An infinite sequence of configurations is called an execution from an initial configuration . Observe that the execution is uniquely determined, once initial configuration , local coordinate systems at time , local memory contents (for non-oblivious robots), and algorithm are fixed.

We say that an algorithm forms a plane from an initial configuration , if, regardless of the choice of initial local coordinate systems of robots and their initial memory contents (if any), for any execution , there exists finite such that satisfies the following three conditions:

(a)

is contained in a plane,

(b)

all robots occupy distinct positions in , and

(c)

for any , .

Because of , gathering the robots to one point (i.e., point formation) is not a solution for the plane formation problem.

2.2 Rotation groups in 3D-space

In 2D-space, the symmetricity of a set of points is defined by the order of its cyclic group, where the rotation center is the center of the smallest enclosing circle of , if . Otherwise, . Then is decomposed into regular -gons with being the common center, where  [23]. (See Figure 1.) Since the robots in the same regular -gon may have the same local observation, no matter which deterministic algorithm they obey, we cannot exclude the possibility that they continue to keep a regular -gon during the execution. This is the main reason that a target pattern is not formable from an initial configuration , if does not divide  [18, 24, 25].

Figure 1: A symmetric initial configuration in 2D-space, whose symmetricity is . Eight robots and their local coordinate systems are symmetric with respect to the center of their smallest enclosing circle. There are two groups consisting of symmetric robots and the robots in each group cannot break their symmetry.

In 3D-space, we consider the smallest enclosing ball and the convex hull of the positions of robots, i.e., robots are vertices of a convex polyhedron. Typical symmetric polyhedra are regular polyhedra (Platonic solids) and semi-regular polyhedra (Archimedean solids). A uniform polyhedron is a polyhedron consisting of regular polygons and all its vertices are congruent. Any uniform polyhedron is vertex transitive, i.e., for any pair of vertices of the polyhedron, there exists a symmetry operation that moves one vertex to the other with keeping the polyhedron as a whole. Intuitively, it makes sense to expect that all vertices (robots) in a uniform polyhedron may have identical local observations and might not break the symmetry in the worst case. The family of uniform polyhedra consists of regular polyhedra (the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron, and the regular icosahedron), 13 semi-regular polyhedra, and other non-convex 57 polyhedra.888 We do not consider Miller’s solid as semi-regular polyhedra though it satisfies the definition because we focus on rotation groups. Actually the rotation group of Miller’s solid is not a polyhedral group but . We do not care for non-convex uniform polyhedra. Contrary to the intuition above, we will show that when robots form a regular tetrahedron, a regular octahedron, a cube, a regular dodecahedron, or an icosidodecahedron, they can break their symmetry and form a plane.

In general, symmetry operations on a polyhedron consists of rotations around an axis, reflections for a mirror plane (bilateral symmetry), reflections for a point (central inversion), and rotation-reflections [9]. But as briefly argued in Section 1. since all local coordinate systems are right-handed, it is sufficient to consider only direct congruent transformations and those keeping the center. They are rotations around some axes that contains the center. We thus concentrate on rotation groups with finite order.

In 3D-space, there are five kinds of rotation groups of finite order each of which is defined by the set of rotation axes and their arrangement [9]. We can recognize each of them as the group formed by rotation operations on some polyhedron. Consider a regular pyramid that has a regular -gon as its base (Figure 2). The rotation operations for this regular pyramid is rotation by for around an axis containing the apex and the center of the base. We call such an axis -fold axis. Let be the rotation by around this -fold axis with where is the identity element. Then, form a group, which is called the cyclic group, denoted by .

A regular prism (except a cube) that has a regular -gon as its base has two types of rotation axes, one is the -fold axis containing the centers of its base and top, and the others are -fold axes that exchange the base and the top (Figure 2). We call this single -fold axis principal axis and the remaining -fold axes secondary axes. These rotation operations on a regular prism form a group, which is called the dihedral group, denoted by . The order of is . When , we can define in the same way, but in the group theory we do not distinguish the principal axis from the secondary one. Indeed, is isomorphic to the Klein four-group, denoted by , which is an abelian group and is a normal subgroup of the alternating group of degree , denoted by . Later we will show that we can recognize the principal axis of from the others because we consider rotations on a set of points.

The rotation axes of a regular polyhedron are classified into three types: The axes that contain the centers of opposite faces (type ), the axes that contain opposite vertices (type ), and the axes that contain the midpoints of opposite edges (type ). For each regular polyhedron, the rotation operations also form a group and the following three groups are called the polyhedral groups.

The regular tetrahedron has four -fold type (and ) axes and three -fold type axes (Figure 2). This rotation group is called the tetrahedral group denoted by . The tetrahedral group is isomorphic to and its order is .

The regular octahedron has four -fold type axes, three -fold type axes, and six -fold type axes (Figure 2). This rotation group is called the octahedral group denoted by . The octahedral group is isomorphic to the symmetric group of degree denoted by and its order is . 999Consider a cube to which we can perform the rotation of . Each rotation permutes the diagonal lines of the cube.

The regular icosahedron has ten -fold type axes, six -fold type axes, and fifteen -fold type axes (Figure 2). This rotation group is called the icosahedral group, denoted by . The icosahedral group is isomorphic to the alternating group of degree denoted by and its order is .

For each regular polyhedron, consider the center of each face. These centers also form a regular polyhedron, which is called the dual of the original regular polyhedron. Any dual polyhedron has the same rotation group as its original polyhedron. The regular tetrahedron is self-dual, the cube and the regular octahedron are dual each other, and so are the regular dodecahedron and the regular icosahedron. Hence we have three polyhedral groups.

Figure 2: Rotation groups: (a) the cyclic group , (b) the dihedral group , (c) the tetrahedral group , (d) the octahedral group , and (e) the icosahedral group . Figures show only one axis for each type.

Table 1 shows for each of the four rotation groups, , , and , the number of elements (excluding the identity element) around its -fold axes ().

Polyhedral group -fold axes -fold axes -fold axes -fold axes Order
3 8 - - 12
6 8 9 - 24
15 20 - 24 60
Table 1: Polyhedral groups. The number of elements around -fold axes (excluding the identity element) and their orders.

Let be the set of rotation groups, where is the rotation group with order 1; its unique element is the identity element (i.e., -fold rotation). When is a subgroup of (), we denote it by . If is a proper subgroup of (i.e., ), we denote it by . For example, we have , , but . If has a -fold axis, if divides .

We now define the rotation group of a set of points in 3D-space.

Definition 1

The rotation group of a set of points is the group that acts on and none of its proper supergroup in acts on .

Clearly, for any given set of points is uniquely determined. For example, when is the set of vertices of a cube, is the octahedral group . The major difference between the symmetricity in 2D-space and the rotation group in 3D-space is that even when the points of are on one plane, its rotation group is chosen from the dihedral groups and cyclic groups. In our context, symmetricity in 2D-space assumes the “top” direction against the plane where the points reside [18, 23, 25], while in 3D-space there is no agreement on the “top” direction.

For any , by and , we denote the smallest enclosing ball of and its center, respectively. From the definition, all rotation axis of contains and is the intersection of all rotation axes of unless . A point on the sphere of a ball is said to be on the ball, and we assume that the interior or the exterior of a ball does not include its sphere. When all points are on , we say that the set of points is spherical. For a ball , we denote the radius of the ball by in the coordinate system to observe .

We say that a set of points is transitive regarding a rotation group if it is an orbit of through some seed point , i.e., for some .101010 For a transitive set of points , any can be a seed point. The vertex-transitivity of uniform polyhedra corresponds to transitivity regarding a 3D rotation group. In the following, we use “vertex-transitivity” for a polyhedron while we use “transitivity” for a set of points. Note that a transitive set of points is always spherical.

Given a set of points , determines the arrangement of its rotation axes. We thus use and the arrangement of its rotation axes in interchangeably. For two groups , an embedding of to is an embedding of each rotation axis of to one of the rotation axes of so that any -fold axis of overlaps a -fold axis of with keeping the arrangement of where divides . For example, we can embed to so that each -fold axis of overlaps a -fold axis of , and each -fold axis of overlaps a -fold axis of . Note that there may be many embeddings of to . There are three embeddings of to depending on the choice of the -fold axis. Observe that we can embed to if and only if . For example, cannot be embedded to , since is not a subgroup of .

In the group theory, we do not distinguish the principal axis of from the other two -fold axes. Actually, since we consider the rotations on a set of points in 3D-space, we can recognize the principal axis of . Consider a sphenoid consisting of congruent non-regular triangles (Figure 3). A rotation axes of such a sphenoid contains the midpoints of opposite edges and there are three -fold axis perpendicular to each other. Hence the rotation group of the vertices of such a sphenoid is . However we can recognize, for example, the vertical -fold axis from the others by their lengths (between the midpoints connecting). The vertex-transitive polyhedra on which only can act are rectangles and the family of such sphenoids and we can always recognize the principal axis. Other related polyhedra are lines, squares, and regular tetrahedra, but acts on a line, acts on a square, and acts on a regular tetrahedron. Hence their rotation groups are proper supergroup of . We can show the following property regarding the principal axis of . See Appendix A for the proof.

Property 1

Let be a set of points. If acts on and we cannot distinguish the principal axis of (an arbitrary embedding of) , then .

Later we will show that the robots can form a plane if they can recognize a single rotation axis or a principal axis. Based on this, we say that the cyclic groups and the dihedral groups are two-dimensional (2D), while the polyhedral groups are three-dimensional (3D) since polyhedral groups do not act on a set of points on a plane.

Figure 3: A sphenoid consisting of congruent isosceles triangles. Its rotation group is . Since the vertices are not placed equidistant positions from the three axes, we can distinguish one axis as the principal axis from the others.

2.3 Basic idea

We first show an stimulating example that shows our idea of the symmetry breaking algorithm and the impossibility of the plane formation problem. From Theorem 1.1, oblivious FSYNC robots can form a plane from an initial configuration where four robots form a regular tetrahedron (i.e., they occupy the vertices of a regular tetrahedron). In such an initial configuration, their local observation may be identical because of the vertex-transitivity of the regular tetrahedron. If each robot proposes one plane, these four planes may be symmetric regarding , and because is three dimensional, these four planes never become identical. They must break their rotation group to form a plane. It is an essential challenge of this paper that the robots solve this symmetry breaking problem by a deterministic algorithm.

We introduce a simple “go-to-midpoint” algorithm for the robots to break the regular tetrahedron. This algorithm makes each robot select an arbitrary edge of the regular tetrahedron which is incident to the vertex it resides and goes along the edge, but stops it before the midpoint, where is of the length of the edge. The selection is somehow done in a deterministic way. We briefly show that this go-to-midpoint algorithm successfully breaks the symmetry of the regular tetrahedron and the robots can form a plane. We could have a better understanding of the execution by illustrating the positions of robots in an embedding of the regular tetrahedron to a cube. Figure 4 shows an initial configuration . Since at least two edges are selected by the four robots, we have the following three cases.

Case A:  Two edges are selected. See Figure 4. The two edges are opposite edges and the robots form skew lines of length , since otherwise, two edges cannot cover the four vertices. The four robots can agree on the plane perpendicular to the line segment containing the midpoints of the skew lines and containing its midpoint.

Case B:  Three edges are selected. See Figures 44 and 4. There is only one pair of robots with distance and the four robots can agree on the plane formed by the midpoint of the two robots with distance and the positions of the remaining two robots.

Case C:  Four edges are selected. If three of the selected edges form a regular triangle (Figure 4), the distance from the remaining robot to two of the three robots is larger than the edge of the regular triangle. Hence, the four robots can agree on the plane containing the regular triangle. Otherwise, the selected edges form a cycle on the original regular tetrahedron (Figure 4). In this case, the four robots form a set of skew lines and can agree on the plane like (A).

In each case, the four robots can land on the foot of the perpendicular line to the agreed plane starting from its current position. They succeed in plane formation since they are FSYNC.

Figure 4: Execution of the go-to-midpoint algorithm from an initial configuration where the four robots form a regular tetrahedron.

One might expect that the go-to-midpoint algorithm could be used to break symmetry of any other regular polyhedra because of the Euler’s equality: For a polyhedron with vertices, edges, and faces, we have . If the go-to-midpoint algorithm is executed in such a configuration, since and hence , there exists at least one edge which is selected by two robots or is not selected by any robot. However, as a matter of fact, the go-to-midpoint algorithm does not work, for example, when the robots form a regular icosahedron. Figure 5 shows an example of a configuration obtained by the go-to-midpoint algorithm from an initial configuration where the robots form a regular icosahedron. The robots cannot agree on a plane in because a 3D rotation group cats on as shown in Figure 5 and the twelve planes that the robots propose are not identical. Later we will show that the robots following any algorithm cannot agree on a plane forever from this configuration irrespective of obliviousness.

The “go-to-midpoint” algorithm shows that the robots can reduce their rotation group by deterministic movement, while in some cases this reduction stops at some subgroup of the rotation group of the initial configuration. Our plane formation algorithm proposed in Subsection 4.2 translates an initial configuration whose rotation group is a 3D rotation group to another configuration whose rotation group is a 2D rotation group. Then robots can agree on a plane that is perpendicular to the single (or principal) axis and contains the center of their smallest enclosing ball. Then they land on the plane.

To show a necessary condition, we characterize the initial configurations from which the robots cannot always form a plane in terms of the rotation group and the number of robots.

Figure 5: An example of a resulting configuration of the go-to-midpoint algorithm from an initial configuration where the robots form a regular icosahedron. (a) A resulting configuration. (b) The rotation group of the resulting configuration is and its -fold axes are illustrated. (We omit the four -fold axes.)

3 Decomposition of the robots

In this section, we will show that the robots can agree on some global properties by using the rotation group of their positions. In a configuration , each robot can obviously calculate from by checking all rotation axes that keep unchanged. Then the group action of decomposes into a family of transitive sets of points and the robots can agree on the ordering of these elements. As we will show in Section 4.1, each of these elements are a set of indivisible robots in the worst case that have the same local observation, move symmetrically, and keep forever. On the other hand, this ordering allows us to control the robots in some order and plays an important role when we design a plane formation algorithm for solvable initial configurations. We start with the following theorem.

Theorem 3.1

Let be a configuration of robots represented as a set of points. Then is decomposed into disjoint sets so that each is transitive regarding . Furthermore, the robots can agree on a total ordering among the elements.

Such decomposition of is unique as a matter of fact and we call this decomposition the -decomposition of . Let us start with the first part of Theorem 3.1.

Lemma 1

Let be a configuration of robots represented as a set of points. Then is decomposed into disjoint sets so that each is transitive regarding .

Proof

For any point , let be the orbit of the group action of through . By definition is transitive regarding . Let be its orbit space. Then is obviously a partition, which satisfies the property of the lemma. Additionally, such decomposition is unique.

Note that () may not hold, while in 2D-space a set of points is decomposed into regular -gons by  [23, 25, 18]. Consider a configuration consisting of the vertices of a regular tetrahedron (4 vertices) and the vertices of a truncated tetrahedron (12 vertices) (Figure 6). Then and the sizes of the elements of the -decomposition of are different.

Figure 6: A set of points consisting of 16 points. Its rotation group is is and the -decomposition of consists of two elements: a set of points forming a regular tetrahedron (of size 4) and a set of points forming a truncated tetrahedron (of size 12).

Let us go on the second part of the theorem. For the robots to consistently compare two elements and of the -decomposition of , each robot computes the “local view” of each robot which is determined only by configuration independently of its local coordinate system , although observes in .

Local views of robots defined in this section satisfy the following properties:

  1. For each (), all robots in have the same local view.

  2. Any two robots, one in and the other in , have different local views, for all .

Then we give an expression of a local view as a sequence of positions of the robots and by using the lexicographic ordering of local views, the robots agree on a total ordering among , i.e., is smaller than if and only if the local view of some is smaller than that of some in the lexicographic order.

To define the local view of a robot, we first introduce amplitude, longitude and latitude. Let be a configuration, where is the position (in ) of . Assume that is not contained in a plane and , because otherwise the plane formation is trivially solvable as we will show later.111111 We note that when the robots are on a plane (especially, when the robots are on a line), we cannot define the local view in the same way. The innermost empty ball is the ball centered at and contains no point in in its interior and contains at least one point in on it. Since , is well-defined. Intuitively, considers as the earth, and the line containing and as the earth’s axis. Recall that can recognize its relative positions from the others, since always holds. The intersection of a line segment and is the “north pole” . Then it chooses a robot not on the earth’s axis as its meridian robot. Indeed, there is a robot satisfying the condition by the assumption that the robots are not on one plane. The meridian robot should be chosen more carefully for our purpose as shown later. Let be the intersection of a line segment and . The large circle on containing and defines the “prime meridian”. Specifically, the half arc starting from and containing is the prime meridian. Robot translates its local observation with geocentric longitude, latitude, and altitude. The position of a robot is now represented by the altitude in , longitude in , and latitude in . Here the altitude of a point on is 0, and that on is . The longitude of is 0, and the positive direction is the counter-clockwise direction. Since the robots are all right-handed they can agree on the counter clockwise direction (i.e., rotating positive -axis to positive -axis) on by using . For example, the robots can agree on the clockwise direction by considering that the negative -axis of their local coordinate systems point to . Finally, the latitudes of the “north pole” , the “equator,” and the “south pole” are 0, , and , respectively.

Now is represented by a triple (or more formally, transforms to ) for all , where by definition. Observe that depends on the choice of the meridian robot and if and only if . See Figure 7 as an example.

Figure 7: Amplitude, longitude, and latitude calculated from ’s local observation. The prime meridian for is drawn by bold arc. The position of is now represented by a triple .

We then use the lexicographic ordering among the positions to compare them: For two positions and , if and only if (i) , (ii) and , or (iii) , and .

Let be a sorted list of the positions , in which the positions and its meridian robot are placed as the first and the second elements and the positions of the other robots are placed in the increasing order, i.e., for all , and where the ties are arbitrarily resolved.

Let us return to the problem of how to choose the meridian robot . As explained, depends on the choice of . Robot computes the robot that minimizes in the lexicographical order and chooses it as the meridian robot , where a tie is resolved arbitrarily. We call this minimum (for chosen in this way) the local view of . Regardless of the choices of meridian robot by robot , the next lemma holds.

Lemma 2

Let and be a configuration of robots represented as a set of points and its -decomposition, respectively. Then we have the following two properties:

  1. For each (), all robots in have the same local view.

  2. Any two robots, one in and the other in , have different local views, for all .

Proof

The first property is obvious by the definitions of -decomposition and local view, since for any there is an element such that .

As for the second property, to derive a contradiction, suppose that there are distinct integers and , such that robots and have the same local view. That is, . Let us consider a function that maps the -th element of to that of . More formally, letting the -th element of (resp. ) be (reps. ), maps to . Then is a congruent transformation that keeps unchanged by the definition of local view, i.e., is a rotation in , which contradicts to the definition of -decomposition.

Corollary 1

Let and be a configuration of robots represented as a set of points and its -decomposition, respectively. Then the robots can agree on a total ordering among these subsets.

Proof

By using the lexicographical ordering of the local views of robots in each element of the -decomposition of .

We now conclude Theorem 3.1 by Lemma 1 and Corollary 1. In the following, we assume that the -decomposition of , is ordered in this way. From the definition, is on , is on , and is in the interior or on the ball that is centered at and contains on it.

We go on to the analysis of the structure of a transitive set of points regarding a 3D rotation group. Recall that a transitive set of points is spherical. Any transitive set of points is specified by a rotation group and a seed point as the orbit of the group action of through , so that holds. Not necessarily holds. For any , we call the folding of . We of course count the identity element of for and holds for all .121212 In group theory, the folding of a point is simply the size of the stabilizers of defined by . Although the lemma is known in group theory (see e.g., [3]), we provide a proof for the convenience of readers.

Lemma 3

Let be the transitive set of points generated by a rotation group and a seed point . If is on a -fold axis of for some , so are the other points and holds. Otherwise, if is not on any axis of , so are the other points and holds.

Proof

We first show that for any . To derive a contradiction, we assume for some . Let (resp. ) be the set of rotations in such that (resp. ) holds for (resp. ). Clearly for any and . Let be a rotation satisfying , which definitely exists by definition. Hence for all , a contradiction, since if , and holds.

Note that the seed point can be taken as in the above proof. Suppose that is on a -fold axis of , then , since the rotations in that move to itself are the rotations around this -fold axis.

Otherwise if is not on a rotation axis of , only the identity element of can move to itself and hence .

Lemma 4

When a set of points is transitive regarding , then we have .

Proof

By Lemma 3, we can compute the cardinality of any transitive set of points for each rotation group.

The tetrahedral group consists of -fold axes and -fold axes, and its order is . If we put a seed on a -fold axis, we obtain a -set as forming a regular octahedron. If we put a seed on a -fold axis, we obtain a -set as forming a regular tetrahedron. If we put a seed not on any axis, we obtain a -set as .

By the same argument, we have the following results: The order of the octahedral group is and the possible cardinalities of are , and . The order of the icosahedral group is and the possible cardinalities of are , and .

By Lemmas 3 and Lemma 4, folding of a point determines the positions of a transitive set of points in the arrangement of rotation axes and these polyhedra are shown in Table 2. When the folding is , a seed point can be taken any point not on any rotation axis and depending on the seed point, infinite number of different polyhedra are obtained.131313 Table 2 does not contain all uniform polyhedra. There are uniform polyhedra consisting of vertices or vertices, such as a rhombitruncated cuboctahedron with vertices and a rhombitruncated icosidodecahedron with vertices. However, they require a mirror plane to induce transitivity and the robots with right-handed local coordinate systems can partition them into two groups. For example, a rhombitruncated cuboctahedron is decomposed into two -sets by its rotation group . We have the following property by the definition of -decomposition of a set of points .

Property 2

Let and be a set of points and its -decomposition, respectively. Then if is a 3D rotation group, is one of the polyhedra shown in Table 2 for .

Rotation group Order Folding Cardinality Polyhedra
3 4 Regular tetrahedron
12 2 6 Regular octahedron
1 12 Infinitely many polyhedra
24 4 6 Regular octahedron
3 8 Cube
2 12 Cuboctahedron
1 24 Infinitely many polyhedra
60 5 12 Regular icosahedron
3 20 Regular dodecahedron
2 30 Icosidodecahedron
1 60 Infinitely many polyhedra
Table 2: Transitive sets of points in 3D-space (i.e., polyhedra) characterized by rotation group and folding.

4 Proofs of Theorem 1.1

We show the proofs of Theorem 1.1 in this section. In Subsection 4.1, we first show the necessity of Theorem 1.1 by showing that any algorithm for oblivious FSYNC robots cannot form a plane from an initial configuration if the initial configuration does not satisfy the condition in Theorem 1.1. Specifically, for any initial configuration that satisfies is in and the size of each element of its -decomposition is in , we construct an arrangement of initial local coordinate systems that makes the robots keep the rotation axes of a 3D rotation group forever so that they never form a plane no matter which algorithm they obey. The orders of , , and are , , and , respectively and when an initial configuration does not satisfy the condition of Theorem 1.1, we can decompose the robots into transitive subsets so that the cardinality of each subset is “full” regarding a 3D rotation group (not necessarily ). Then we show that there exists an arrangement of local coordinate systems that is also transitive regarding the selected rotation group so that the robots continue symmetric movement forever. The impossibility proof holds for non-oblivious robots because starting from such a symmetric initial configuration , the contents of memory at robots in the same element are kept identical and if the initial memory content of the robots are identical, they cannot break the symmetry. Thus we obtain the necessity of Theorem 1.1.

In Subsection 4.2, we show the sufficiency of Theorem 1.1 by presenting a plane formation algorithm for oblivious FSYNC robots. When of an initial configuration is a 2D rotation group, the robots are on one plane or they can agree on the plane that is perpendicular to the single rotation axis (or the principal axis). Actually, the robots can land on such a plane without making any multiplicity. On the other hand, when is a 3D rotation group, the condition of Theorem 1.1 guarantees that there exists an element in the -decomposition of that forms a regular tetrahedron, a regular octahedron, a cube, a regular dodecahedron, or an icosidodecahedron (Table 2). The proposed algorithm adopts the “go-to-center” strategy, which is very similar to the “go-to-midpoint” algorithm in Subsection 2.3. Then we show that after the movement, the rotation group of the robots’ positions is not a 3D rotation group any more intuitively because the candidates of next positions form a transitive set of points, while the number of the robots is not sufficient to select a set of points with 3D rotation group from such set of points. Because their rotation group is a 2D rotation group, the robots can form a plane. Clearly non-oblivious FSYNC robots can execute the proposed algorithm and we obtain the sufficiency of Theorem 1.1.

4.1 Necessity

Provided , we first show that when a set of points is a transitive set of points regarding a 3D rotation group, there is an arrangement of local coordinate system for each robot such that the execution from keeps a 3D rotation group forever no matter which algorithm the oblivious FSYNC robots obey.

Lemma 5

Consider oblivious FSYNC robots with . Then the plane formation problem is unsolvable from an initial configuration if is a transitive set of points regarding a 3D rotation group.

Proof

Let be an initial configuration of robots that is transitive regarding .

To derive a contradiction, we assume that there is an algorithm that enables the robots to solve the plane formation problem for any choice of initial arrangement of local coordinate systems of robots. We will show that there is an initial arrangement of local coordinate systems such that the robots move symmetrically and keep the axes of rotation group forever, where is given as follows depending on :

We first claim that there is always an embedding of to . The claim obviously holds when . Suppose . Then , since otherwise (i.e., is either or ), by Table 2 and by the definition of . If , then by the definition of . Since , the claim holds.

We fix an arbitrary embedding of to . For any point , we next claim and is the order of , i.e., . Obviously the claim holds when from the definition. Suppose that . Then , and by the argument above. If , all points in are on -fold axes of from Table 2, but there is no embedding of to that makes the rotation axes of overlap -fold axes of . That is, regarding is . Otherwise if , like the above case, all points in are on -fold axes of from Table 2, but there is no embedding of to that makes the rotation axes of overlap -fold axes of . That is, of is .

Now we define a local coordinate system for each by using , the local coordinate system of , so that any algorithm produces an execution such that is a subgroup of for all . We define and is specified by . Let , where is the position of at time . For each , there is an element such that , and this mapping between and is a bijection between and , i.e., if , and because . Thus is the identity element. Local coordinate system is specified by the positions of its origin , , and in . That is, we can specify by a quadruple . Define as the coordinate system specified by a quadruple , for . 141414Recall that here means at time 0.

Then for and outputs the same value in every robot as its next position. Let be this output at observed in . Then we have . That is, is the orbit of through and obviously is a subgroup of . By an easy induction, we can show that is a 3D rotation group for .

We finally address multiplicity during any execution of . Algorithm may move some robots to one point at some time . Because , all robots gather at one point. However, since further needs to move the robots to distinct positions by the definition of the plane formation problem, must hold, that is, outputs a point that is different from the current position (i.e., the origin of ) as the next position and these destinations form a transitive set of points regarding or its supergroup in . Thus the robots never form a plane.

Lemma 5 considers an arbitrary transitive initial configurations regarding a 3D rotation group. We next extend it to handle general initial configurations, which may not be transitive. Let be the -decomposition of an initial configuration . Intuitively, we wish to specify for in the same way as the proof of Lemma 5 for each (). We however need to take into account the cases in which and for is different from the one for . For example, consider a configuration consisting of a regular icosahedron ( points) and a truncated icosahedron ( points), where . Then the -decomposition of consists of the regular icosahedron and the truncated icosahedron , and for is , while it is for . In this case, we make use of the -decomposition (instead of the -decomposition) of and apply Lemma 5 to each element of the -decomposition of . Then we show that any execution keeps the rotation axes of forever.

Theorem 4.1

Let and be an initial configuration and the -decomposition of , respectively. Then the plane formation problem is unsolvable from for oblivious FSYNC robots, if is a 3D rotation group and for .

Proof

Let be the -decomposition of an initial configuration . We define the rotation group by:

We show that there exists an arrangement of local coordinate systems of robots that makes the robots keep the rotation axes of forever regardless of the algorithm they obey.

By Table 2, or is a subgroup of and there is an embedding of to . We fix an arbitrary embedding of to , and consider the -decomposition of which is defined in the same way as the -decomposition. Formally, consider the orbit space regarding .

For example, let be a cuboctahedron embedded in a truncated cube as illustrated in Figure 8. Then . The -decomposition is , where the cardinalities of the elements are 12 and 24. By definition, . The -decomposition of is , that is obtained with seed points , and , and the orbit regarding through for . (See Figure 8.)

Figure 8: (a) consists of a cuboctahedron and a truncated cube so that its . (b) Three seed points . (c) of through , whose cardinality is 12. (d) of through , whose cardinality is 12. (e) of through , whose cardinality is 12.

We first show that for each (), , thus for any , regarding . Let be the set of sizes of the elements of the -decomposition of , that is, . Observe that , since while implies , there is no transitive set of points with regarding by Lemma 4. Hence . Depending on , we have the following three cases.

Case A: The case is trivial. When , we must consider the following two cases.

Case A1:  When and . Then . Let be any point for . By definition regarding for some . Observe that under an arbitrary embedding of to , is not on any rotation axis of , since otherwise, regarding is or , and is or . Consequently, there is no point in that is on a rotation axis of any embedding of to . Thus we have for .

Case A2:  When and . Then . The proof is exactly the same as (A1), except that, in this case, we observe that there is no point in that is on a rotation axis of any embedding of to , since otherwise is or . Thus we have for .

Case B: Then and we have the following two cases.

Case B1:  When . Then and