Combinatorics of Bricard's octahedra
CCombinatorics of Bricard’s octahedra
Matteo Gallet ∗ , (cid:5) Georg Grasegger ∗ ,. Jan Legerský ◦ Josef Schicho ∗ , ◦ We re-prove the classification of flexible octahedra, obtained by Bricardat the beginning of the XX century, by means of combinatorial objectssatisfying some elementary rules. The explanations of these rules rely onthe use of a well-known creation of modern algebraic geometry, the modulispace of stable rational curves with marked points, for the description ofconfigurations of graphs on the sphere. Once one accepts the objects andthe rules, the classification becomes elementary (though not trivial) andcan be enjoyed without the need of a very deep background on the topic.
Cauchy proved [Cau13] that every convex polyhedron is rigid, in the sense that itcannot move keeping the shape of its faces. Hence flexible polyhedra must be concave,and indeed Bricard discovered [Bri97, Bri26, Bri27] three families of concave flexibleoctahedra. Lebesgue lectured about Bricard’s construction in 1938/39 [Leb67], andBennett discussed flexible octahedra in his work [Ben12]. In recent years, there hasbeen renewed interest in the topic; see the works of Baker [Bak80, Bak95, Bak09],Stachel [Sta87, Sta14, Sta15], Nawratil [Naw10, NR18], and others [Con78, BS90,Mik02, Ale10, AC11, Nel10, Nel12, CY12]. The goal of this paper is to re-proveBricard’s result by employing modern techniques in algebraic geometry that hopefullymay be applied to more general situations.The three families of flexible octahedra are the following (see Figures 1 and 2): ∗ Supported by the Austrian Science Fund (FWF): W1214-N15, project DK9. ◦ Supported by the Austrian Science Fund (FWF): P31061. (cid:5)
Supported by the Austrian Science Fund (FWF): Erwin Schrödinger Fellowship J4253. . Supported by the Austrian Science Fund (FWF): P31888. a r X i v : . [ m a t h . M G ] A p r igure 1: Flexible octahedra of Type I and Type II found by Bricard. Type I.
Octahedra whose vertices form three pairs of points symmetric with respectto a line.
Type II.
Octahedra whose vertices are given by two pairs of points symmetric withrespect to a plane passing through the last two vertices.
Type III.
Octahedra all of whose pyramids have the following property: the two pairsof opposite angles are constituted of angles that are either both equal or bothsupplementary; moreover, we ask the lengths ‘ ij of the edges to satisfy threelinear equations of the form: η ‘ + η ‘ + η ‘ + η ‘ = 0 ,η ‘ + η ‘ + η ‘ + η ‘ = 0 ,η ‘ + η ‘ + η ‘ + η ‘ = 0 , where η ij ∈ { , − } and in each equation we have exactly two positive η ij andtwo negative ones.Animations of the motions of each of the three families can be found at https://jan.legersky.cz/project/bricard_octahedra/ . Here by “pyramid” we mean a 4-tuple of edges sharing a vertex. See Definition 2.4 for formalspecification and notation. Here by “angle” of a pyramid we mean the angle formed by two concurrent edges belonging to thesame face. Here we label the vertices of the octahedron by the numbers { , . . . , } . A and A and two circles, anddraw the tangent lines to the circles passing through the points A i . Theselines determine four other points B , B , C and C , which together with A and A define a flat realization of a Type III octahedron.The fact that an octahedron with line symmetry is flexible is well-known (see, forexample, [Sch10, Section 5]) and follows from a count of the free parameters versusthe number of equations imposed by the edges. A similar argument also shows thatplane-symmetric octahedra are flexible. Proving that Type III octahedra are flexibleis more complicated, and for this we refer to the proof given by Lebesgue (see [Leb67]).The technique we adopt to analyze flexible octahedra is to reduce to the case of flexiblespherical linkages, and to use the tools developed in our previous work [GGLS19] toderive the classification. More precisely, our work consists of two parts: in the firstpart, we prove some elementary facts about flexible octahedra and we provide theirclassification by using combinatorial objects called octahedral and pyramidal bonds ,and rules that relate them; in the second part, we explain the rules via the theorydeveloped in [GGLS19] on flexible graphs on the sphere. The first part is rathernontechnical and aimed at a general public; the second part involves more technicalitiesand requires some acquaintance with the material from [GGLS19] for the detailedjustification of the arguments. By splitting the text in such a way we hope to widenthe possible readership to those readers who may not be extremely interested in thespecific details of the algebro-geometric part of the proof but are fascinated by this oldtopic; at the same time, we hope to convince them that the techniques we introduceby employing objects from modern algebraic geometry may be well-suited for these3lassical questions, and may have the chance to shed light on related topics that havenot been fully investigated yet.The paper is structured as follows. Section 2 reports elementary results on flexibleoctahedra. Section 3 provides the classification of flexible octahedra by introducingoctahedral and pyramidal bonds, and by setting up, in an axiomatic way, the rulesthat guide their behavior. The constraints imposed by these rules are then used toclassify flexible octahedra. Section 4 describes how realizations of octahedra in thespace determine realizations on the unit sphere of the graph whose vertices are theedges of the octahedron, and whose edges encode the fact that edges of the octahedronlie on the same face. This opens the way to the use of the methods developed by theauthors in [GGLS19], namely to the study of flexible graphs on the sphere. Section 5provides the precise background for the notion of bonds, and justifies the rules inSection 3 via the techniques from [GGLS19]. In this section we collect some results about the motions of octahedra, which we usein later sections. The results are known and elementary; we report them here mainlyfor self-containedness.We represent the combinatorial structure of an octahedron by the graph G oct withvertices { , . . . , } and edge set E oct given by all unordered pairs { i, j } where i, j ∈{ , . . . , } except for { , } , { , } , and { , } (see Figure 3). Figure 3: The octahedral graph G oct . Definition 2.1. A realization of an octahedron is a map ρ : { , . . . , } −→ R . A labeling is a map λ : E oct −→ R > ; we use the notation λ { i,j } for λ ( { i, j } ). A real-ization ρ is compatible with a labeling λ if k ρ ( i ) − ρ ( j ) k = λ { i,j } for all { i, j } ∈ E oct .4 labeling λ is called flexible if there exist infinitely many non-congruent realizationscompatible with λ . Two realizations ρ and ρ are called congruent if there exists anisometry σ of R such that ρ = σ ◦ ρ .We formalize the intuitive notion of motion of an octahedron by exploiting the factthat being compatible with a labeling imposes polynomial constraints on realizations.Notice that, since we can always apply rotations or translations to a given realizationto obtain another compatible realization, once we have a compatible realization, weactually have a 6-dimensional set of compatible ones. Therefore motions are asked tobe 7-dimensional objects, in order to encode octahedra that move with one degree offreedom , up to rotations and translations. Definition 2.2.
Given a flexible labeling of an octahedron, the set of realizationscompatible with it is an algebraic variety. Indeed, all realizations are a real vectorspace of dimension 3 × motion of the octahedron is a 7-dimensional irreduciblecomponent of this algebraic variety.The goal of this paper is to classify flexible octahedra satisfying the following genericityassumption. Assumption G.
No two faces of the octahedron are coplanar for a general realizationin a motion.
Remark 2.3.
An elementary, but tedious, inspection of all possibilities shows that ifa triangle in an octahedron degenerates, then the octahedron does not have flexiblelabelings. Moreover, as a corollary of the assumption we have that no two vertices ofthe octahedron coincide for a general realization in a motion.A key object in our proof of the classification of octahedra are pyramids . Definition 2.4.
Pyramids are subgraphs of G oct induced by a vertex and its fourneighbors. The pyramid determined by v is denoted by v . Realizations of pyramids,their congruence, and flexibility are defined analogously as for octahedra. The samehappens for motions. Here we make a similar request as in Assumption G, namely wedo not allow motions for which two triangular faces of a pyramid are coplanar in ageneral realization of that motion.Flexible pyramids come in four families (here by an angle of a pyramid v we meanan angle between edges of the form { u, v } and { w, v } , where u and w are neighbors):5 eltoids: here two disjoint pairs of adjacent angles are constituted of angles that areeither both equal or both supplementary; rhomboids: here the two pairs of opposite angles are constituted of angles that areeither both equal or both supplementary; lozenges: here none of the angles equals π/ ; general: here are flexible pyramids such that not all angles equal π/ Remark 2.5.
Given these definitions, we can say that a Type III octahedron is anoctahedron where all pyramids are rhomboids or lozenges, and for which the equationson the edge lengths from the introduction hold.
Hereafter we list some elementary properties of pyramids, in particular concerningtheir planar (also called flat ) realizations.
Fact 2.6.
Deltoids and rhomboids have two flat realizations. Lozenges have three flatrealizations. In the case of deltoids and lozenges, in these positions three vertices ofthe pyramid are collinear. When all the angles are π/
2, any motion of such a pyramid is degenerate, namely two triangularfaces stay coplanar during the motion. v v Figure 5: The three flat realizations of a lozenge.
Fact 2.7.
If a realization of a rhomboid has one dihedral angle between its triangularfaces which is or π , then the realization is flat; vice versa, if a realization of such apyramid is flat, then all dihedral angles are or π . Fact 2.8.
Consider a non-degenerate motion of a lozenge. The flat realizations insuch a motion are precisely the ones where all dihedral angles are or π . Remark 2.9.
For deltoids and lozenges, there are non-flat realizations where onedihedral angle between its triangular faces is 0 or π . Notice that these non-flat real-izations appear, for example (though not only, in the case of deltoids), in degenerate motions, namely when two pairs of faces stay always coplanar during the motion. Definition 2.10.
A dihedral angle between two triangular faces of a flexible pyramidis simple with respect to a motion of the pyramid if, once we fix a general value forthat angle, there exists a unique (up to isometries) realization in that motion for whichthe angle has the given value.Notice that a lozenge has four simple dihedral angles, while a deltoid has two simpledihedral angles. Fact 2.11.
A deltoid is in a flat realization if and only if one of its simple dihedralangles between triangular faces are or π . Proposition 2.12.
If a flexible octahedron has two neighbor rhomboids or lozengesamong its pyramids, then it admits flat realizations. Here, two pyramids v and w are neighbors if the vertices v and w are connected by an edge.Proof. Suppose that the two neighbor pyramids are 1 and 3 . Suppose we are in arealization that is flat for 1 . Then the dihedral angle between the planes 135 and 136is 0 or π . Hence, by Facts 2.7 and 2.8 this realization is also flat for 3 . Remark 2.13.
By Proposition 2.12, Type III octahedra admit two flat realizations. Recall that we always consider non-degenerate motions of pyramids. Classification of flexible octahedra
In this section we provide the classification of flexible octahedra, reproving the knownresults by Bricard. We do this by attaching combinatorial objects to flexible octahedraand prescribing rules for these objects. Eventually, the rules determine constraints onedge lengths and angles, which can be grouped in four cases. By analyzing each ofthese cases, we classify flexible octahedra into the three families introduced by Bricardand described in the introduction.The justification for the rules is provided in Section 5, and requires the algebro-geometric notion of moduli space of rational stable curves with marked points , togetherwith the theory developed by the authors in [GGLS19] about flexible graphs on thesphere. Once the rules are established, however, the derivation of the classificationis combinatorial in nature, and uses the elementary facts about octahedra reportedin Section 2. We believe that inserting this combinatorial “extra-layer” in the proofhas two advantages: it helps highlighting the structure of the proof and separatinglogically independent units, and facilitates readers that may not be interested in thealgebro-geometric technicalities to follow the proof of the classification.
We are going to introduce two combinatorial objects that will guide the classification,called octahedral and pyramidal bonds . These are graphical representations of “pointsat infinity” of the space of realizations of an octahedron. Key ingredients in ourconstructions are quadrilaterals in G oct ; they are induced subgraphs isomorphic to thecycle C on four vertices. Notation.
There exist exactly three quadrilaterals of G oct : they are those inducedby the vertices { , , , } , { , , , } , and { , , , } . Each quadrilateral is completelyspecified by the pair of vertices not appearing in it, which form a non-edge in G oct .Therefore we can label the quadrilaterals by 12, 34, and 56. Octahedral bonds
Octahedral bonds are quadrilaterals of G oct together with a choice of orientation forall edges in the quadrilateral. There are, therefore, 16 octahedral bonds supported ona given quadrilateral in G oct . Octahedral bonds come with some multiplicity, whichwe call the µ -number . 8 yramidal bonds A pyramidal bond is a pyramid v , together with a direction on two edges incidentto v that are not in the same triangle subgraph (see Figure 6). There are hence 8pyramidal bonds supported on a given pyramid v in G oct . Figure 6: A pyramidal bond defined by vertex 5.We fix a standard representation of a pyramid v by specifying which vertex is drawnwhere. More precisely, we draw the pyramid as a square with the vertex v in themiddle. Then we take the clockwise neighbor of v in the drawing of Figure 3 to beon the bottom right corner of the square. The other vertices are drawn accordingly tothe clockwise order (see Figure 7). vw w w w w w w w v Figure 7: A pyramid in standard representation.The standard representation of pyramids provides a standard way to represent pyra-midal bonds:
P P P P P P P P
9f we want to specify that a bond is supported on a pyramid v , we put the symbol v as superscript, as for example in P v . We introduce the rules that are satisfied by octahedral and pyramidal bonds; theirformal justification will be provided in Section 5. First of all, we fix a motion of anoctahedron. This induces motions also on all the pyramids of the octahedron. Themotion of the octahedron carries a certain number of octahedral bonds, and similarlythe motions of the pyramids carry a certain number of pyramidal bonds. “To carry”a bond means that its µ -number is positive for a given motion. If the µ -number ofa bond is zero, we say that the motion “does not have” that bond. Pyramidal andoctahedral bonds associated to a motion must satisfy the following rules. R1 : Depending on their bonds, flexible pyramids can be distinguished into five fam-ilies: general ( g ), even deltoids ( e , with two subfamilies), odd deltoids ( o , withtwo subfamilies), rhomboids ( r , with four subfamilies), and lozenges ( l , withfour subfamilies). The family/subfamily is completely determined by the kindof bonds arising (with only one ambiguity: rhomboids/lozenges), as specified byTable 1. A deltoid v is even if the dihedral angles at its even edges are simple,where even edges are determined by Figure 8. Figure 8: Assignment for even and odd edges of pyramids. Only three pyramids areshown, since the assignment for the other three can be deduced as follows: { , a } is even/odd if and only if { , a } is so, and analogously for the othertwo pairs (3 ,
4) and (5 , g o coincide 1 1 1 0antipodal 1 1 0 1 e coincide 1 0 1 1antipodal 0 1 1 1 r Type 1 1 0 1 0Type 2 0 1 1 0Type 3 1 0 0 1Type 4 0 1 0 1 l Type 1 1 0 1 0Type 2 0 1 1 0Type 3 1 0 0 1Type 4 0 1 0 1
Definition 3.1.
We choose the orientation of the edges of G oct as in Figure 9 anddenote this oriented graph by ~ G oct . Notice that this choice is equivariant under cyclicpermutations of the vertices (1 , , , , , λ : E oct −→ R > , and given an oriented edge ( i, j ) in ~ G oct , we definethe number ‘ ij to be the length λ { i,j } . We define the number ‘ ji to be − ‘ ij . R2 : For a motion having an octahedral bond with oriented edges ( t , s ), ( t , s ),( t , s ), ( t , s ), the following relation among the edge lengths of the octahedronholds: ‘ t s + ‘ t s + ‘ t s + ‘ t s = 0 . (1)From Rule R2 we can already infer some properties of bonds of flexible octahedra.We show that only some octahedral bonds may arise for a motion.11
452 3 6
Figure 9: Fixed orientations in the graph G oct . We call this oriented graph ~ G oct . Lemma 3.2.
Consider an octahedral bond for a flexible octahedron, with an orienta-tion ( t , s ) , . . . , ( t , s ) . Equation (1) from Rule R2 has non-trivial solutions only ifexactly two of the oriented edges ( t , s ) , . . . , ( t , s ) coincide with the oriented edgesinduced by ~ G oct (see Figure 10). Figure 10: Orientations of the edges of a quadrilateral in ~ G oct (left) and those inducedby a bond of a flexible octahedron (right). Green edges describe edgeswhere the orientation coincides and red ones where they are opposite. Proof.
If all (or no) oriented edges in the quadrilateral coincide with the ones inducedby ~ G oct , then in Equation (1) we have that the sum of four positive quantities is zero,a contradiction. If one (or three) oriented edges in the quadrilateral coincide with theones induced by ~ G oct , then we obtain a relation of the form ‘ = ‘ + ‘ + ‘ , whereall quantities ‘ k are positive. This implies that all the vertices of the quadrilateralare collinear in a general realization of the flexible octahedron; hence, some faces arecoplanar, and we excluded this possibility. Then the only situation left is the one fromthe statement.A simple inspection provides the following result.12 roposition 3.3. Out of the possible octahedral bonds supported on a quadrilateralin G oct , only fulfill the condition of Lemma 3.2. They come in three pairs, where twoorientations are in the same pair if one can be obtained from the other by reversingthe orientations of all edges. One of these pairs is constituted of orientations with thefollowing property: if ( t , s ) , . . . , ( t , s ) are the oriented edges, then (cid:12)(cid:12)(cid:12)[(cid:8) { t k , s k } : ( t k , s k ) is a directed edge of ~ G oct , k ∈ { , . . . , } (cid:9)(cid:12)(cid:12)(cid:12) = 4 . This means that those edges that are oriented as in ~ G oct span the vertices of thequadrilateral. These two special orientations are depicted as case X in Figure 11. Notation.
We use the following notation for the 6 possible octahedral bonds ona given quadrilateral as described by Proposition 3.3. Let 12, 34, and 56 be thethree quadrilaterals of G oct . The six possible bonds associated to the quadrilateral ij are denoted B ijX , B ij ¯ X , B ijY , B ij ¯ Y , B ijZ , B ij ¯ Z according to the following criterion. Aswe mentioned in Definition 3.1, the orientation in ~ G oct is equivariant under cyclicpermutations of the indices (1 , , , , , B X , B Y , and B Z as the bonds inducingthe orientations as in Figure 11. B X B Y B Z Figure 11: Three of the six possible octahedral bonds on the quadrilateral 56.The bonds B X , B Y , and B Z are defined to be the bonds inducing the reversedorientations with respect to the three previous ones. The two bonds B X and B X havethe special property mentioned in Proposition 3.3. By applying cyclic permutationsto the previous 6 orientations, we obtain 36 quadrilaterals with oriented edges. Thenotation symbols for these bonds are obtained by applying cyclic permutations to theindices appearing in the symbols for the bonds B • , where • ∈ { X, Y, Z, ¯ X, ¯ Y , ¯ Z } , andthen by applying the following rules: B ijX = B ji ¯ X , B ijY = B jiY , B ij ¯ Y = B ji ¯ Y , B ijZ = B jiZ , B ij ¯ Z = B ji ¯ Z . otation. We denote the µ -number of the octahedral bond B ijX by µ ijX , and similarlyfor the other bonds. Notation.
If a pyramid v has a pyramidal bond P , we write µ v = 1; otherwise wewrite µ v = 0. Similarly we define µ -numbers for the other pyramidal bonds.The next rule explains what happens when we reverse directed edges in a bond. R3 : The µ -number of a pyramidal or octahedral bond coincides with the µ -numberof the bond obtained by reversing the directed edges. In particular, if a mo-tion carries a bond then it also carries the corresponding bond with reverseddirections.To start the classification, we need one last rule, linking µ -numbers of octahedral bondsto µ -numbers of pyramidal bonds. This rule, however, works only under an assumptionon the pyramids of the octahedron, called simplicity. Definition 3.4.
Consider a flexible octahedron. We say that a pyramid is simple if,given a general realization of the pyramid for the induced motion, there is exactly onenon-degenerate realization of the octahedron that extends the one of the pyramid.We can now state the last rule and then we start the classification in the case of simplepyramids. Afterwards, we deal with the situation of non-simple pyramids. R4 : Given a motion of a flexible octahedron, suppose that all pyramids are simple.Then we have relations between the µ -numbers of the possible pyramidal bondssupported on v and the µ -numbers of the 18 possible octahedral bonds givenby the following graphical rule (see Figure 12). We consider a possible pyramidalbond, for example P on 1 . We draw the orientation of the two edges specifiedby P on the representation of G oct as in Figure 3. The µ -number of P is thenequal to the sum of the µ -numbers of the octahedral bonds that “extend” the twooriented edges of P ; in this case, we have a unique way to extend them, namelyby B Y . Hence, we get the relation µ = µ Y . If we start, instead, from P again on pyramid 1 , we have two ways to extend it, namely by B X , and B Z .Therefore, the relation is µ = µ X + µ Z .By applying the graphical procedure to all pyramidal bonds, and taking into14 B Y P B X B Z Figure 12: Graphical derivation of the linear relations among µ -numbers of pyramidaland octahedral bonds.account the equalities from Rule R3 , we obtain the following linear system: µ = µ Z µ = µ Z µ = µ Z µ = µ X + µ Y µ = µ X + µ Y µ = µ X + µ Y µ = µ Y µ = µ Y µ = µ Y µ = µ X + µ Z µ = µ X + µ Z µ = µ X + µ Z µ • = µ • µ • = µ • µ • = µ • (2)where • is any of the symbols { , , , } .With the rules at hand, we are ready to attack the classification. From now on, we suppose that the hypothesis in Rule R4 holds, namely that we aregiven a motion of an octahedron and that all pyramids are simple. At the end of thesection we analyze the cases when some pyramids are not simple. We distinguish fourcases, parametrized by the sums of the µ -numbers of octahedral bonds.15 efinition 3.5. For each quadrilateral ij in G oct , we define µ ij to be the quantity: µ ij := µ ijX + µ ijY + µ ijZ + µ ij ¯ X + µ ij ¯ Y + µ ij ¯ Z R3 = 2( µ ijX + µ ijY + µ ijZ ) . Lemma 3.6.
There are only possibilities (up to swapping quadrilaterals) for thenumbers ( µ , µ , µ ) : ( µ , µ , µ ) ∈ (cid:8) (4 , , , (4 , , , (4 , , , (2 , , (cid:9) . Proof.
By Table 1 from Rule R1 , we have 1 ≤ µ v + µ v ≤ v ∈ { , . . . , } ,and similarly for µ v + µ v . It follows by Equation (2) from Rule R4 that µ ij ∈ { , } for all ij ∈ { , , } . The statement is then proven.Now we analyze the cases from Lemma 3.6 one by one. Case (4 , , : From Equation (2), we know that for all quadrilaterals ij in G oct µ ijX + µ ijY = µ k ∈ { , } for a suitable k ,µ ijX + µ ijZ = µ ‘ ∈ { , } for a suitable ‘ . Moreover, by assumption we have2( µ ijX + µ ijY + µ ijZ ) = 4 . This implies µ ijX = 0 , µ ijY = µ ijZ = 1 . The equations on the edge lengths from Rule R2 imposed by the fact that µ ijY = µ ijZ = 1 are, in the case ij = 56: ‘ − ‘ − ‘ + ‘ = 0 ,‘ + ‘ − ‘ − ‘ = 0 . This implies that ‘ = ‘ and ‘ = ‘ . Namely, opposite edges in the threequadrilaterals of G oct have the same length (see Figure 13). Now notice that aparameter count shows that an octahedron whose opposite edges in each quadri-lateral have equal length possesses a line-symmetric motion. Since all pyramidsare simple, there is exactly one way in the motion under consideration to ex-tend a realization of a pyramid. Since all pyramids are general, each of themadmits exactly one motion. Therefore, such unique extension must be in theline-symmetric motion. 16 Figure 13: The edge length situation in Case (4 , , Case (4 , , : From Equation (2) and Table 1 from Rule R1 , we infer that the pyra-mids 5 and 6 are general, while 1 and 2 are odd deltoids and 3 and4 are even deltoids. Moreover, from the fact that µ = µ = 4, we deduceas in Case (4 , ,
4) that the opposite edges in the quadrilaterals 12 and 34 havethe same length. We now show that the opposite edges in the quadrilateral 56have the same length, so as in Case (4 , ,
4) we conclude that we have a Type Iflexible octahedron. Consider a realization for which the pyramid 1 is flat; thenwe have that 1, 3, and 4 are collinear. Let us now look at the pyramid 3 forthat realization: we would like to conclude that 3 is flat as well. Since 1 isflat, we have that the dihedral angle between the faces 135 and 136 is either 0or π ; however, this is a simple angle for 3 , hence by Lemma 2.11 also 3 isflat. Therefore, the vertices 1, 2, 3, and 4 are collinear in that realization, andall the vertices are coplanar. Then the quadrilateral 34 is, in that realization,a parallelogram or an antiparallelogram (see Figure 14). Thus, the footpointof the midpoint of the diagonal { , } on the line 1234 is the midpoint of thediagonal { , } . By considering the quadrilateral 12, we get that the footpointof the midpoint of the diagonal { , } on the line 1234 is the midpoint of thediagonal { , } . Hence we obtain ‘ = ‘ and ‘ = ‘ . Thus this case is a special case of a Type I flexible octahedron allowing a flatrealization.
Case (4 , , : Here we see that the pyramids 3 and 4 are even deltoids, and the17
Figure 14: Flat realization in Case (4 , , π , henceit is flat as well. This implies that we have two flat realizations for the octahedronas a whole. Since we have deltoids, as in Case (4 , ,
2) we have collinearities ina flat realization, namely the following triples of vertices are collinear (keep intoaccount that 3 and 4 are even deltoids, while 5 and 6 are odd deltoids): { , , } { , , } { , , } { , , } . Therefore, all the vertices are collinear, unless in this special flat realization wehave that 1 and 2 coincide . If the vertices are collinear in this special realization,then all the triangular faces are degenerate, and so all vertices are collinear inany realization of the motion, but in this case the octahedron cannot move atall. Hence only the situation where 1 and 2 coincide can happen (see Figure 15).For this situation to happen, we must have ‘ = ‘ , ‘ = ‘ , ‘ = ‘ , ‘ = ‘ . Moreover, the fact that µ = 4 implies, as in Case (4 , , ‘ = ‘ and ‘ = ‘ . Altogether, this implies that for a general realization in this motion the vertices3, 4, 5, and 6 are coplanar and that 1 and 2 are symmetric with respect to the Recall that we forbid two vertices to coincide for a general realization in a motion, but they areallowed to coincide in special realizations.
45 61 = 2
Figure 15: Global flat realization of an octahedron in Case (4 , , R1 we get that for the two odd deltoids µ = 1 and µ = 0 , while for the two even deltoids µ = 1 and µ = 0 . Therefore by Equations (2) from Rule R4 , we obtain µ Z = 1 , µ X = µ Y = 0 ,µ Y = 1 , µ X = µ Z = 0 . By using Rule R2 we get the constraints − ‘ − ‘ + ‘ + ‘ = 0 and ‘ + ‘ − ‘ − ‘ = 0 . Taking into account the previous relations between lengths, these imply theequalities ‘ = ‘ and ‘ = ‘ . Altogether, these equations imply that, if the quadrilateral 12 is an antiparallel-ogram, then the projection of the vertices 1 and 2 on the plane spanned by 3,4,5,and 6 lies, for all realizations of the motion, on the symmetry line of the antipar-allelogram. Hence we get a Type II flexible octahedron. If the quadrilateral 1219ere a parallelogram, then the projection of the vertices 1 and 2 would be at theintersection of its two symmetry axes; but then we would have a convex flexibleoctahedron, and this conflicts with Cauchy’s theorem.
Case (2 , , : In this case, all the 6 pyramids are rhomboids or lozenges. Moreover,we have µ ijX + µ ijY + µ ijZ = 1for any ij ∈ { , , } , and so exactly one of these three quantities equals 1,while the other two are zero. We hence obtain three linear constraints for theedge lengths, one for each of the three quadrilaterals in G oct . Therefore we havean octahedron of Type III.The classification when all the pyramids are simple is then completed. We concludethis section by showing that we can always reduce to the simple case. Let us describethis reduction procedure as follows. Reduction.
Suppose that a pyramid, say , is not simple. This means that thereexist at least two realizations of the octahedron extending a general realization of .This implies that in all those realizations the points , , , and must be coplanar.Then we construct another octahedron by substituting the realization of vertex withthe mirror of the realization of the vertex with respect to the plane spanned by , , , and ; see Figure 16. By the hypothesis on the initial octahedron we get that thenew octahedron is flexible, and it has the further property that pyramids and are simple. Here the fact that and are simple is ensured by Assumption G,which prevents different vertices from having the same realization. We claim that we can repeat this procedure finitely many times (actually, three times)and obtain a situation where all the pyramids are simple. In fact, notice that the re-duction process preserves coplanarity in the following sense. Suppose that pyramid 1is not simple and apply the reduction. This means that vertices 3 , , , , , , , , , , , , v with vertices v onthat plane can only be deltoids or lozenges.To refine the by-product stated in the last paragraph, we introduce the notion of multiplicity of an edge of the octahedron. A specific rule discusses the behavior ofedge multiplicity. Definition 3.7.
Consider a motion of an octahedron. The multiplicity of an edge ofthe octahedron is the number (up to rotations and translations) of realizations of theoctahedron that have the same general value of the dihedral angle between the twotriangular faces adjacent to the edge. R5 : Edges may have multiplicity 1, 2, or 4. Two opposite edges of a pyramid v incident to v have the same multiplicity; hence all the edges in a quadrilateralof G oct have the same multiplicity. The multiplicity of two neighboring edgesincident to v in a pyramid v may at most differ by a factor of 2. A general pyra-mid has all edges of multiplicity 2 or 4. The edges of a deltoid have multiplicity(2 ,
4) or (1 , , , , , , , , , β π − β β π − β Figure 17: Flat positions of an octahedron obtained by applying the reduction processtwice. This case, actually, does never occur.case can never happen. In fact, we prove that the constraints derived from the flatrealization, together with the fact that we have six lozenges, are not compatible withthe orthogonality of the planes 1256 and 3456. To show this, we focus on the dihedralangle at the edge 23: using the constraints from the flat realization, we can fix (up toscaling) vertex 2 to be at (0 , , b, ,
0) (for some b ∈ R > \ { } ) forthe whole motion; we parametrize vertex 5 as (cid:0) , − r cos( t ) , r sin( t ) (cid:1) for some r ∈ R > ,22o vertex 6 has coordinates (cid:0) b − b , br − b , (cid:1) ; see Figure 18. However, a computation ββ Figure 18: To show that the case of two reductions does not occur, we focus n thedihedral angle at edge 23: in the flat position, the angles d
325 and d
326 areequal, and the vertices 3, 5, and 6 are collinear.shows that the inner product between two normal vectors to the planes 356 and 256during the motion is not 0 for any choice of b and r . This is not compatible with thefact that the planes 1256 and 3456 are orthogonal.Assume now that only 3 , , , Case A.
All edges are simple. Then by Rule R5 we are again in Case (2 , , , , R6 : In the situation of Case A, the plane quadrilateral 12 is either an antipar-allelogram or a parallelogram.Consider the flat realization of the octahedron where the four vertices 3, 4, 5,and 6 are collinear. Because the plane quadrilateral 12 is an antiparallelogram ora parallelogram, it follows that the edges 35 and 46 are equal in length. Becausethe pyramid 1 is a rhomboid or a lozenge, it follows that the angles at 1 in thetwo triangles 135 and 146 are equal — they could not be supplementary becausethis would contradict collinearity of 3 , , ,
6. Hence the triangles 135 and 146have one side in common, the opposite angle in common, and the normal heightin common. It follows that the two triangles are congruent. It follows that, for all23onfigurations, the footpoint of vertex 1 to the plane lies on the symmetry line ofthe antiparallelogram or in the midpoint of the parallelogram, depending whetherthe plane quadrilateral 12 is an antiparallelogram or a parallelogram. Then thefootpoint of vertex 1 lies in the symmetry line of the plane antiparallelogram 12or in the midpoint of the parallelogram, also for the original octahedron. Thesame holds for the footpoint of vertex 2, analogously. It follows that the originaloctahedron, before the reduction process, is plane-symmetric.
Case B.
The four edges are double. By Rule R5 we have four deltoids and tworhomboids or lozenges, thus we are in the (4 , ,
2) case. Then the plane quadri-lateral 12 is an antiparallelogram, and the footpoint of vertex 1 to the plane lieson the symmetry line of the antiparallelogram. Say we had before reduced byreplacing 2 by the mirror of 1 at the plane 3456. Then the footpoint of vertex 1lies in the symmetry line of the plane antiparallelogram 12, also for the originaloctahedron. The same holds for the footpoint of vertex 2, analogously. It followsthat the original octahedron is plane-symmetric.
Now that we showed that the classification of flexible octahedra can be achieved oncewe accept the rules introduced in Section 3, we are left with the task of explainingwhy the rules are correct.We start by reducing the problem of flexibility of octahedra to a problem of flexibilityof graphs on the sphere, as in [Izm17, Kok33, Sta10]. For each realization in 3-spaceof an octahedron compatible with a given edge labeling, the normalized vectors of theedges define a configuration of points on the unit sphere. For any triangular face ofthe octahedron, the angle between two edge vectors is determined by the edge lengthsof the octahedron. Let us define G edg to be the graph whose vertices are the edgesof G oct , and where two vertices are connected by an edge when the correspondingedges in G oct belong to the same triangular face of the octahedron (see Figure 19).From the previous discussion we get that a labeling for the edges of G oct induces alabeling of the edges of G edg given by the cosine of the angles between edge vectorsbelonging to the same face. In formulas, if λ is the labeling for G oct , then the inducedlabeling for G edg is the map:( { i, j } , { m, j } )
7→ − λ { i,m } − λ { i,j } − λ { m,j } λ { i,j } λ { m,j } . Hence there is a bijective correspondence, modulo translations, between realizations of24 , }{ , } { , }{ , } { , }{ , } { , }{ , } { , } { , }{ , } { , } Figure 19: The graph G edg : its vertices are the edges of the octahedron, and twovertices are adjacent if they come from the same face of the octahedron.the octahedron in 3-space compatible with λ and realizations of the edge graph G edg on the unit sphere compatible with the labeling induced by λ .The choice of the normalized vector corresponding to an edge in G oct is not uniqueand depends on an orientation of the edges of the octahedron (any orientation is, inprinciple, fine). Recall that we have already fixed an orientation in Definition 3.1; fromnow on, we will always refer to this choice of orientation. Hence, given a realization ρ : { , . . . , } −→ R of G oct , for each edge { i, j } ∈ E oct we define the point q { i,j } inthe unit sphere to be the one such that ρ ( i ) − ρ ( j ) = ‘ ij q { i,j } , where we recall from Definition 3.1 that ‘ ij > i, j ) is an oriented edge in ~ G oct ,and ‘ ij = − ‘ ji . Hence, if ρ is a realization of G oct compatible with a labeling λ thenthe map that associates { i, j } 7→ q { i,j } for all { i, j } ∈ E oct = V edg is the induced realization of G edg on the unit sphere.Notice that, once we have a triangle in the octahedron, the labeling induced on theunit vectors of the edges forces the three points on the unit sphere to lie on the samegreat circle. Therefore, realizations of G edg induced by realizations of G oct look likethe one in Figure 20.The paper [GGLS19] contains necessary criteria for the flexibility of any graph ona sphere, as well as a detailed analysis of spherical quadrilaterals; these arise in the25igure 20: A realization of G edg in S (on the right) induced by one of G oct in R (onthe left).current paper as the edges of a pyramid incident with the vertex. The techniquein [GGLS19] requires to extend to the complex numbers many notions we encoun-tered so far: realization, flexibility, and also the unit sphere. Therefore, from now onrealizations of G oct will be maps ρ : { , . . . , } −→ C , and two realizations will beconsidered congruent if they differ by a complex isometry, which is given by the actionof a complex orthogonal matrix followed by a complex translation. Flexibility for agraph labeling will always mean admitting infinitely many compatible non-congruentrealizations, where now congruence is meant over the complex numbers, but we stillconsider real-valued labelings. Compatibility of a realization ρ with a labeling λ nowmeans that h ρ ( i ) − ρ ( j ) , ρ ( i ) − ρ ( j ) i = λ { i,j } for all { i, j } ∈ E oct , where h· , ·i is considered just as a quadratic form, and not a scalar product. Thecomplexification of the unit sphere will be denoted by S C = (cid:8) ( x, y, z ) ∈ C : x + y + z = 1 (cid:9) , so realizations of G edg will be maps V edg −→ S C . Two such realizations will be congru-ent if they differ by a complex orthogonal matrix. As in the spatial case, labelings arereal-valued functions E edg −→ R . Compatibility of a realization in S C with a labelingis again tested via the standard quadratic form h· , ·i , which in the real setting givesthe cosine of the angle between two unit vectors. As we see from their definition, theconstruction of the points q { i,j } starting from a realization of G oct carries over the26omplex numbers. Recall however that the numbers ‘ ij are always real. Flexibility ofgraphs on the complex sphere is defined analogously to flexibility in C .From the discussion and the construction in this section, we then obtain that a labelingof G oct is flexible in C if and only if the corresponding induced labeling for G edg isflexible in S C . This section provides geometric counterparts of the notions of octahedral and pyrami-dal bonds introduced in Section 3.1, and gives justifications for the rules in Sections 3.2and 3.3. The needed theory is the one developed by the authors in [GGLS19] aboutflexibility of graphs on the sphere, together with a new finding related to Rule R2 ,namely to the connection between bonds and linear conditions on the edge lengths offlexible octahedra. We recall here the main concepts of [GGLS19] and refer to thatwork for proofs and precise constructions.The geometric concept of bond arises as follows. First of all, we define what we meanby configuration space of a flexible labeling of a graph. This notion makes it possibleto consider “realizations up to isometries” as an algebraic variety, and so it makes itpossible to use the tools of algebraic geometry to study it. It turns out that thesevarieties are not compact, and there are several possible ways to compactify them. Bydoing this, we add “points at infinity” to the configuration space, namely points thatdo not correspond to realizations. These points are the bonds. Although they do notcorrespond to realizations, they still carry deep geometric information: by extractingit, we will be able to explain the rules we stated in Section 3.We now describe the notion of configuration space and its compactification for realiza-tions of graphs on the sphere, as it is introduced in [GGLS19]. This is accomplishedby noticing that it is possible to associate to each general n -tuple of points in S C a2 n -tuple of points in P C in such a way that two n -tuples on the sphere differ by a com-plex rotation (namely, by an element in SO ( C )) if and only if the corresponding two2 n -tuples in P C are P GL(2 , C )-equivalent. The association works as follows: consider S C as the affine part of a smooth quadric in P C , which is covered by two families oflines; given a point O ∈ S C , we can consider the two projective lines in S C passingthrough O ; each of these two lines intersects the plane at infinity in a single point;the two points that we obtain are called the left and right lift of O , respectively. Theleft and right lift belong to the intersection of the projective closure of S C with theplane at infinity, which is a smooth plane conic, hence isomorphic to P C . This means27hat we can consider general realizations on the complex unit sphere, up to complexrotations, as points in the moduli space M , n of 2 n distinct points on the projectiveline. Moreover, one notices that constraints in terms of spherical distances on S C can be translated into relations among the lifts in P C in terms of their cross-ratios.Therefore, one can encode realizations of graphs on the sphere compatible with a givenlabeling by algebraic subvarieties of M , n . This moduli space is non-compact, anda possible (projective) compactification is provided by the so-called moduli space ofrational stable curves with marked points , introduced by Knudsen and Mumford, anddenoted M , n . In this way, it is possible to assign to each graph G = ( V, E ), togetherwith a labeling λ : E −→ R , a projective variety C G inside M , | V | whose intersectionwith M , | V | encodes the realizations of G in S C compatible with λ , up to rotations.Given this premise, we can define the notion of bond of motion of a graph. Definition 5.1.
Given a graph G = ( V, E ) and a labeling λ : E −→ R , the projectivevariety C G ⊆ M , | V | is called the configuration space of realizations of G in S C compatible with λ . Since the labeling λ takes real values, the variety C G is real aswell. The components of C G that intersect M , | V | nontrivially are called motions of G . The points in C G ∩ ( M , | V | \ M , | V | ) are called the bonds of G , and if K ⊆ C G is a motion, bonds of G that lie in K are called bonds of K . Since C G is a real varietyand there are no real points on M , | V | \ M , | V | , bonds come in complex conjugatepairs.The following is one of the main results of [GGLS19]. Proposition 5.2.
A graph G with a flexible labeling λ on S C admits at least a bond. It is interesting to notice that also Connelly, in the introduction of [Con78], highlightsthe fact that extending the field to the complex numbers and “going to infinity” (aswe do here with bonds) may help understanding the geometric properties of flexibleobjects.
Justification of the objects
Now we are ready to explain why we introduced octahedral and pyramidal bonds inSection 3.From Section 4 we know that an octahedron has a flexible labeling if and only if theinduced labeling for the graph G edg is flexible on the sphere. This means that whenwe have a flexible octahedron, we get bonds for G edg . The presence of bonds imposescombinatorial restrictions to graphs in terms of colorings , which arise as follows. Let28s again consider an arbitrary graph G = ( V, E ) on n vertices. The boundary M , n \ M , n is constituted of divisors (i.e., subvarieties of codimension 1) that are denotedby D I,J , where (
I, J ) is a partition of the set { P , . . . , P n , Q , . . . , Q n } of marked points— we denote the marked points in this way to recall that we interpret them as left andright lifts of points on S C . If the configuration curve C G meets a divisor D I,J , then thepartition (
I, J ) induces a coloring on the graph G as follows: an edge { i, j } of G is red if at least three of { P i , P j , Q i , Q j } belong to I ; it is blue otherwise. Properties of themoduli space M , n imply that in each of these colorings there is no path of length 3in which the colors are alternated. For this reason, these coloring are called NAP (forNot Alternating Path) if they are surjective. The main result about NAP-coloringsin [GGLS19] is that their presence characterizes flexibility on the sphere: a graph G admits a flexible labeling λ if and only if G admits a NAP-coloring.Let us now describe the NAP-colorings of the graph G edg . We will see that thesecolorings are in bijection with quadrilaterals in G oct . To make the notation easier, fromnow on and for the rest of the paper we denote the marked points on the stable curvesof M , not by P u , Q u for u ∈ { , . . . , } , but rather by P ij , P ji for { i, j } ∈ E oct ,with i < j , since the vertices of G edg are labeled by pairs of indices. Definition 5.3.
Each of the three quadrilaterals in G oct determines a NAP-coloringof G edg as follows. Let i, j, k, ‘ be the vertices of the quadrilateral. There are exactlyfour vertices of G edg given by unordered pairs of elements in { i, j, k, ‘ } . A directinspection shows that those four vertices form a disconnecting set for G edg , namely ifthey are removed the resulting graph has two connected components. One then getsa NAP-coloring by coloring all the edges with their endpoints in the same componentby the same color; see Figure 21.Figure 21: The three NAP-colorings of G edg induced by the three quadrilateralsin G oct .By sorting out all the cases, helped by the fact that there are several triangles in G edg ,which must be monochromatic in a NAP-coloring, one proves the following result.29 roposition 5.4. The only NAP-colorings of G edg are those induced by the threequadrilaterals in G oct as in Definition 5.3. Remark 5.5.
There are 16 divisors D I,J inducing the same NAP-coloring. For ex-ample, if we consider the quadrilateral { , , , } , then one of these divisors is givenby I = { P ∗ , P ∗ , P , P , P , P } J = { P ∗ , P ∗ , P , P , P , P } , where ∗ takes all the values in { , , , } . The other divisors are obtained by swappingthe pairs ( P , P ), ( P , P ), ( P , P ), and ( P , P ). Hence we get a total of48 = 16 × M , of aflexible octahedron.By examining the shape of the partitions ( I, J ), we get the following graphical descrip-tion of the divisors D I,J . Proposition 5.6.
Let { m, n } be any of { , } , { , } , { , } and let Q be the quadri-lateral in G oct with vertices { , . . . , } \ { m, n } . There is a bijection between thedivisors D I,J inducing the NAP-coloring determined by { , . . . , } \ { m, n } and theset of orientations of the edges of Q . The bijection works as follows. Write I = { P m ∗ , P ∗ m , P t s , . . . , P t s } , then { t , s } , . . . , { t , s } are the edges of the (undirected) -cycle Q . We then declare that D I,J determines the orientations ( t , s ) , . . . , ( t , s ) of the edges of Q ; see Figure 10 for the quadrilateral corresponding to the example inRemark 5.5. Hence Proposition 5.6 explains why we defined octahedral bonds in Section 3 in thatway: oriented quadrilaterals of G oct codify the divisors D I,J that may be intersectedby the configuration curve C G edg of flexible labeling of G edg . The µ -number of anoctahedral bond reports the sum of the intersection multiplicities between C G edg anda divisor D I,J , and is defined as the degree of the divisor on C G edg cut out by D I,J .To explain the origin of pyramidal bonds, notice that if we apply the reduction from thespace to sphere described in Section 4 to a pyramid v , taking into account only theedges incident to v , we obtain a 4-cycle on the sphere. If we take as graph G a 4-cycle,whose vertices are { , , , } and whose edges are (cid:8) { , } , { , } , { , } , { , } (cid:9) , thenthe moduli space where its configuration space lives is M , . Let us, for a moment,switch back to the notation P u , Q u for the marked points of stable curves, just to makethe notation in this particular case less heavy. There are four divisors D I,J in M , I = { P , Q , P , P } , J = { P , Q , Q , Q } ,I = { P , Q , P , P } , J = { P , Q , Q , Q } ,I = { P , Q , P , Q } , J = { P , Q , Q , P } ,I = { P , Q , P , Q } , J = { P , Q , Q , P } . By swapping the P ’s with the Q ’s in the previous partitions, we obtain four otherdivisors, which are the complex conjugates of the previous ones. Let us focus on the I -part of the partition: we see that we always have a pair P k , Q k . If k is even we saythat the divisor is even (e), while we say that it is odd (o) if k is odd . Moreover, we seethat in the I -part we have another pair of marked points of the form either ( P i , P j )or ( P i , Q j ). In the first case we say that the divisor is unmixed (u), while in the secondcase we say that the divisor is mixed (m). Hence, to specify one of these four divisorsit is enough to specify whether it is even or odd, and unmixed or mixed. Therefore,we denote these divisors by D om , D ou , D em , and D eu .Now we can go back to our usual notation for marked points, and discuss the situationfor all pyramids in the octahedron. Recall that in Figure 8 we fixed the conven-tion about even and odd edges of the six pyramids of the octahedron, which corre-spond to the six quadrilaterals in G edg . This convention is summarized in Figure 22;one can notice that it is equivariant with respect to cyclic permutations of the ver-tices (1 , , , , ,
6) of the octahedron. With these choices, we see for example that if { , }{ , } { , }{ , } { , }{ , } { , }{ , } { , } { , }{ , } { , } eoe o oeo e e oeo o eoe oeoeo eoe G edg .we consider the pyramid 1 , then the odd mixed divisor D is given by the following31artition: I = ( P , P , P , P ) , J = ( P , P , P , P ) . The notion of pyramidal bond then codifies the information contained in the I -part ofthe partition determined by one of the four divisors associated with a pyramid in thefollowing way. As we saw, the I -part of such a partition associated to a pyramid v is of the form: I = (cid:16) P va , P av , P vu or P uv , P vw or P wv (cid:17) If b is the vertex such that { a, b } is a non-edge of G oct , then the pyramid v is the oneinduced by the vertices v, a, b, u, w . The two oriented edges of this subgraph, formingthe pyramidal bond, are then ( v, u ) (or ( u, v )) and ( v, w ) (or ( w, v )). For example, thepyramidal bond associated to the divisor D is P . Justification of Rule R1
This rule summarizes the content of [GGLS19, Section 4.1]. In fact, flexible pyra-mids determine flexible quadrilaterals on the sphere, and the reference describes theirbehavior, concerning in particular the intersection of their motions with the divisorsin M , . Justification of Rule R2
We want to obtain necessary conditions for the edge lengths of a flexible octahedron.Notice that Mikhalëv in [Mik01] obtains the same conditions for any suspensions whoseequator is a cycle . The first author who discussed these relations for octahedra was,to our knowledge, Lebesgue.Let us suppose, for simplicity, that a motion of a flexible octahedron meets the divi-sor D I,J in M , described in Remark 5.5. Let q { i,j } be the point in the sphere S C determined by the edge { i, j } in G oct as described in Section 4. Let us first clarifythe relation between q { i,j } and the two marked points P ij and P ji corresponding to itin the stable curves with marked points of C G edg . When the marked points belong toa stable curve that is not in the boundary of M , , we can recover the coordinatesof q { i,j } from the ones of P ij and of P ji . Let us suppose that P ij = ( u ij : v ij ) and P ji = ( u ji : v ji ) (here we think about them as points in P C ). The point q { i,j } is A suspension is a polyhedron whose combinatorial structure is the one of a double pyramid. P ij , P ji ) under the Segre embedding P × P −→ P . We needto be a little cautious here, since we should not use the “standard” map( u ij : v ij ) , ( u ji : v ji ) ( u ij u ji : u ij v ji : v ij u ji : v ij v ji )but rather( u ij : v ij ) , ( u ji : v ji ) ( u ij u ji : v ij v ji : u ij v ji + v ij u ji : u ij v ji − v ij u ji ) . In fact, our choice of coordinates should be such that the points where P ij = P ji correspond to the plane at infinity (this justifies the choice of the last coordinates),and moreover the origin should be the polar of the plane at infinity with respect tothe polarity induced by the quadric that is the image of P × P . Hence we get thefollowing expression for the vector q { i,j } : q { i,j } = (cid:18) u ij u ji u ij v ji − v ij u ji , v ij v ji u ij v ji − v ij u ji , u ij v ji + v ij u ji u ij v ji − v ij u ji (cid:19) . (3)Since the quadrilateral { , , , } in G oct forms a closed loop, we get the followingcondition, where ‘ ij is the (signed) length of the edge { i, j } : ‘ q { , } + ‘ q { , } + ‘ q { , } + ‘ q { , } = 0 . (4)Our goal is to express the condition of Equation (4) in local coordinates of the modulispace M , obtained by forgetting all marked points of the form P , ∗ , P ∗ , , P , ∗ , and P ∗ , , where ∗ takes all the values in { , , , } . Once we have done that, we can restrictthe equation to the (projection of the) divisor D I,J and obtain a necessary conditionon the numbers ‘ ij . We make the following choice of local coordinates for M , : P = (1 : 0) , P = (0 : 1) , P = ( x : 1) , P = ( z : x ) ,P = (1 : 1) , P = ( z : 1) , P = ( x : 1) , P = ( z : x ) . Notice that, with this choice of coordinates, { z = 0 } is a local equation for the projec-tion of the divisor D I,J on M , . By using this choice of coordinates in Equation (3)and by substituting the expressions for the { q { i,j } } in Equation (4), we get three equa-tions given by rational functions in z, x , . . . , x and the lengths ‘ , . . . , ‘ . Cleaningthe denominators and saturating by them the obtained polynomial equations yieldsequations that can be restricted to the projection of the divisor D I,J by imposing z = 0. Once we eliminate the variables z, x , . . . , x , we are left with a single equa-tion, namely ‘ + ‘ + ‘ + ‘ = 0 . This equation is precisely the one prescribed by Rule R2 . This and the previous operations can be performed by a computer algebra system such as Maple,Mathematica, or Sage. ustification of Rule R3 This rule follows from the fact that the configuration curve C G edg is a real variety, andso the degree of the divisor cut out on C G edg by a divisor D I,J equals the degree ofthe divisor cut out on C G edg by its conjugate D I,J . One then notices that complexconjugation interchanges the marked points P ij and P ji , and so D I,J determines thesame quadrilateral of D I,J but with opposite orientation.
Justification of Rule R4
Fix a motion K ⊆ C G edg and a pyramid v . The pyramid v defines a quadri-lateral Q in G edg . By assumption, we know that the pyramid v is simple. Let π Q : M , −→ M , be the projection that forgets all marked points except the onesrelated to Q . The fact that the pyramid v is simple implies that the restriction π Q | K is birational. As recalled in Section 4, there are 4 divisors (together with their complexconjugates) that are relevant for us, namely { D v om , D v ou , D v em , D v eu } . For each of them,we can use the following elementary fact from algebraic geometry: if f : X −→ Y isa birational morphism between projective curves, and E is a divisor on Y , then thedegree of E equals the degree of the pullback of E via f . By applying this fact to eachof the four divisors, we get the following equations: X π Q ( D I,J )= D v deg( D I,J | K ) = deg( D v | π Q ( K ) ) , for each D v ∈ { D v om , D v ou , D v em , D v eu } . Thus we obtain equations linking sums of µ -numbers of octahedral bonds to µ -numbers of pyramidal bonds. Because of the choicein the notation we made so far, which is equivariant under cyclic permutations of theindices (1 , , , , , × D , in M , we need to compute the divisors D I,J in M , that project to D v om via π Q . Since D is given by the partition I om = ( P , P , P , P ) , J om = ( P , P , P , P ) , it is enough to compute all the partitions ( I, J ) of the set { P ij , P ji : { i, j } ∈ E oct } of vertices of G edg that extend the partition ( I om , J om ). There is exactly one suchpartition: I = ( P ∗ , P ∗ , P , P , P , P ) , J = ( P ∗ , P ∗ , P , P , P , P ) . M , are reported in Table 2. From thistable we see that the rule yielding the equations is the same as the graphical procedurein Rule R4 . Justification of Rule R5
This follows from [GGLS19, Section 4.1]: the multiplicities of the edges in this papercorrespond to the degrees of the maps r k‘i in the reference. Justification of Rule R6
Assuming that vertices 3, 4, 5, 6 are coplanar and that the vertices 1 and 2 lie sym-metric with respect to that plane, we claim that the plane quadrilateral 12 is eitheran antiparallelogram or a parallelogram. To show this claim, let C O be the configu-ration space of the octahedron for the motion we are considering, let C and C beconfiguration spaces of the pyramids 1 and 2 , and let C be the configurationspace of the plane quadrilateral 12. There are natural projection maps f : C O −→ C , f : C O −→ C , g : C −→ C , g : C −→ C , and an isomorphism h : C −→ C defined by reflecting the vertex 1 at the plane of the quadrilateral 12. We claim alsothat g is an isomorphism (and g = g ◦ h is also an isomorphism). Assume, for acontradiction, that g is a 2 : 1 map. Then there is an automorphism s : C −→ C flipping the two points of any fiber of g . We can define two proper subsets X, Y ⊂ C O in the following way: X is the set of all x ∈ C O such that f ( x ) = h (cid:0) f ( x ) (cid:1) , and Y is the set of all y ∈ C O such that f ( y ) = s (cid:0) h ( f ( y )) (cid:1) . These are two closed subsetswhich cover C O . This is a contradiction to the irreducibility of C O , so g and g areisomorphisms.To conclude the proof, we want to show that all four vertices of the plane quadri-lateral 12 are simple, where the definition of the multiplicity of the vertex of a planequadrilateral parallels Definition 3.7 for multiplicity of dihedral angles. In fact, a planequadrilateral in which all vertices are simple is a parallelogram or an antiparallelogram.It suffices to show that vertex 3 is simple. Let D be the configuration space of the pyra-mid 3 ; let D be configuration space of the subgraph with vertices 1 , , ,
6, whichis determined by the dihedral angle at the edge 13; let D be configuration space ofthe subgraph with vertices 2 , , ,
6, which is determined by the dihedral angle at theedge 23; let D be configuration space of the subgraph with vertices 3 , ,
6, whichis determined by the angle at 3. Then by the same argument as before, the naturalprojections D −→ D and D −→ D are isomorphisms. By the simplicity of the35able 2: Derivation of graphical procedure in Rule R4 . P a r t i t i o n i n M , P a r t i t i o n ( s ) i n M , P y r a m i d a l b o nd E x t e n s i o n ( s )t o o c t a h e d r a l b o nd ( s ) µ - r e l a t i o n s ( P , P , P , P )( P ∗ , P ∗ , P , P , P , P )
15 2 615 2 6 Z µ = µ Z ( P , P , P , P )( P ∗ , P ∗ , P , P , P , P )
15 2 615 2 6 Y µ = µ Y + µ X ( P ∗ , P ∗ , P , P , P , P )
15 2 6¯ X ( P , P , P , P )( P ∗ , P ∗ , P , P , P , P )
14 2 314 2 3 Y µ = µ Y ( P , P , P , P )( P ∗ , P ∗ , P , P , P , P )
14 2 314 2 3 X µ = µ X + µ Z ( P ∗ , P ∗ , P , P , P , P )
14 2 3 Z C −→ D is an isomorphism. Because the pro-jections commute, it follows that the projection C −→ D is also an isomorphism.Hence the angle at 3 is simple and so the quadrilateral 12 is a parallelogram or anantiparallelogram.Figure 23: An example of a motion which is an instance of all three Bricard types. Remark 5.7.
The case where the quadrilateral 12 is a parallelogram is probablypurely hypothetical, but the case where the quadrilateral 12 is an antiparallelogramreally does exist. An example is an octahedron with edge lengths ‘ = ‘ = ‘ = ‘ = 20 , ‘ = ‘ = ‘ = ‘ = 13 , ‘ = ‘ = 11 , ‘ = ‘ = 21 . This flexible octahedron is an instance of all three Bricard types. It has two planesymmetries, one by the plane through 3, 4, 5, 6, and another by the plane intersectingorthogonally in the symmetry line of the antiparallelogram, which makes it plane-symmetric. The line reflection making it line-symmetric is the composition of the twoplane reflections. See Figure 23 for an example.
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Email address: [email protected] (JL, JS) Johannes Kepler University Linz, Research Institute for Sym-bolic Computation (RISC)
Email address: [email protected] , [email protected] (JL) Department of Applied Mathematics, Faculty of Information Tech-nology, Czech Technical University in Prague(JL) Department of Applied Mathematics, Faculty of Information Tech-nology, Czech Technical University in Prague