AA Note on Toroidal Maxwell–Cremona Correspondences
Patrick LinUniversity of Illinois, Urbana-Champaign
Abstract
We explore toroidal analogues of the Maxwell–Cremona correspondence. Erickson and Lin [4]showed the following correspondence for geodesic torus graphs G : a positive equilibrium stress for G , an orthogonal embedding of its dual graph G ∗ , and vertex weights such that G is the intrinsicweighted Delaunay graph of its vertices. We extend their results to equilibrium stresses that arenot necessarily positive, which correspond to orthogonal drawings of G ∗ that are not necessarilyembeddings. We also give a correspondence between equilibrium stresses and parallel drawings ofthe dual. The Maxwell–Cremona correspondence on the plane establishes an equivalence between the followingstructures:• An equilibrium stress on G is an assignment of non-zero weights to the edges of G , such that theweighted edge vectors around every interior vertex p sum to zero: (cid:88) p : pq ∈ E ω pq ( p − q ) = (cid:129) (cid:139) • A (orthogonal) reciprocal diagram for G is a straight-line drawing of the dual graph G ∗ , in whichevery edge e ∗ is orthogonal to the corresponding primal edge e .Erickson and Lin [4] considered graphs on the torus, and established a weaker correspondence fortorus graphs: Every positive equilibrium graph on any flat torus is affinely equivalent a graph admittingan embedded (orthogonal) reciprocal diagram on some flat torus. Erickson and Lin’s focus on positiveequilibria was motivated by equivalence between G admitting an orthogonally embedded dual and G being a coherent subdivision (or weighted Delaunay complex ) of its point set.Our first result generalizes the correspondence to non-positive equilibria: Every (not necessarilypositive) equilibrium graph on any flat torus is affinely equivalent a graph admitting an (not necessarilyembedded) orthogonal reciprocal diagram on some flat torus.Maxwell [5–7] derived his reciprocal diagrams with dual edges drawn orthogonally to their corre-sponding primal edges so as to obtain an equivalence to a third structure, namely, a polyhedral liftingof G in (cid:82) so that G is the projection of this polyhedral lift, and G ∗ is the projection of its polar dualaround the unit paraboloid. But both Rankine [9, 10] beforehand and Cremona [2, 3] afterwards drewtheir reciprocal diagrams such that dual edges were parallel to their corresponding primal edges; indeedCremona derived parallel reciprocal diagrams via what Crapo [1] called “skew polarity.” This skew polarcan be obtained by taking the standard polar dual through the unit paraboloid, a quarter turn aroundthe z -axis, and then a reflection through the xy -plane.1 a r X i v : . [ m a t h . M G ] S e p Patrick Lin
On the plane there is no functional difference between a reciprocal diagram in which every edge e ∗ is orthogonal to the corresponding primal edge e , and one in which every edge e ∗ is parallel to thecorresponding primal edge; indeed, one is just a rotation of the other.On the torus, however, we find a noticeable difference between orthogonal reciprocal diagrams andparallel reciprocal diagrams. Our second result is a necessary and sufficient condition for an equilibriumtorus graph G to admit a parallel reciprocal diagram. We assume familiarity with the notations and conventions in Erickson and Lin [4]. We define newconcepts and recall particularly important ones below. Major differences from Erickson and Lin arehighlighted in red. A drawing of a graph G on a torus (cid:84) is a continuous function from G as a topological space to (cid:84) .An embedding is an injective drawing, mapping vertices of G to distinct points and edges to interior-disjoint simple paths between their endpoints. The faces of an embedding are the components of thecomplement of the image of the graph; in particular, embeddings are cellular, i.e., all faces are opendisks. We will refer to any drawing of a graph G that is homotopic to an embedding of G as a torusgraph .In any embedded graph, left ( d ) and right ( d ) denote the faces immediately to the left and right ofany dart d . Similarly, in any drawing homotopic to the embedding, left ( d ) and right ( d ) refer to the cycles bounding the corresponding faces in the embedding. Conventionally, the dual graph G ∗ of an embedded torus graph has the following natural embedding:every vertex f ∗ of G ∗ lies in the interior of the corresponding face f of G , each edge e ∗ of G ∗ crosses onlythe corresponding edge e of G , and each face p ∗ of G ∗ contains exactly one vertex p of G in its interior.We regard any drawing of G ∗ on (cid:84) (cid:131) to be rotated dual to G if its image is homotopic to the clockwiserotation by a quarter circle of the image of the aforementioned natural embedding of G ∗ on (cid:84) (cid:131) . Moregenerally, a drawing of G ∗ is rotated dual to G on some flat torus (cid:84) if the image of G ∗ on (cid:84) (cid:131) is rotateddual to the image of G on (cid:84) (cid:131) .Geodesic graphs G and G ∗ that are dual to each other are orthogonal reciprocal if every edge e in G is orthogonal to its dual edge e ∗ in G ∗ . We emphasize the G and G ∗ need not be embedded. Anequilibrum stress ω is an orthogonal reciprocal stress for G if there is an orthogonal reciprocal drawingof its dual G ∗ on the same flat torus so that ω e = | e ∗ | / | e | for each edge e .A geodesic graph G and a geodesic rotated dual G ∗ are parallel reciprocal if every edge e in G isparallel to its dual edge e ∗ in G ∗ . An equilibrium stress ω is a parallel reciprocal stress for G if there isa parallel reciprocal graph G ∗ on the same flat torus so that ω e = | e ∗ | / | e | for each edge e . We make frequent use of the following lemmas, whose proofs can be found in the paper of Erickson andLin [4].
Note on Toroidal Maxwell–Cremona Correspondences Lemma 2.1 ( [4, Lemma 2.1]).
Fix a geodesic drawing of a graph G on (cid:84) (cid:131) with displacement matrix ∆ .For any circulation φ in G , we have ∆ φ = Λ φ = [ φ ] . Lemma 2.2 ( [4, Lemma 2.4]).
Fix an essentially simple, essentially 3-connected graph G on (cid:84) (cid:131) , a × E matrix ∆ , and a positive stress vector ω . Suppose for every directed cycle (and therefore any circulation) φ in G , we have ∆ φ = Λ φ = [ φ ] . Then ∆ is the displacement matrix of a geodesic drawing on (cid:84) (cid:131) thatis homotopic to G . If in addition ω is a positive equilibrium stress for that drawing, the drawing is anembedding. Let Λ be the × E matrix whose columns are homology vectors of edges in G . Let λ and λ denotethe first and second rows, respectively, of Λ . Lemma 2.3 ( [4, Lemma 2.5]).
The row vectors λ and λ describe cocirculations in G with cohomologyclasses [ λ ] ∗ = (
0, 1 ) and [ λ ] ∗ = ( −
1, 0 ) . The following lemma forms the analogue to Lemma 2.3, for rotated duals.
Lemma 2.4.
The row vectors λ and λ describe rotated cocirculations in G with cohomology classes [ λ ] ∗ = (
1, 0 ) and [ λ ] ∗ = (
0, 1 ) . Proof:
Without loss of generality assume G is embedded on the flat square torus (cid:84) (cid:131) . Let γ and γ denote directed cycles in (cid:84) (cid:131) ( not on G ) induced by the boundary edges of (cid:131) , oriented respectivelyrightward and upward.Let d , d , . . . , d k − be the sequence of darts in G that cross γ from left to right, indexed by theupward order of their intersection points. As shown in Erickson and Lin [4], these darts dualize in theto a closed walk d ∗ , d ∗ , . . . , d ∗ k − in G ∗ that, in the natural embedding, can be continuously deformedto γ ; when rotated clockwise by a quarter circle, this closed walk can instead be continuously deformedto γ , so it has the same homology class as γ , i.e., [ λ ] ∗ = (
1, 0 ) . See Figure 1. G dualize −−−−−→ G* rotate −−−−−→ Figure 1.
Proof of Lemma 2.4: The darts in G crossing either boundary edge of the fundamental square dualize to a closed walk in G ∗ parallel to the rotation of that boundary edge. Symmetrically, the darts crossing γ upward define a closed walk in G ∗ with the same homologyclass as γ , so [ λ ] ∗ = (
0, 1 ) . (cid:131) For an essentially simple, essentially -connected geodesic graph G on the square flat torus (cid:84) (cid:131) and anequilibrium stress ω , we let ∆ be the × E displacement matrix for G , and let Ω be the E × E matrixwhose diagonal entries are Ω e , e = ω e and whose off-diagonal entries are all . Define α = (cid:88) e ω e ∆ x e , β = (cid:88) e ω e ∆ y e , γ = (cid:88) e ω e ∆ x e ∆ y e . (3.1) Patrick Lin
These values are the entries of the covariance matrix ∆Ω∆ T = (cid:0) α γγ β (cid:1) .Section 5 of Erickson and Lin [4] consider the case of embedded geodesic torus graphs with positiveequilibrium stresses. It turns out that their analysis also applies to the case of a drawing of a torusgraph G that is not necessarily an embedding and the stress is not necessarily positive, with minimalchanges. For completeness we give the relevant restatements of the main results: Theorem 3.1.
Let G be a geodesic drawing on (cid:84) (cid:131) homotopic to an embedding with a (not necessarilypositive) equilibrium stress ω . Let α , β , and γ be defined as in Equation (3.1) . If αβ − γ = , then ω is an orthogonal reciprocal stress for the image of G on (cid:84) M if and only if M = σ R (cid:0) β − γ (cid:1) for any rotationmatrix R and any real number σ > . On the other hand, if αβ − γ (cid:54) = , then ω is not a reciprocal stressfor the image of G on any flat torus. The result can be reinterpreted in terms of force diagrams as follows:
Lemma 3.2.
Let G be a geodesic drawing on (cid:84) M homotopic to an embedding, and let ω be a (not necessarilypositive) equilibrium stress for G . The orthogonal force diagram of G with respect to ω lies on the flat torus (cid:84) N , where N = J M ∆Ω∆ T J T . The main differences between this setting and the setting of positive stress vector ω (and thusembedded G ) are as follows.When ω is positive, then αβ − γ = (cid:80) e , e (cid:48) ω e ω e (cid:48) (cid:12)(cid:12) ∆ x e ∆ y e ∆ x e (cid:48) ∆ y e (cid:48) (cid:12)(cid:12) > , so in fact the requirement αβ − γ = is just a scaling condition: given any positive stress vector ω , the stress vector ω/ (cid:112) αβ − γ is a positive stress vector that satisfies said requirement. If ω is non-positive, however, it is possible that αβ − γ < , in which case no scaling of ω is an orthogonal reciprocal stress on any flat torus (in termsof force diagrams, no scaling of ω can result in M being equal to N ).Furthermore, when ω is positive, the force diagram is embedded on (cid:84) N , and every face of the forcediagram is a convex polygon; if ω is not necessarily positive, then the force diagram still lies on (cid:84) N , butis not necessarily embedded, and faces may self-intersect. Consider the symmetric embedding of K shown in Figure 2. The edges fall into one of three equivalenceclasses, with slopes , / , − / and lengths (cid:112) / , (cid:112) / , (cid:112) / , respectively. Assigning the edgesof slope a stress of , the edges of slope / a stress of − , and the edges of slope − / a stress of ,we can verify that this indeed induces an equilibrium stress, and furthermore αβ − γ = , so we canfind a × matrix M such that the image of G on (cid:84) M has an orthogonal reciprocal diagram. However,this reciprocal diagram has coincident vertices and overlapping edges and self-intersecting faces; seeFigure 3.If we instead assign the edges of slope and / a stress of , and the edges of slope − / a stressof − , then this is an equilibrium stress, but αβ − γ = − , so no scaling of this stress vector can be areciprocal stress for G on any flat torus. In this section we give an analogous version of the previous section, but for parallel reciprocality. Majordifferences from Erickson and Lin [4] are highlighted in red.Fix an essentially simple, essentially 3-connected geodesic graph G on the square flat torus (cid:84) (cid:131) , alongwith a (not necessarily positive) equilibrium stress ω for G . In this section, we describe simple necessaryand sufficient conditions for ω to be a parallel reciprocal stress for G . Note on Toroidal Maxwell–Cremona Correspondences –1 –1 –1 –1–1 –1 –12 2 22 22 23 3 33 33 3 Figure 2.
The symmetric embedding of K with weights − , , and . (a) (b) Figure 3. (a) The reciprocal diagram of the image of K on (cid:84) M . Overlapping vertices and edges are drawn as being distinct, but verticesclose to and pointing to each other actually occupy the same position. (b) One of the faces of the reciprocal diagram. Let ∆ be the × E displacement matrix of G , and let Ω be the E × E matrix whose diagonal entriesare Ω e , e = ω e and whose off-diagonal entries are all . The results in this section are phrased in termsof the covariance matrix ∆Ω∆ T . We first establish necessary and sufficient conditions for ω to be a parallel reciprocal stress for G on the square flat torus (cid:84) (cid:131) , in terms of the covariance matrix ∆Ω∆ T . Lemma 4.1. If ω is a parallel reciprocal stress for G on (cid:84) (cid:131) , then ∆Ω∆ T = I . Proof:
Suppose ω is a parallel reciprocal stress for G on (cid:84) (cid:131) . Then there is a geodesic drawing of thedual graph G ∗ on (cid:84) (cid:131) where e (cid:107) e ∗ and | e ∗ | = ω e | e | for every edge e of G . Let ∆ ∗ = ( ∆Ω ) T denotethe E × matrix whose rows are the displacement row vectors of G ∗ .Recall from Lemma 2.3 that the first and second rows of Λ describe cocirculations of G with coho-mology classes (
1, 0 ) and (
0, 1 ) , respectively. Applying Lemma 2.1 to G ∗ implies θ ∆ ∗ = [ θ ] ∗ for anycocirculation θ in G . It follows immediately that Λ∆ ∗ = (cid:0) (cid:1) = I .Because the rows of ∆ ∗ are the displacement vectors of G ∗ , for every vertex p of G we have (cid:88) q : pq ∈ E ∆ ∗ ( p (cid:1) q ) ∗ = (cid:88) d : tail ( d )= p ∆ ∗ d ∗ = (cid:88) d : left ( d ∗ )= p ∗ ∆ ∗ d ∗ = (
0, 0 ) . (4.1)It follows that the columns of ∆ ∗ describe circulations in G . Lemma 2.1 now implies that ∆∆ ∗ = ∆Ω∆ T = I . (cid:131) Patrick Lin
Lemma 4.2.
Fix an E × matrix ∆ ∗ . If Λ∆ ∗ = I , then ∆ ∗ is the displacement matrix of a geodesic drawingon (cid:84) (cid:131) that is dual to G . Moreover, if that drawing has a positive equilibrium stress, it is actually anembedding. Proof:
Let λ and λ denote the rows of Λ . Rewriting the identity Λ∆ ∗ = I in terms of these rowvectors gives us (cid:80) e ∆ ∗ e λ e = (
1, 0 ) = [ λ ] ∗ and (cid:80) e ∆ ∗ e λ e = (
0, 1 ) = [ λ ] ∗ . Extending by linearity, wehave (cid:80) e ∆ ∗ e θ e = [ θ ] ∗ for every cocirculation θ in G ∗ . The result now follows from Lemma 2.2. (cid:131) Lemma 4.3. If ∆Ω∆ T = I , then ω is a parallel reciprocal stress for G on (cid:84) (cid:131) . If ω is a positive equilibriumstress, then the parallel reciprocal diagram is in fact embedded on (cid:84) (cid:131) . Proof:
Set ∆ ∗ = ( ∆Ω ) T . Because ω is an equilibrium stress in G , for every vertex p of G we have (cid:88) q : pq ∈ E ∆ ∗ ( p (cid:1) q ) ∗ = (cid:88) q : pq ∈ E ω pq ∆ Tp (cid:1) q = (
0, 0 ) . (4.2)It follows that the columns of ∆ ∗ describe circulations in G , and therefore Lemma 2.1 implies Λ∆ ∗ = ∆∆ ∗ = ∆ ( ∆Ω ) T = ∆Ω∆ T = I .Lemma 4.2 now implies that ∆ ∗ is the displacement matrix of a drawing G ∗ dual to G . Moreover, thestress vector ω ∗ defined by ω ∗ e ∗ = /ω e is an equilibrium stress for G ∗ : under this stress vector, the dartsleaving any dual vertex f ∗ are dual to the clockwise boundary cycle of face f in G . Thus if ω is positive,then G ∗ is in fact an embedding. By construction, each edge of G ∗ is parallel to the corresponding edgeof G . (cid:131) The results of the previous section have a more physical interpretation that may be more intuitive. Let G be any geodesic graph on the unit square flat torus (cid:84) (cid:131) . Recall that any equilibrium stress ω on G inducesan equilibrium stress on its universal cover (cid:101) G , which in turn induces a (parallel) reciprocal diagram ( (cid:101) G ) ∗ by the classical Maxwell–Cremona correspondence. This infinite plane graph ( (cid:101) G ) ∗ is doubly-periodic,but in general with a different period lattice from the universal cover (cid:101) G .Said differently, we can always construct another geodesic torus graph H that is combinatorially dualto G , such that for every edge e of G , the corresponding edge e ∗ of H is parallel to e and has length ω e ·| e | ;however, this torus graph H does not necessarily lie on the square flat torus. (By construction, H is theunique torus graph whose universal cover is ( (cid:101) G ) ∗ , the reciprocal diagram of the universal cover of G .)We call H the parallel force diagram of G with respect to ω . The parallel force diagram H lies on thesame flat torus (cid:84) (cid:131) as G if and only if ω is a parallel reciprocal stress for G . Lemma 4.4.
Let G be a geodesic graph in (cid:84) (cid:131) , and let ω be a (not necessarily positive) equilibrium stressfor G . The parallel force diagram of G with respect to ω lies on the flat torus (cid:84) M , where M = ∆Ω∆ T . Proof:
As usual, let ∆ be the displacement matrix of G . Let ∆ ∗ denote the displacement matrix of theforce diagram H ; by definition, we have ∆ ∗ = ( ∆Ω ) T = Ω∆ T . Equation (4.2) implies that the columnsof ∆ ∗ are circulations in G . Thus, Lemma 2.1 implies that Λ∆ ∗ = ∆∆ ∗ = ∆Ω∆ T .Set M = ∆∆ ∗ = ∆Ω∆ T . We immediately have Λ∆ ∗ = M = M T and therefore Λ∆ ∗ ( M T ) − = I .Lemma 4.2 implies that ∆ ∗ ( M T ) − is the displacement matrix of a homotopic drawing of G ∗ on (cid:84) (cid:131) . Itfollows that ∆ ∗ is the displacement matrix of the image of G ∗ on (cid:84) M . We conclude that H is a translationof the image of G ∗ on (cid:84) M . (cid:131) Note on Toroidal Maxwell–Cremona Correspondences In the case of orthogonal reciprocal diagrams, Erickson and Lin [4] established necessary and sufficientconditions for an equilibrium graph to be reciprocal on some flat torus.In contrast, we find that in the case of parallel reciprocal diagrams, an equilibrium graph is reciprocalon (cid:84) (cid:131) if and only if it is reciprocal on every flat torus. This is perhaps unsurprising: orthogonality relieson the conformal structure of the flat torus (cid:84) ; parallelism is an affine property.
Lemma 4.5. If ω is a parallel reciprocal stress for a geodesic graph G on (cid:84) M for some non-singular matrix M , then ∆Ω∆ T = I . Proof:
Suppose ω is a parallel reciprocal stress for G on (cid:84) M . Then there is a geodesic drawing of thedual graph G ∗ on (cid:84) M where e (cid:107) e ∗ and | e ∗ | = ω e | e | for every edge e of G .We will consider the geometry of G and G ∗ on the reference torus (cid:84) (cid:131) . (The drawinsg of G and G ∗ on the reference torus (cid:84) (cid:131) are still dual, but not necessarily reciprocal.) Let ∆ denote the × E reference displacement matrix for G , whose columns are the displacement vectors for G on the squaretorus (cid:84) (cid:131) . Then the columns of M ∆ are the native displacement vectors for G on the torus (cid:84) M . Thus,the native displacement row vectors of G ∗ are given by the rows of the E × matrix ( M ∆Ω ) T . Finally,let ∆ ∗ = ( M ∆Ω ) T ( M T ) − denote the reference displacement row vectors for G ∗ on the square torus (cid:84) (cid:131) .We can rewrite this definition as ∆ ∗ = ( M ∆Ω ) T ( M T ) − = Ω∆ T M T ( M T ) − = Ω∆ T . (4.3)Because the rows of ∆ ∗ are the displacement vectors for G ∗ , equation (4.1) implies that the columns of ∆ ∗ describe circulations in G , and therefore ∆∆ ∗ = Λ∆ ∗ = (cid:0) (cid:1) = I by Lemmas 2.1 and 2.3. Weconclude that ∆Ω∆ T = ∆∆ ∗ = I . (cid:131) Lemma 4.6. If ∆Ω∆ T = I , then ω is a parallel reciprocal stress for G on (cid:84) M where M is any non-singular × matrix. Moreover, if ω is a positive equilibrium stress, then the reciprocal diagram is embedded on (cid:84) M . Proof:
Suppose ∆Ω∆ T = I . Fix an arbitrary × non-singular matrix M . Let ∆ denote the × E reference displacement matrix for G on the square flat torus (cid:84) (cid:131) , and define the E × matrix ∆ ∗ =( M ∆Ω ) T ( M T ) − .Derivation (4.3) in the proof of Lemma 4.5 implies ∆ ∗ = Ω∆ T . It follows that ∆∆ ∗ = ∆Ω∆ T = I . Because ω is an equilibrium stress in G , for every vertex p of G we have (cid:88) q : pq ∈ E ∆ ∗ ( p (cid:1) q ) ∗ = (cid:88) q : pq ∈ E ω pq ∆ Tp (cid:1) q = (
0, 0 ) . (4.4)Once again, the columns of ∆ ∗ describe circulations in G , so Lemma 2.1 implies Λ∆ ∗ = ∆∆ ∗ = I .Lemma 4.2 now implies that ∆ ∗ is the displacement matrix of a homotopic drawing of G ∗ on (cid:84) (cid:131) , and if ω is positive, said drawing is in fact an embedding. It follows that ( M ∆Ω ) T = ∆ ∗ M T is the displacementmatrix of the image of G ∗ on (cid:84) M . By construction, each edge of G ∗ is parallel to its corresponding edgeof G . We conclude that ω is a parallel reciprocal stress for G . (cid:131) Our main theorem now follows immediately.
Patrick Lin
Theorem 4.7.
Let G be a geodesic graph on (cid:84) (cid:131) with an equilibrium stress ω . If ∆Ω∆ T = I , then ω is aparallel reciprocal stress for the image of G on (cid:84) M for any non-singular matrix M ; furthermore, if ω is a positive equilibrium stress, then the parallel reciprocal diagram is embedded on (cid:84) M . On the other hand,if ∆Ω∆ T (cid:54) = I , then ω is not a parallel reciprocal stress for the image of G on any flat torus. In terms of force diagrams:
Lemma 4.8.
Let G be a geodesic graph on (cid:84) M , and let ω be a positive equilibrium stress for G . The parallelforce diagram of G with respect to ω lies on the flat torus (cid:84) N , where N = M ∆Ω∆ T . Proof:
We argue exactly as in the proof of Lemma 4.4. Let ∆ be the reference displacement ma-trix of (the image of) G on (cid:84) (cid:131) . Then the native displacement matrix of the force diagram is ∆ ∗ =( M ∆Ω ) T = Ω∆ T M T . Equation (4.4) and Lemma 2.1 imply that Λ∆ ∗ = ∆Ω∆ T M T .Now let N = M ∆Ω∆ T . We immediately have N T = Λ∆ ∗ and thus Λ∆ ∗ ( N T ) − = I . Lemma 4.2implies that ∆ ∗ ( N T ) − is the displacement matrix of a homotopic drawing of G ∗ on (cid:84) (cid:131) . It follows that ∆ ∗ is the displacement matrix of the image of G ∗ on (cid:84) N . (cid:131) Consider the symmetric embedding of K on the square flat torus (cid:84) (cid:131) shown in Figure 4. Symmetryimplies that G is in equilibrium with respect to the uniform stress ω ≡ . Straightforward calculationgives us ∆Ω∆ T = (cid:0) (cid:1) for this stress vector. Thus, Lemma 4.1 immediately implies that ω is not aparallel reciprocal stress for G ; rather, by Lemma 4.4, the parallel force diagram of G with respect to ω lies on the torus (cid:84) M , where M = ∆Ω∆ T = (cid:0) (cid:1) . Figure 4.
The symmetric embedding of K with the uniform equilibrium stress ω ≡ . Erickson and Lin [4] show that the scaled uniform stress ω (cid:48) ≡ / (cid:112) is an orthogonal reciprocal stressfor G on the flat torus (cid:84) M where M = (cid:112) (cid:0) − (cid:112) (cid:1) . On the other hand, Theorem 4.7 implies that ω ≡ and its scalings are never parallel reciprocal stresses. By working with orthogonal reciprocality, Erickson and Lin [4] were able to extend the correspondenceto coherent subdivisions: a geodesic torus graph G admits an orthogonal reciprocal diagram if and onlyif it is coherent, i.e., the weighted Delaunay graph of its vertices with respect to some vector of weights.Comparing our results with the results of Erickson and Lin, we find that parallel reciprocalityand orthogonal reciprocality coincide exactly when the torus is square, and so parallel reciprocality Note on Toroidal Maxwell–Cremona Correspondences square flat tori.We pose as a line of inquiry whether there exists a more general correspondence between parallelreciprocality and some variant of coherence, though as previously noted, parallel reciprocality is anaffine property (whereas orthogonal reciprocality is a conformal property), and so any form of coherencecorresponding to parallel reciprocality will also need to be an affine property.
Acknowledgements.
We thank Jeff Erickson for helpful comments, especially about skew polarity,and for help with generating figures.
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