Periodic Steiner Networks Minimizing Length

We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices K4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_4$$\end{document}. For comparison we consider a competing topological class, also with a quotient on four vertices, and determine the minimizing ths networks.


Introduction
Given a finite set of points, the Steiner problem is to find a tree of minimal length connecting them [8].While this is a classical problem for the plane, the case of dimension 3 and higher has received less attention.Trees minimizing length usually have further vertices which necessarily are of degree 3, where the incident edges are coplanar and meet at 120 • -angles.This is valid for any dimension, and we call this the Steiner condition.
Here we consider infinite graphs without terminal vertices in Euclidean space which are multiply periodic and have a finite quotient.We call these graphs networks or, if all vertices have degree 3 and the Steiner condition is met, Steiner networks.We make these notions precise in Section 2.
We are interested in the case of dimension 3, which has a strong motivation by surface theory.There are various self-assembling biological and chemical systems which give rise to triply periodic interfaces.As pointed out for instance in [3], the most prevalent geometry is the gyroid, a triply periodic embedded surface with the body-centred cubic lattice and quotient surface of genus 3. Alan Schoen discovered the gyroid minimal surface in the 1970's in terms of a Steiner network [10] (see also [5]).By calling this network srs, a name which refers to the strontium silicide SrSi 2 crystal, we follow a crystallographic convention (see [7] and also rcsr.net).Other names for the network are Laves [2] or (10, 3)-a [15] (see also [11]).
The srs network, shown in Figure 1, is highly symmetric, with symmetry group I4 1 32.Its quotient under the body-centred cubic lattice is the complete graph on 4 vertices K 4 , see Figure 2.However, Steiner networks with 4 vertices in the quotient exist for arbitrary lattices (see Theorem 2.3).Since nature neither assumes symmetries nor the choice of a particular graph, this raises the question: Is there a simple property distinguishing the I4 1 32-symmetric srs network from all other networks?Date: April 26, 2018.2010 Mathematics Subject Classification.05C10; 53A10, 49Q05.This question is significant in the context of surface theory.Indeed, the gyroid minimal surface has been compared numerically with other explicitely known minimal surfaces.Compared with these particular surfaces, the gyroid has certain optimal features: among them network length, surface area per fundamental cell, Gauss curvature variance, or channel diameter variance [4,12].Nevertheless, it seems completely out of reach to show optimality in comparison to arbitrary surfaces, in particular without prescribing a lattice.
The present paper identifies the srs network as the unique network minimzing length in a sense we describe now, and which is illustrated by Figure 4 for the two-dimensional case.Let Λ be the lattice of a triply periodic network N ⊂ R 3 .Then the fundamental domain R 3 /Λ is a flat 3-torus with volume V , and the network quotient N/Λ has a length L. We usually refer to V and L as the volume and length of the network N itself.Since scaling can reduce the length of N , a well-posed variational problem is: Minimize the network length L under the constraint V = 1.Equivalently, one can minimize the scale-invariant ratio L 3 /V .Our main result answers the above question: Theorem A. The length and volume of a triply periodic network N in R 3 satisfy Equality holds exactly for the srs network, where the lattice Λ is body-centered cubic.
Here, the terminlogy is as in Section 2; in particular uniqueness is always up to (unnecessary) vertices of degree 2.
Let us indicate how the results of our paper combine to prove Theorem A. First, for a fixed lattice we establish the existence of a minimizer of L 3 /V in Theorem 2.3.It must be an embedded Steiner network on 4 vertices.Since a minimizer cannot have loops in the quotient graph (Lemma 2.6) only the two graphs shown in Figure 2 can arise.Theorems 5.4 and 6.3 then give sharp estimates for the ratio L 3 /V of networks with arbitrary lattice for the two cases of quotient graph.These estimates imply (1) as well as the characterization of the equality case.This establishes Theorem A for arbitrary triply periodic networks.
An obvious approach to prove the estimates of Theorems 5.4 and 6.3 would be to minimize the ratio L 3 /V for a given lattice, and thereafter minimize over all lattices.However, Steiner networks for a given lattice are not unique, and lattices are inconvenient to parameterize.So we use a different approach: In Lemmas 5.2 and 6.2 we show that it is possible to parameterize the space of Steiner networks covering each of the underlying graphs by their six edge lengths alone (plus an angle parameter in one case).Then not only the length L but also the volume V become explicit functions of these parameters (Lemma 5.2 and 6.2).Thus to prove Theorems 5.4 and 6.3 we need to solve a finite dimensional optimization problem under constraints: On our parameter space, we minimize the total length L under the constraints that V = 1 and the length parameters be positive.Then it turns out that lattice generators are linear functions of our parameters.
The graph different from K 4 which arises in the proof of Theorem A is D 1 D 2 , the Cartesian graph product of the dipole graphs D 1 and D 2 (see Figure 2).In Theorem 5.4 we determine the 1-parameter family which attains the minimal length ratio for this quotient graph (see Figure 1).We call these minimizers ths networks, again making use of a crystallographic name, this time refering to the thorium silicide ThSi 2 crystal (it is also known as (10, 3)-b [15]).Our result for networks with quotient D 1 D 2 is: Equality holds exactly for a one-parameter family of (non-similar) Steiner networks, all with the same lattice Λ, for instance generated by (1, 0, 0), (0, 1, 0), (1/2, 1/2, √ 3/2).Moreover, each of these minimizing ths networks can be homotoped into a Steiner network of smaller length covering K 4 , such that the length is non-increasing and the lattice remains fixed.
The homotopy is established in Theorem 7.1 and visualized in Figure 3.We take it as evidence for why in nature ths networks play a minor role compared to srs.
Our results should be compared with the work of Sunada and Kotani [9] (see also [14,13]).Instead of minimizing the length L = m i=1 x i of a network with edge lengths x 1 , . . ., x m , they minimize the quadratic energy E = m i=1 (x i ) 2 ; they also impose a volume constraint.For various combinatorial types of networks, Sunada and Kotani determine energy minimizers, essentially by solving a linear algebra problem.Unlike for our setting, minimizers are unique for given combinatorics.The srs network is also the energy minimizer, while for the case of D 1 D 2 , Sunada and Kotani exhibit a unique energy minimizer, not contained in our length minimizing ths family.The underlying abstract graph of the first two networks is the dipole graph of order 3 (cf.Figure 6).The first network, with the hexagonal lattice, minimizes length for given area of its fundamental domain.The network on the right has the quotient D 1 D 2 .
The energy E can be understood to model a physical crystal by harmonic oscillators along the edges of the network.Nevertheless, for the modelling of many other real world problems, in particular those relating to surface theory, the network length L seems more appropriate than the quadratic energy E. Note the analogy to the case of curves and surfaces, where variational problems involving length and area are natural, while the minimization of energy is usually simpler to handle mathematically.At this place we would like to recommend Sunada's book [14] as a comprehensive introduction to the geometric theory of networks and graphs, in particular to a systematic treatment of their topology.
It remains open to determine optimal Steiner networks in higher dimensions n ≥ 4. While our reasing generalizes in principle, the number of admissible combinatorial graphs strongly increases with n.In the forthcoming paper [1] we pursue another direction: There we study length minimizing n-periodic networks with vertices of prescribed degree d ≥ 4 in R 3 and R n .
The results presented in this paper are part of the first author's PhD thesis in progress.

Steiner networks
For our purposes, it is convenient to use the term network in the following sense: Definition 2.1.An (n-periodic) network N is a connected simple graph, immersed with straight edges of positive length into R n , subject to the following: • N is invariant under a lattice Λ of rank n.
• The quotient N/Λ is a finite graph Γ, possibly with loops and multiple edges.
We call V = V (R n /Λ) the (spanned) volume of N and L = L(N/Λ) the length of N .
Let us explain our terminology.If a graph is mapped injectively to R n then we call the map an embedding.We call it an immersion if for each vertex the restriction to the union of the incident edges is injective, i.e., the star of each vertex is embedded.
where the vectors g 1 , . . ., g n ∈ R n are linearly independent.The ambient space quotient R n /Λ is an n-dimensional flat torus.It can be represented by a parallelepiped spanned by the vectors g i .We refer to the quotient or its representing epiped as a fundamental domain.
Conversely, we can start with an abstract finite graph Γ.Our networks can then be described as immersions of certain abelian coverings of Γ.For the cases considered in the present paper, the first homology group H 1 (Γ, Z) has n closed cycles as generators (that is, the first Betti number is n).Thus for N to be nperiodic, each such cycle must map to a generator of the lattice.See Sunada [14] for a detailed account of the covering theory of graphs.
We wish to minimize the length L = L(N/Λ) of the quotient network N/Λ, subject to the constraint that the n-dimensional volume V = V (R n /Λ) of a fundamental domain is fixed to 1. Equivalently, we may minimize the scaling-invariant length ratio L n /V .Suppose a vertex p of a network is connected with edges to the three vertices q 1 , q 2 , q 3 .If the network is critical for length the first variation formula gives Equivalently, the three edges incident to p meet at 120 • -angles.We refer to (2) as the Steiner condition or as balancing.

Definition 2.2.
A Steiner network is an n-periodic network where all vertices have degree 3 (the network is 3-regular) and satisfy the Steiner condition (2) at each vertex.
Steiner networks result from minimizing the length ratio: Theorem 2.3.Among n-periodic networks in R n with fixed lattice Λ, there exists a network minimizing the length L. Any such minimizing network N is an embedded Steiner network such that N/Λ has 2n − 2 vertices, up to vertices of degree 2 with opposite incident edges.
Remark.If a minimizer has vertices of degree 2 then the incident edges must be opposite, and so such vertices can always be removed without changing L. From now on we assume this is the case.
For the proof, we need the notion of the circuit rank of a connected finite graph Γ, rank Γ := 1 − #vertices of Γ + #edges of Γ .
Note that a tree has circuit rank 0; that for any connected graph the circuit rank is a non-negative integer; and that for a 3-regular graph we have The circuit rank is precisely the number of generators of H 1 (N/Λ, Z).To verify this, consider a spanning tree T ⊂ N/Λ of a quotient network N/Λ, so that H 1 (T, Z) is trivial.Reinsert the edges one by one to see that both the circuit rank of T , as well as the number of generating cycles in T , increases by 1 in each step.
For a network N , we define the circuit rank as the rank of its quotient, rank N := rank(N/Λ).The following two lemmas serve to show that we can assume a minimizing sequence of networks to have rank n and be 3-regular.Lemma 2.4.Let N be an n-periodic network with lattice Λ ⊂ R n .If rank N > n then there exists an n-periodic network N , also with lattice Λ, which has smaller length, L(N ) < L(N ), and rank N = n.
The graph G 0 is chosen as a spanning tree of the quotient network N/Λ, while for i = 1, . . ., n the graph G i is the union of G i−1 with a single edge e i ∈ (N/Λ) \ G i−1 , subject to the requirement that G i lifts to a network N i ⊂ N with a lattice of rank i.Then N := N n has circuit rank n and is n-periodic.Since rank N > n, the network N has fewer edges than N , and so L(N ) < L(N ).Proof.Clearly we can decrease length by successively removing all vertices of degree d = 1 from N/Λ together with their incident edges.If the resulting network contains a vertex of degree 2 with non-opposite edges we can replace these edges by a single edge to reduce length.
If the resulting graph contains a vertex p of degree d ≥ 4 we use a well-known argument to reduce length (see for instance [8, p. 120f.]).The star at p contains two (non-collinear) edges with endpoints q 1 , q 2 which make an angle of less than 120 degrees.To define N , we replace these two edges in N/Λ with a tripod which connects the triple p, q 1 , q 2 with a further point in the same plane, chosen such that the length decreases.Thereby the degree at p changes from d to d − 1. Upon iteration we can reduce the degree to d ≤ 3 at all vertices of N/Λ.
Our operations preserve n-periodicity.However, for d = 2 and d ≥ 4 they possibly do not preserve the immersion property.So assume incident to p there are two or more edges in the same direction.We replace the initial portion, up to the first vertex, by a single edge, thereby reducing length.Iteration of this construction yields a network of shorter length, all of its stars are embedded.
Finally, we merge opposite edges incident to a vertex of degree 2 to form a single edge, leaving L unchanged, and let N be the resulting network.Note that all our operations preserve the circuit rank.Hence N has the properties claimed.
Proof of the theorem.Consider a length minimizing sequence (N k ), that is, lim k→∞ L(N k ) = inf L(N ), where the infimum is taken over all n-periodic networks with fixed lattice Λ.
Applying Lemma 2.4 we may assume that rank N k = n, and applying Lemma 2.5 thereafter we may assume N k still has rank N k = n but is 3-regular.By (3) the number of vertices then is 2n − 2. Combinatorially, there are only finitely many such graphs.Thus by passing to a subsequence we can assume all N k /Λ have the same combinatorial type Γ.
The set of 2n − 2 vertices of N k /Λ is compact in R n /Λ, and the connecting edges are geodesics with uniformly bounded length.Hence vertices and edges of a further subsequence of (N k ) converge to a limit N .We claim that all edges of N attain positive length.To see this, note that if an edge attains length 0, then N has a vertex with degree d ≥ 4. By Lemma 2.5, however, N cannot be a minimizer of length over all combinatorial types, contradicting the fact that (N k ) is a minimizing sequence.
Let us show N is embedded.Suppose two edges intersect in an interval of positive length.Then comparison with a network where the intersection set is replaced by a single edge shows that N cannot be minimizing.Similarly, supposing N has an isolated point of intersection we compare N with a network where this point is a vertex of degree d ≥ 4. Then N cannot minimize by Lemma 2.5.Finally, since N is a minimizer with positive edge lengths the first variation formula (2) shows N is Steiner.
Remark.The proof indicates that our results do not change if we drop the connectivity assumption in the definition of n-periodic networks (but still require that the cycles of the underlying possibly disconnected graph span a lattice of rank n).Indeed, if a minimizer was disconnected, we could use translations to move one component as to intersect the other.Again this contradicts Lemma 2.5.
We now show that Steiner networks cannot contain loops, thereby constraining the combinatorial types further.For instance the graphs shown in Figure 5 are impossible for networks critical for length with n = 3. Lemma 2.6.Let N ⊂ R n be an n-periodic Steiner network.Then N/Λ contains no loops.Proof.A loop based at a vertex p ∈ N/Λ corresponds to a straight edge of the lift N .Thus the lift contains a vertex with opposite incident edges, thereby contradicting balancing.
With Proposition 5.3 we will derive yet another constraint on the combinatorial graph of a length minimizing Steiner network: it must be simple, i.e., it cannot contain double edges.
Remark.The combinatorial graph of a minimizer may depend on the lattice.Indeed, as a result of [1] triply periodic networks with two degree-5 vertices which minimize length have different combinatorial types for different prescribed lattices.However, for the Steiner case, the homotopy of Theorem 7.1 implies that a minimizer for a fixed lattice always has K 4 as a quotient.

Maclaurin's inequality for elementary symmetric polynomials
In the cases we will consider, the volume V of a given network N with m labelled edges of length (x 1 , . . ., x m ) is a polynomial P (x 1 , . . ., x m ).Thus the task to minimize the quotient L 3 /V is equivalent to maximizing the polynomial P under the length constraint L = 1, where L(x 1 , . . ., x m ) := x 1 + . . .+ x m .
In the most symmetric case, P is the elementary symmetric polynomial of degree k, We also set P 0 (x 1 , . . ., x m ) := 1.We can estimate these polynomials by the length: where equality holds if and only if x 1 = . . .= x m .
In particular, for degree k ≥ 2 the elementary symmetric polynomial P k takes its maximum under the length constraint L = 1 exactly at ( 1 m , . . ., 1 m ).One way to prove Maclaurin's inequality is to use Newton's inequality, see [6].We present a more direct proof here, inspired by our application.
Proof.We prove (4) by induction over m.The base case is m = k, where 4) is the estimate on geometric and arithmetic mean.
For the step suppose m > k ≥ 2. We claim (4) holds strictly if some but not all x i vanish.In view of the symmetry of (4) we may assume x m = 0. Note that P k (x 1 , . . ., x m−1 , 0) is an elementary symmetric polynomial of degree k in m − 1 variables, and so the induction hypothesis gives We estimate the right hand side, using the strict inequality If not all x i vanish this yields strict inequality in (4), as claimed.
Since ( 4) is scaling invariant, it is sufficient to prove this inequality under the length constraint L = 1.The continuous function P k attains a maximum over the compact set L −1 (1) ⊂ [0, ∞) m at some point z = (z 1 , . . ., z m ).Note that z = 0, that we have equality in (4) for x = ( 1 m , . . ., 1 m ), and that the induction hypothesis, in form of the claim, gives strict inequality in (4) on ∂([0, ∞) m ) \ {0}.Thus z must be an interior point of [0, ∞) m .Since z is critical for P k under the smooth constraint L = 1 we obtain the necessary condition with λ ∈ R a Lagrange multiplier.It remains to show this implies z 1 = . . .= z m .Then since z assigns equality to (4) and z was chosen maximally, the proof of the induction step is completed.
Since P k is elementary symmetric, for i = j we can express P k at any point x = (x 1 , . . ., x m ) as where Q 0 , Q 1 , Q 2 are polynomials in m − 2 variables, independent of x i and x j .From (5) we conclude ∂ i P k (z) = ∂ j P k (z) for all 1 ≤ i, j ≤ m, so that Moreover, since k ≥ 2 and z i > 0 for all i, the polynomial Q 0 cannot vanish at z. Thus indeed z i = z j for all i, j.

Doubly periodic Steiner networks
We find it instructive to present the case of dimension n = 2 before studying the more involved case n = 3.We first determine the topology of the quotient graph of a minimizer for prescribed lattice.By Theorem 2.3 it has 2 vertices, and by Lemma 2.6 it has no loops.The only connected 3-regular graph on 2 vertices without loops is D 3 , see Figure 6.Hence we obtain: Lemma 4.1.A doubly periodic network N ⊂ R 2 minimizing the length ratio L 2 /A for prescribed lattice Λ is Steiner on 2 vertices with the dipole graph D 3 as a quotient.
Since the edges of a Steiner network enclose 120 • -angles, a minimizing network can be described in terms of three edge lengths alone: Lemma 4.2.Up to isometry, a doubly periodic Steiner network N ⊂ R 2 with quotient D 3 is uniquely determined by its three edge lengths x 1 , x 2 , x 3 > 0. Its length and spanned area are Proof.The two vertices of D 3 correspond to a vertex p 0 ∈ N and the incident vertices p 1 , p 2 , p 3 ∈ N , where the labelling relates to the lengths as in Figure 6.
Then the lattice Λ of N is spanned, for instance, by g 1 := p 1 − p 3 and g 2 := p 2 − p 3 .Specifically, we may assume that up to isometry we have and so the lattice Λ has area As might be expected, the optimal doubly periodic network is given by the tesselation of R 2 with regular hexagons: Proposition 4.3.For each doubly periodic network N ⊂ R 2 we have Equality holds if N has the quotient D 3 and the three edge lengths of N are equal; then the lattice is hexagonal.
Proof.For a prescribed lattice Λ, Lemma 4.1 asserts the existence of a Steiner network N 0 with quotient D 3 which is a minimizer, (L 2 /A)(N ) ≥ (L 2 /A)(N 0 ), where the inequality is strict unless N has quotient D 3 .According to Lemma 4.2, the edge lengths x 1 , x 2 , x 3 > 0 determine N 0 , and the area A(N 0 ) is a multiple of the elementary symmetric polynomial of degree 2 in three variables.Thus Maclaurin's inequality (4) implies (6) for N 0 : In particular, (6) follows for N .To discuss the equality case, note that for N with quotient D 3 and x 1 = x 2 = x 3 , equality in ( 6) is obvious.But by the above and Lemma 3.1 the equality can only hold for this case.

Triply periodic Steiner networks covering D 1 D 2
Our approach to triply periodic Steiner networks is similar to the doubly periodic case.However, as pointed out in the Introduction, Theorem 2.3 and Lemma 2.6 allow exactly two distinct topologies of minimizing Steiner networks:

Lemma 5.1. The combinatorial graph of a triply periodic Steiner network, minimizing length for a prescribed lattice
We analyze the case D 1 D 2 first since our analysis of the more prominent K 4 -case makes use of it.In both cases we can parameterize the space of networks by the edge lengths x 1 , . . ., x 6 of the six edges e 1 , . . ., e 6 in the quotient N/Λ; for D 1 D 2 there is a further angle parameter.This will follow from considering the tangent planes at the vertices; note that the Steiner condition implies that each vertex is coplanar with its three neighbours.In dimension n = 3, the angles between the different tangent planes turn out to be independent of the edge lengths.
With respect to a labelling as in Figure 7 we state: Lemma 5.2.Let N ⊂ R 3 be a triply periodic Steiner network with quotient D 1 D 2 .Then, up to isometry, the network N is uniquely determined by its six edge lengths x 1 , . . ., x 6 > 0 and an angle α ∈ (0, π).Moreover, N has length L = i x i and, for a labelling of the edge lengths as in Figure 7, the spanned volume is We will see that all edges of N are contained in two sets of parallel planes which make an angle α to be chosen independently of the edge lengths.The limiting cases α = 0 and π relate to a doubly periodic network.
Proof.Consider a connected subgraph Ñ ⊂ N with seven vertices p 0 , . . ., p 6 as in Figure 7 such that p 0 , p 4 , p 6 , as well as p 3 , p 5 are identified in the quotient.We may assume p 1 is the orgin, p 2 is on the x-axis, and p 0 , p 4 are in the xy-plane.Balancing then implies The tangent plane at p 2 must be a rotation about the x-axis of the tangent plane at p 1 by an angle α ∈ [0, 2π).Let A α ∈ SO(3) denote such a rotation.The Steiner condition then implies that p 3 − p 2 points in the same direction as p 1 − p 0 rotated by A α .The same applies to p 5 − p 2 and p 1 − p 4 .That is, Triple periodicity implies α = 0 mod π and changing α to α ± π corresponds to a change of numbering of the vertices p 3 and p 5 .So we may assume α ∈ (0, π).Using the Steiner condition we see that for a pair of vertices which are doubly connected in N/Λ the normals must agree.Since p 6 and p 0 are identified in N/Λ the vector p 6 − p 3 points in the same direction as p 2 − p 1 .So we have ( 10) The three vectors span the lattice Λ; indeed, an inspection of Figure 7 shows they correspond to minimal cycles in the abstract graph.Then | det(g 1 , g 2 , g 3 )| can be computed to (7).
As an aside, we use the reasoning of Lemma 5.2 to show that a minimizer N ⊂ R n of L n /V for prescribed lattice Λ cannot contain double edges for n ≥ 3.According to Theorem 2.3 the network N is an n-periodic Steiner network.Let p 0 , q 0 ∈ N be two adjacent vertices, and suppose they project onto doubly connected vertices p, q ∈ N/Λ.Denote by r 0 the neighbour of p 0 which does not project to q, and by s 0 the neighbour of q not projecting to p, see Figure 8. Then the Steiner condition shows the vectors r 0 − p 0 and s 0 − q 0 are parallel and point into opposite directions.Now move p 0 and q 0 simultanously in one of these directions: For 0 ≤ t < 1, replace p 0 by p t 0 := p 0 + t(r 0 − p 0 ) and q 0 by q t 0 := q 0 + t(r 0 − p 0 ), and similarly so for all other lifts of p, q.We obtain an n-periodic Steiner network N t with the same lattice Λ and L(N t ) = L(N ).The limiting network N 1 with lattice Λ has length L(N 1 ) = L(N ) and so is again minimizing.However, N 1 has one vertex of degree 4, thereby contradicting Lemma 2.5.Our reasoning proves: Proposition 5.3.An n-periodic minimizer of L n /V with n ≥ 3 for prescribed lattice covers a simple graph on 2n − 2 vertices of degree 3. Remark.The number of connected 3-regular simple graphs on 2n − 2 vertices, i.e., cubic graphs, is rapidly growing in n ≥ 3, see oeis.org.
The proposition implies that a triply periodic minimizer can only have the quotient K 4 .Thus if we are merely interested in establishing Theorem A it may appear that we do not need the estimate for the quotient D 1 D 2 , stated in the next theorem.However, a limiting case of (12) below will enter the proof of Theorem 6.3, and the equality result will also be used in Section 7.
To determine optimal networks with quotient D 1 D 2 we now solve a standard calculus problem, namely we maximize the function V under a constraint for L. Interestingly enough, up to similarity of R 3 there is a one-parameter family of optimal networks, meaning that these networks are not strictly stable: where equality holds if and only if In the equality case the lattice is generated, up to similarity, by (0, 1, 0), (0, 0, Proof.Admitting vanishing edge lengths, we will show the inequality in a form implying (12), namely 6 and α ∈ (0, π) , (14) with equality precisely for (13).
For fixed x = (x 1 , . . ., x 6 ) clearly L is independent of α, while (7) gives that V is maximal exactly at α = π/2.Moreover, both V and L depend on x 5 , x 6 only through y := x 5 + x 6 .Thus in order to establish (14) we may fix α to π/2 and consider the functions induced by L and V on the domain [0, ∞) 5 (x 1 , x 2 , x 3 , x 4 , y).For the remainder of the proof we denote these continuous functions again by L and V .
We claim that ( 14) holds along the boundary of [0, ∞) 5 .Trivially, this is true at 0. Otherwise let (x 1 , x 2 , x 3 , x 4 , y) be a point where at least one coordinate vanishes.In case y = 0 the volume is The right-hand side contains the elementary symmetric polynomial of degree k = 3 in m = 4 variables and so indeed, by Maclaurin's inequality (4), The other case is that some x i vanishes for i = 1, 2, 3, or 4. In view of the symmetry of V and L it suffices to consider the case x 1 = 0.Under this assumption Maclaurin's inequality gives Then the claim follows from the estimate on geometric and arithmetic mean, We now proceed as in the proof of Maclaurin's inequality.The continuous function V attains its maximum on the compact set L −1 (1) ⊂ [0, ∞) 5 \ {0} at some point z := (x 1 , . . ., x 4 , y).
We have shown there is a unique critical point z ∈ (0, ∞) 5 for V under the constraint L = 1, where V attains its maximal value V (z) = 4/81.This implies the inequality (14) first for L = 1, and then, by the scaling invariance of L 3 /V , in general.Finally, the uniqueness of z implies that in general equality holds if and only if (13) holds; to verify the lattice vectors use (11).

The srs network covering the K 4 graph
We discuss the network related to the gyroid.Each vertex of a Steiner network has a well-defined affine tangent plane, containing the edge vectors to the incident vertices; each vertex in N/Λ defines a tangent plane up to translation.(We avoid the usage of normal vectors since the tangent planes are unoriented.)For a Steiner network with quotient K 4 we use balancing and the fact that each pair of vertices in K 4 is connected with an edge to show that the four tangent planes are perpendicular to the four space diagonal directions: Lemma 6.1.Let N ⊂ R 3 be a triply periodic Steiner network with quotient graph K 4 .Then the four tangent planes of N/Λ are tangent to the four faces of a regular tetrahedron.Consequently, up to isometry of R 3 , the network N is uniquely defined by its six edge lengths x 1 , . . ., x 6 > 0.
Proof.From N we pick a connected subgraph which contains a vertex p 0 and its three neighbours p 1 , p 2 , p 3 , representing the vertices of N/Λ.Without loss of generality we may assume p 0 to be the origin, p 1 to lie on the x-axis and p 2 , p 3 to lie in the xy-plane.That is, we assume where x i > 0 is the edge length of the edge incident to p i .
Let p 6 = p 0 be a vertex incident to p 1 , compare Figure 9. Copying the reasoning of the proof of Lemma 5.2 we find, in terms of some rotation A β about the x-axis, where −π < β < π: Then min |β|, π − |β| represents the dihedral angle between the two tangent planes at p 0 and p 1 .
In the quotient N/Λ, the vertex p 6 must be identified with one of the four vertices p 0 , . . ., p 3 .Since the shortest cycle in K 4 consists of three edges this vertex must be either p 2 or p 3 .Suppose p 6 is identified with p 2 .The tangent planes at these two points agree as vector spaces.Hence the balancing equation ( 2) implies that the vectors p 2 − p 0 and p 6 − p 1 enclose 120 degrees, and the sum of the two unit vectors pointing into these directions must be a unit vector: The other case is that p 6 is identified with p 3 .Then, similarly, From both cases we conclude | cos(β)| = 1/3, and so the dihedral angle of the tangent planes at p 0 and p 1 is the tetrahedral angle arccos(1/3) ≈ 70.53 • .
In K 4 , any pair of vertices is connected by an edge, and so the same argument applies to any pair of vertices p i , p j of N/Λ.But four planes in R 3 can only have pairwise dihedral angles arccos(1/3) if they are parallel to the faces of a regular tetrahedron.
Finally, lengths and tangent planes determine a Steiner network completely up to isometry.
For the next statement we choose to label the six edges e 1 , . . ., e 6 , such that the edges e i and e i+3 do not have endpoints in common, see Figure 9.We let x i be the length of e i .Lemma 6.2.Let N be a triply periodic Steiner network with quotient K 4 .Then N has length L = i x i and the spanned volume is The sum extends over all possible products of three edge lengths except for those relating to triples of concurrent edges.
Remark.By Lemma 6.1, lengths and tangent planes determine a Steiner network completely up to isometry.Up to rigid motion, however, there are two different Steiner networks covering K 4 with the same edge lengths.The isometry mapping the two networks onto another is a reflection which corresponds to a sign change of β.
Note that the four tangent planes at the vertices of a network are the tangent planes of a regular tetrahedron.Hence, the choice of any two tangent planes determines the other two.
Proof.After isometry of R 3 we may assume the coordinates are as in (16).For i = 1, 2, 3 let A i β ∈ SO(3) be the rotation fixing p i with an angle β = arccos(1/3).In view of Remark 6, possibly by replacing β by −β, the three vectors are linearly independent and span the lattice Λ.To verify (17), calculate Theorem 6.3.Let N be a triply periodic Steiner network in R 3 with quotient K 4 .Then where equality holds if and only if all edge lengths of N coincide and the lattice is body-centered cubic.
Proof.We follow the strategy of the proof of Theorem 5.4.For the present case, L and V are functions of the six edge lengths, see Lemma 6.2.
We first verify the strict inequality L 3 > (27/ √ 2)V along the boundary of [0, ∞) 6 without the point 0. Assume that at least one x i vanishes.By symmetry of V and L in all variables we may assume x 6 = 0. Then the volume V becomes V = 1 √ 2 (x 1 x 2 x 4 +x 1 x 2 x 5 +x 1 x 3 x 4 +x 1 x 3 x 5 +x 1 x 4 x 5 +x 2 x 3 x 4 +x 2 x 3 x 5 +x 2 x 4 x 5 ) .
Figure 10.The graph shown in the middle arises as a limit of the graph K 4 (left) or of D 1 D 2 (right) when the dashed edge is contracted.
Therefore x 1 = x 2 = x 3 = x 6 and, using x 1 x 4 = x 2 x 5 = x 3 x 6 , these must agree with x 4 = x 5 .This proves the claim.Reasoning literally as in the proof of Theorem 5.4 concludes the proof.
Remark.The proof of Theorem 6.3 asserts that if x 6 = 0 the length and volume of the srs network and the ths network with α = arccos(1/3) agree.In particular, the combinatorial graphs of the networks agree, see Figure 10.
We would like to draw another consequence of Lemma 6.1.

Figure 1 .
Figure 1.We identify the srs network (top) as the length minimizer in the class of all triply periodic networks.As indicated by the colouring, the quotient has four vertices and is the graph K 4 .Triply periodic Steiner networks on four vertices can also have the graph D 1 D 2 as a quotient; a minimizing ths network is depicted on the bottom.Observe that the long edges define zigzag curves which are contained in perpendicular planes.The short edges are contained in lines of intersection of these planes.

Figure 2 .
Figure 2. The two graphs with degree 3 on four vertices without loops: K 4 (left) and D 1 D 2 (right).

Figure 3 .
Figure 3.A ths network (a) can continuously be deformed into a triply periodic srs-network (c).The homotopy preserves the lattice but decreases length.The change of topology occurs when two vertices coincide (b).

Figure 4 .
Figure 4. Doubly periodic Steiner networks and their fundamental domains.The underlying abstract graph of the first two networks is the dipole graph of order 3 (cf.Figure6).The first network, with the hexagonal lattice, minimizes length for given area of its fundamental domain.The network on the right has the quotient D 1 D 2 .

Lemma 2 . 5 .
Suppose an n-periodic network N with lattice Λ contains a vertex with degree d ≥ 4 or d = 1, or with degree d = 2 and non-opposite edges.Then there exists an n-periodic network N with lattice Λ and rank N = rank N , such that N is 3-regular with smaller length, L(N ) < L(N ).

Figure 5 .
Figure 5.All connected 3-regular graphs with loops on 4 vertices.By Lemma 2.6, none of these graphs can be the quotient graph of a minimizer.

Figure 6 .
Figure 6.The dipole graph D 3 and a network covering it.

6 Figure 7 .
Figure 7.The graph D 1 D 2 and its covering network schematically.

Figure 8 .
Figure 8.A Steiner network with double edges.The stars of the two doubly connected vertices lie in a common plane.

3 Figure 9 .
Figure 9.The graph K 4 and the labelling of the network covering it.

Figure 11 .
Figure 11.A triply periodic Steiner network with quotient K 4 where the lattice is primitive.The eight vertices shown in red correspond to the eight vertices of a cube.