## Abstract

Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in

## 1. Introduction

Line congruences, that is, two-parameter families of lines, constitute fundamental objects in classical differential geometry [1]. In particular, normal and Weingarten congruences have been studied in great detail [2]. Their importance in connection with the geometric theory of integrable systems has been well documented (see [3] and references therein). Recently, in the context of integrable discrete differential geometry [4], attention has been drawn to *discrete line congruences* [5], that is, two-parameter families of lines which are (combinatorially) attached to the vertices of a

Here, we are concerned with the extension of discrete line congruences in a three-dimensional complex projective space *discrete line complexes*. In the following, attention could be restricted to discrete line complexes in a real projective space

This paper is based on the observation that a system of algebraic equations which arises in various avatars in the theory of both continuous and discrete integrable systems may be interpreted without reference to its origins as a system of integrable discrete equations. Thus, in §§2 and 3, we set down this privileged discrete integrable system (*M-system*) and indicate how it is related to, for instance, the important Fundamental and Darboux transformations of classical differential geometry [13,14], the theory of conjugate lattices [4,5] and the hexahedron recurrence which has been proposed in the context of cluster algebras and dimer configurations [15]. The latter connection is made by interpreting the *M*-system as discrete evolution equations for the minors of a matrix, which, in turn, provides a link with the ‘principal minor assignment problem’ [16].

In §4, we confine ourselves to the case of a 5×5 matrix which depends on three discrete independent variables. On use of the classical correspondence between lines in a three-dimensional projective space and points in the four-dimensional Plücker quadric [17,18], we demonstrate that the corresponding *M*-system governs privileged integrable line complexes in *fundamental line complexes*, which are termed rectilinear congruences in [5], are characterized by the property of intersecting neighbouring lines and a particular planarity property of the points of intersection. It is noted that the planarity property holds automatically in the case of the aforementioned line complexes in *M*-system. In order to show this, we use a geometric construction of fundamental line complexes which owes its existence to Desargues' classical theorem of projective geometry [19]. In fact, this construction reveals that any ‘elementary cube’ of a fundamental line complex should be regarded as being embedded in a classical spatial point-line configuration (15_{4} 20_{3}) of 15 points and 20 lines [20]. It is observed that some of the theorems about fundamental line complexes set down in this section turn out to be important in the construction of supercyclidic nets [21].

In §5, we conclude the paper by characterizing fundamental line complexes in terms of correlations [18,22] of the ambient projective space

## 2. A three-dimensional discrete integrable system

The geometries presented in this paper are integrable in that they are multi-dimensionally consistent in the sense of integrable discrete differential geometry [4]. In algebraic terms, this is readily verified once one has established that their algebraic incarnations constitute canonical reductions of a fundamental system of discrete integrable equations. Thus, we consider three finite sets of ‘lower’ and ‘upper’ indices
*M-system*. Here, and in the following, we suppress the arguments of any discrete function *f* and indicate increments of the independent variables by lower indices. For instance, if *N*=3, then
*M*-system are two-dimensional. For the same reason, the *M*-system may be extended consistently to a system of equations of the same type on any larger lattice *M*-system constitutes a multi-dimensionally consistent three-dimensional (integrable) system. It is noted that, in order to avoid a relabelling of the upper indices on the functions *M*^{ik}, we have assumed without loss of generality that

As indicated below, there exists a variety of connections with the geometric and algebraic theory of integrable systems both discrete and continuous which illustrates the universal nature of the *M*-system (2.3).

### (a) The Darboux and fundamental transformations

It is well known that composition of the classical Darboux transformation [24] and its adjoint leads to a compact binary Darboux transformation formulated in terms of ‘squared eigenfunctions’ [25]. Thus, if (*ϕ*^{k},*ϕ*^{l}) and (*ψ*^{i},*ψ*^{l}) are pairs of solutions of the time-dependent Schrödinger equation and its adjoint
*u*=*u*(*x*,*t*) constitutes a given potential, then, up to additive constants of integration, bilinear potentials *M*^{αβ} are uniquely defined by the compatible pairs
*α*∈{*i*,*l*}, *β*∈{*k*,*l*} and the quantities

The new squared eigenfunction *ϕ*^{β} and *ψ*^{α} as carrying a ‘hidden’ index and sets *M*^{0β}=*ϕ*^{β}, *M*^{α0}=*ψ*^{α}, then (2.8) is precisely of the same form as (2.10).

The algebraic structure of the superposition formula (2.10) coincides with that of the discrete dynamical system (2.3). In fact, if one regards the lower index *l* in (2.10) as a shift on a lattice and iterates the above binary Darboux transformation then one may generate solutions of system (2.3). The transformation formulae (2.8) and (2.10) are universal in that they apply to a large class of linear evolution equations in 1+1 dimensions with potentials *M*^{αβ} being defined by suitable analogues of the pair (2.7) [26]. Moreover, if one considers vector-valued solutions of the hyperbolic equation
_{1} represents nothing but the classical fundamental transformation [13,14] for conjugate nets encoded in (2.11). The transformation formulae (2.8)_{1} and (2.10) have also been shown to apply in the case of the canonical discrete analogue of the fundamental transformation [5,27].

### (b) Conjugate lattices

We now consider the case *U*^{l}={0,1,…,*N*}, *U*^{r}={1,…,*N*+*d*} and introduce the vector notation
*M*-system (2.3) then translates into
**M**^{l} may be regarded as the ‘tangent vectors’ of an *N*-dimensional quadrilateral lattice
*d*-dimensional Euclidean space. Moreover, system (2.13)_{2} implies that the quadrilaterals are planar and, hence, * r* constitutes a conjugate lattice [4]. Importantly, it may be shown that all conjugate lattices may be obtained in this manner (cf. §4

*b*(iii)). In view of the previous section, the appearance of conjugate lattices is consistent with the classical and well-known observation that iteration of the classical fundamental transformation generates planar quadrilaterals [28] and therefore conjugate lattices. Thus, one may regard system (2.3) as the algebraic essence of the classical fundamental transformation without reference to geometry. Important algebraic properties of the fundamental

*M*-system are presented below.

## 3. Minors, **τ**-function and the hexahedron recurrence

It turns out that the link between the algebraic and geometric perspectives adopted in this paper is provided by algebraic identities for the minors of the matrix *M*^{ik}. Thus, for any two multi-indices *A*=(*a*_{1}·*a*_{s}) and *B*=(*b*_{1}·*b*_{s}), where the entries of each multi-index are assumed to be distinct elements of *U*^{l} and *U*^{r}, respectively, we may define the minors
*M*^{∅,∅}=1.

### (a) Jacobi-type identities and identification of a metric

In terms of the above minors, Jacobi's classical identity for determinants [29] may be expressed as

The identity

A degenerate case of the identity (3.6) is obtained by (temporarily) assuming that the rows of the matrix *a* and

Here, (3.12) plays the role of the Jacobi identity (3.2). It is evident that, for reasons of symmetry, the identity

### (b) Evolution of the minors and the **τ**-function

For any *l*∈*L* which is not contained in two multi-indices *A* and *B*, a Laplace expansion of *M*^{lA,lB} with respect to the row or column labelled by *l* shows that
*A* and *B* constitute simple indices, then the *M*-system (2.3) is retrieved. Accordingly, system (2.3) may be reinterpreted as the integrable evolution (3.15) of the minors of the matrix *M*^{A,B}, because the compatibility condition *a* realization of the quantities *M*^{A,B} in terms of the minors of a matrix

In the particular case *U*^{l}=*U*^{r}=*L*, the above evolution of the minors gives rise to a compact formulation of the evolution of a *τ*-function associated with the *M*-system. Thus, it is readily verified that the compatibility conditions (*τ*_{i})_{k}=(*τ*_{k})_{i}, which guarantee the existence of a function *τ* defined according to
*M*-system (2.3). In fact, it turns out that
*i* and *k*. In general, if we consider the multi-index *A*=(*a*_{1}·*a*_{α}) with distinct entries, then the evolution (3.15) immediately implies that

### (c) The hexahedron recurrence

We conclude this section with an application of the *τ*-function. A so-called hexahedron recurrence has been proposed by Kenyon & Pemantle [15] which admits a natural interpretation in terms of cluster algebras and dimer configurations [33]. Here, it is demonstrated that the hexahedron recurrence is but another avatar of the *M*-system for *U*^{l}=*U*^{r}=*L*={1,2,3}. Thus, four functions
_{4} may be cast in the form
*h* in terms of the functions
_{1,2,3} may be written as
_{1} shifted in the direction *n*_{k} and (3.27)_{2} yields
_{2} shifted in the direction *n*_{l}, *l*≠*i*,*k* and matched with (3.28) produces
*M*^{ik} evolve according to
*M*-system (2.3) for *U*^{l}=*U*^{r}=*L*={1,2,3} and the function
*τ*-function obeying the linear system (3.16) and its higher-order implications (3.17) and (3.19).

One of the intriguing properties of the hexahedron recurrence (3.21) is that the solution remains positive if the initial data are positive. Unfortunately, this important property is hidden in the *M*-system avatar. However, the advantage of the latter formulation is that multi-dimensional consistency is automatically guaranteed. In this connection, it is noted that the cube recurrence, that is, the discrete BKP (Miwa) equation, may be formulated in such a manner that positivity is guaranteed but this property does not hold on higher-dimensional lattices [34].

The hexahedron recurrence admits a reduction to a single third-order ‘recurrence’ obtained by Kashaev [35] in the context of star-triangle moves in the Ising model. The existence of this recurrence is readily verified if one exploits its formulation in terms of the *M*-system. Indeed, it is evident that the *M*-system considered above admits the reduction *M*^{ik}=*M*^{ki} corresponding to a symmetric matrix *M*^{ik} between the linear system (3.17) and (3.19) has been shown to lead to the discrete CKP (dCKP) equation [36]

## 4. Integrable discrete line complexes in C P 3

In the remainder of this paper, we are concerned with the case *N*=3 and *U*^{l}=*U*^{r}={1,2,3,4,5}. Hence, the associated *M*-system (2.3) governs the evolution of the matrix
*n*_{1},*n*_{2},*n*_{3}. More precisely, if we prescribe the Cauchy data
*S*^{ik}, then

### (a) From algebra to geometry

In order to reveal the geometry encoded in the matrix

#### (i) Identification of a Plücker quadric

If we define the vector-valued function
*A*=*B*=∅ and *Q*^{4} embedded in a five-dimensional complex projective space *Q*^{4} with the complexification of the classical Plücker quadric [18] then, on use of the Plücker correspondence (cf. §4*b*(i)), we may interpret any point [V(* n*)]∈

*Q*

^{4}as a line

*(discrete) line complex*

#### (ii) Incidence of lines

In order to uncover the geometric properties of the line complex l defined by V, we first observe that a shift of V in any lattice direction may be expressed elegantly in terms of higher-order minors of *A*=*B*=∅ and _{l} intersect. Thus, the line complex l has the property that the two lines which are combinatorially attached to the two vertices of any edge of

The above geometric property implies that the four lines l,l_{l},l_{m} and l_{lm} with *l*≠*m* form a (skew) quadrilateral (figure 2). Its diagonals l^{l,m} and l^{m,l} may be obtained by solving the linear system
^{*,*},V^{*,*}〉=0. On raising indices by means of (3.15), this linear system may be formulated as
*C* and *D* such that the relevant minors are well defined. Up to scaling, the two null vector solutions turn out to be
*A*=*B*=∅ and *A*=*l*, *B*=*m* coincides with 〈V^{l,m},V〉=0 and 〈V^{l,m},V^{lm,lm}〉=0, respectively. On the other hand, 〈V^{l,m},V^{l,l}〉=0 due to the identity (3.13) for *A*=*B*=∅. Similarly, the remaining vanishing inner product 〈V^{l,m},V^{m,m}〉=0 is a consequence of the identity (3.14). Finally, for reasons of symmetry, the same arguments apply in the case of the solution V^{m,l}.

#### (iii) Coplanarity and concurrency properties

In order to analyse the properties of the diagonals l^{l,m}, we label the points of intersection of any two neighbouring lines by
*b*. It is noted that, for brevity, we make no distinction between a point of intersection

### Lemma 4.1

*The diagonal* l^{l,m} *passes through the points of intersection* p^{l} *and*

It is natural to associate the point of intersection of any two neighbouring lines with the edge connecting the corresponding vertices of the ^{l,m},l^{l,p} and ^{l,m} and ^{l,m} and l^{p,m} lie in the plane spanned by the lines l and l_{m}, the former lines likewise intersect. Algebraically, this is confirmed by

### Definition 4.2

A line complex *fundamental* if any neighbouring lines l and l_{l} intersect and the points of intersection p^{l} enjoy the *coplanarity property*, that is, for any distinct *l*,*m* and *p*, the quadrilaterals ^{l} and *concurrency property*, that is, the four lines connecting the points

### Remark 4.3

It is readily verified that if any one of the six coplanarity and concurrency conditions associated with an elementary cube of a line complex is satisfied then the other five conditions automatically hold.

The analysis presented in the preceding may therefore be summarized as follows.

### Theorem 4.4

*Any solution* *of the M-system* (2.3) *with N*=3 *and U*^{l}*= U*^{r}={1,2,3,4,5} *parametrizing a function* V *according to* (4.4) *encapsulates a fundamental line complex* l *via the Plücker correspondence*

### (b) From geometry to algebra

Here, we demonstrate that the converse of theorem 4.4 is also true, that is, all fundamental line complexes are algebraically represented by the *M*-system (2.3).

#### (i) The Plücker correspondence

The classical Plücker correspondence is established by considering a lift a∧b of a line l in *γ*^{μν} are defined by the subdeterminants
*γ*^{μν} obey the classical Plücker identity
*Q*^{4}. Specifically, in the generic case, we may make the choice
*M*^{44},*M*^{55},*M*^{54} and *M*^{45} are merely labels of the ‘non-trivial’ homogeneous coordinates of the points a and b.

We now focus on the lines of a fundamental line complex l. Because any neighbouring lines l and l_{l} intersect, the points a,b and a_{l},b_{l} must be coplanar. This may be expressed as
*N*^{l4} and *N*^{l5} are functions to be determined and Δ_{l}*f*=*f*_{l}−*f*. Accordingly, the points of intersection p^{l} are given by
*M*-system as stated in theorem 4.4. In this case, the evolution of the matrix (4.3) may be formulated as
*a*, it is then straightforward to verify that the line passing through the points of intersection p^{l} and ^{l,m} as stated in lemma 4.1.

#### (ii) Fundamental line complexes in C P 4

In order to make the transition from a fundamental line complex to a solution of the *M*-system, we first observe that, as pointed out in [5], the inclusion of the coplanarity property or, equivalently, the concurrency property in the definition of a fundamental line complex is due to the dimensionality of the ambient space of the line complexes discussed here. Thus,

### Definition 4.5

A line complex *fundamental* if any neighbouring lines l and l_{l} intersect.

Indeed, the two sets of four lines {l,l_{1},l_{2},l_{12}} and {l_{3},l_{13},l_{23},l_{123}} span two (three-dimensional) hyperplanes which, generically, intersect in a (two-dimensional) plane. The latter plane contains the points ^{3} automatically enjoys the coplanarity property.

It turns out that fundamental line complexes in

### Theorem 4.6

*A line complex in* *is fundamental if and only if it may be regarded as a projection onto a hyperplane of a fundamental line complex in*

### Proof.

Given a fundamental line complex in _{123} and associated points of intersection. We begin by choosing a generic projection *π* and prescribing five (black) points *l*,*m*)≠(1,3) as depicted in figure 4. These points define the four lines _{23},l_{2},l_{12},l_{1}, respectively. Because [p^{1}]∈l_{1} and [p^{2}]∈l_{2}, the lines _{3}. Once again, because _{13}.

As discussed in §4*c*, the six lines *c*. It remains to observe that the above arguments also apply (iteratively) to complete fundamental line complexes in light of the Cauchy problems for fundamental line complexes formulated in the same section. ▪

#### (iii) A conjugate lattice connection

For any pair of fundamental line complexes related by a projection in the sense of theorem 4.6, we identify the associated hyperplane with the ‘hyperplane at infinity’. Hence, the transition from a fundamental line complex in *same* functions *N*^{l4} and *N*^{l5}. The latter may be resolved by introducing discrete ‘tangent vectors’ **M**^{l} according to
* a* and

*constitute Combescure transforms of each other [39].*

**b**The planarity of the quadrilaterals of * a* and

*may be expressed as*

**b**

**M**^{1},

**M**^{2},

**M**^{3}, we therefore conclude that, in particular,

**M**^{l}were associated with a line complex in

*φ*

^{l}. These may be used to scale the coefficients

*I*

^{ml}to unity by applying the gauge transformation

*I*

^{ml}=1. The remaining compatibility conditions then reduce to the nonlinear system

*N*

^{lm},

**M**^{l}of the discrete Darboux system and the linear system (4.31), two Combescure transforms are given by

*and*

**a***, where the coefficients*

**b***N*

^{l4}and

*N*

^{l5}are solutions of the same linear system

The final step in the identification of the *M*-system in question is based on the observation that the discrete Darboux system admits the ‘conservation laws’
*M*^{ll} defined by the linear equations
*M*-system
*M*^{6k} are merely auxiliary functions which represent the transition of a fundamental line complex from

### Theorem 4.7

*Any fundamental line complex in* *gives rise to a solution* *of the M-system (2.3) with N=3 and U*^{l}*=U*^{r}*={1,2,3,4,5} via the lift* a*∧*b *of the lines* l *encapsulated in (4.22).*

### (c) Geometric construction of fundamental line complexes

An elementary cube of a fundamental line complex l in _{1},l_{2},l_{3} and l_{12},l_{23},l_{13} (figure 5). On the assumption that these triples are in general position, the lines l and l_{123} are then uniquely determined by the requirement that these pass through the relevant triple of lines. An entire fundamental line complex is uniquely determined by prescribing a ‘plane’ of hexagons as Cauchy data, that is, by specifying the set of lines

In the case of fundamental line complexes in _{1},l_{2},l_{3} may be interpreted as three generators of a unique quadric. The line l then constitutes an element of the second one-parameter family of generators of the quadric. An analogous interpretation is valid in the case of the second triple of lines l_{12},l_{23},l_{13} and its transversal line l_{123}. As shown in [5], the line l_{123} is uniquely determined by the coplanarity property and a fixed choice of the line l. In fact, the existence of the line l_{123} may be traced back to the classical Desargues theorem of projective geometry [19].

### Theorem 4.8

*Given seven lines in* *which are combinatorially attached to seven vertices of an elementary cube and intersect each other ‘along edges’, there exists a unique eighth line such that the eight lines constitute an elementary cube of a fundamental line complex.*

### Proof.

Without loss of generality, we consider the seven (short black) lines l,l_{1},l_{2},l_{3} and l_{12},l_{23},l_{13} depicted in figure 6. The associated nine (black) points of intersection are given by _{12}. Similarly, the (grey) points _{123}, we focus on a subset of nine lines, namely, for instance, the lines l_{1},l_{12},l_{13} and the lines passing through the pairs of points of intersection _{3}) configuration of 10 points and 10 lines, because the tenth line passing through the points

### Remark 4.9

Theorem 4.8 implies that, for any given ‘hexagon’ of six lines l_{1},l_{2},l_{3} and l_{23},l_{13},l_{12} of an elementary cube of a fundamental line complex, the planarity property gives rise to a unique map between the lines l and l_{123} contained in the hyperboloids defined by l_{1},l_{2},l_{3} and l_{23},l_{13},l_{12}, respectively.

### Remark 4.10

The complete set of eight lines of an elementary cube of a fundamental line complex and the 12 associated diagonals together with the 12 points of intersection of the lines and the three points of concurrency of the diagonals give rise to a spatial point-line configuration (15_{4} 20_{3}) of 15 points and 20 lines with four lines through each point and three points on each line (figure 9). This configuration constitutes the frontispiece to Baker's first volume of *Principles of geometry* [20] and was used (but not displayed) in Coxeter's monograph *Projective geometry* [19] in connection with the proof of the converse of Desargues' theorem.

As in the case of fundamental line complexes in

## 5. Fundamental line complexes and correlations

It turns out that fundamental line complexes in *d*-dimensional projective space is an incidence-preserving transformation which maps *k*-dimensional projective subspaces to *d*−*k*−1-dimensional projective subspaces [18,22]. In particular, in three dimensions, the points of a line are mapped to planes which meet in a line. Here, we represent a correlation by a map
*κ* for the representation of this map in terms of homogeneous coordinates. Any correlation is then encoded in a complex 4×4 matrix B such that

In the previous section, it has been demonstrated that, for any given line of an elementary cube of a fundamental line complex in ^{1},…,x^{6} as displayed in figure 10. The planes spanned by any three successive vertices x^{i−1},x^{i},x^{i+1}, where indices are taken modulo 6, are denoted by *π*^{i}. The condition for a correlation *κ* to map any line (extended edge) of the hexagon to its opposite line may therefore be expressed as

As in the case of projective transformations of *κ* of the above type is unique if it exists and is necessarily involutive, because it acts as an involution on the hexagon. The corresponding matrix B must then be either skew-symmetric or symmetric. In the former case, any point lies in its image, which contradicts the assumption of the hexagon being in general position. Hence, the correlation constitutes a polarity [18,22] with respect to the quadric x^{T}Bx=0 defined by the symmetric matrix B and it is convenient to define the inner product

In order to demonstrate that the above system of linear equations admits a solution (which is unique up to scaling), we assume without loss of generality that
*B*^{02} and *B*^{13} turn out to be two Plücker coordinates of the line passing through a and B, namely

### Theorem 5.1

*For any hexagon in* *in general, position, there exists a unique correlation which maps any line (extended edge) of the hexagon to its opposite line. The correlation is involutive and constitutes a polarity.*

We now combinatorially attach the lines of the hexagon to six vertices of an elementary cube and the points of intersection x^{i} to the corresponding edges as depicted in figure 11. The line passing through the vertices x^{i} and x^{i+1} is denoted by l^{ii+1}. There exists a one-parameter family of lines l which intersect the lines l^{12},l^{34} and l^{56}. Any fixed line l is mapped by the correlation *κ* to a line l′ which intersects the lines l^{23},l^{45} and l^{61}. Accordingly, the eight lines l,l^{12},l^{23},l^{34},l^{45},l^{56},l^{61} and l′ form an elementary cube of a line complex with the usual property of lines intersecting along edges. On the other hand, as pointed out in the preceding, the hexagon and the line l uniquely determine via the planarity property an eighth line

We denote by x^{ii+1} the point of intersection of a line l or its counterpart l′ and the line l^{ii+1} as indicated in figure 11. One may construct the one-parameter family of lines l by parametrizing the points

and solving the collinearity condition
*μ*^{3},*μ*^{4},*μ*^{5},*μ*^{6} in terms of *μ*^{1} and *μ*^{2}. A brief calculation reveals that
^{23},l^{45} and l^{61} may be obtained on use of the parametrization
*ν*^{6},*ν*^{1},*ν*^{2},*ν*^{3} in terms of *ν*^{4} and *ν*^{5}. The latter two parameters are fixed (up to scaling) by the condition that l′ be the image of l under the correlation *κ*. This is equivalent to demanding that, for instance,
*ν*^{i}=*μ*^{i}. Finally, one may directly verify that, for instance,
^{2},x^{34},x^{5},x^{61} are seen to be coplanar as indicated in figure 11. This implies that, remarkably, the lines l^{ii+1} and l,l′ form an elementary cube of a fundamental line complex in

### Theorem 5.2

*Fundamental line complexes in* *are line complexes with the property that neighbouring lines intersect and opposite lines of any elementary cube are interchanged by a correlation which, necessarily, constitutes a polarity. Furthermore, the planarity and concurrency properties associated with any elementary cube of a fundamental line complex are interchanged by the corresponding correlation in the sense that any four coplanar diagonals are mapped to four concurrent diagonals and vice versa.*

## 6. Conclusion

The superposition principle (2.10) for the squared eigenfunctions *M*^{ik} associated with binary Darboux transformations is standard in the algebraic and geometric theory of continuous and discrete integrable systems. In this paper, we interpret this superposition principle as a stand-alone discrete integrable system, namely the *M*-system (2.3), and discuss its algebraic and geometric properties. In order to highlight its universality, we have briefly indicated its direct connection with conjugate lattices and the hexahedron recurrence. The main aim of this paper is to introduce a novel correspondence between the *M*-system and fundamental line complexes represented as lattices in the Plücker quadric. In fact, the Plücker coordinates of the lines turn out to be the entries *M*^{ik} and, more generally, the minors of the matrix *M*^{ik}=*M*^{ki} is compatible with the *M*-system. In geometric terms, this means that one considers the intersection of the Plücker quadric with a hyperplane, resulting in a three-dimensional quadric which one may identify with the Lie quadric of Lie circle geometry. In this manner, the intersecting lines of the fundamental line complexes are contained in a linear complex and may be reinterpreted as touching-oriented circles. As a consequence, these circle complexes are governed by the dCKP equation alluded to in §3. An analogous approach is also available in the context of Lie sphere geometry. A key ingredient in the geometric treatment of these special fundamental line complexes is the characterization of fundamental line complexes in terms of the correlations discussed in §5.

## Author contributions

The authors contributed in an equal manner to all components of the research reported in this paper.

## Funding statement

This research was supported by the DFG Collaborative Research Centre SFB/TRR 109 Discretization in Geometry and Dynamics and the Australian Research Council (ARC).

## Conflict of interests

We have no competing interests.

- Received October 20, 2014.
- Accepted January 14, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.