A novel generalization of Cliffords classical point-circle configuration. Geometric interpretation of the quaternionic discrete Schwarzian KP equation

A no v el generaliz ati o n of C lifford’s classical p oin t-circle co n figuration. Geometric i n terpretatio n of the quaternioni c discre t e Sc h w arzian KP equation B y W. K. Schief 1 , 2 and B. G. Konopelchenko 3 1 Institut f¨ ur Mathematik, T e chnische Un iversit¨ at Berlin, Str aße des 17. Juni 136, D-10623 Berlin, Germany 2 Austr alian R ese ar ch Council Centr e of Exc el lenc e for Mathematics and Statistics of Complex S ystems, Scho ol of Mathematics, The University of New South Wales, Sydney, N S W 2052, Austr alia 3 Dip artimento di Fisic a, Universit` a del Salento and INFN, Sezione di L e c c e, 73100 L e c c e, Italy The algebraic and geo metr ic prop erties of a nov el generalizatio n of Clifford’s clas - sical C 4 po int -circ le configuration are a nalysed. A co nnec tion with the int egr able quaternionic discrete Sc h warzian Kadom tsev-Petviashvili equation is rev ealed. Clifford configurations; quaternions; multi-rat ios; integrable systems; discrete differen tia l geometry 1. I ntro duction In his seminal paper Cliffo r d’s chain and its analo gu es in r elation to the higher p olytop es , Longuet-Higgins (1972) asserts that ”a chain o f theo r ems ... has ex- erted a p eculiar fascinatio n for mathematicians since its disc overy by Cliffo r d in 1871” . Indeed, v arious genera lizations and analog ues in higher dimension of Clif- ford’s p oint-circle configurations C n (Clifford 18 71) asso ciated with such luminaries as de Long echamps (1877), Cox (1 891), Grace (18 98), Brown (19 54), Coxeter (19 56) and Longuet-Higg ins (1972 ) hav e b e e n reco rded and analy sed in detail. The original celebrated chain o f ‘circle theorems’ may b e stated as follows: Given four stra ight line s on a plane, the four circumcircles of the four triangles so formed ar e concurre n t in a po int Q 4 , s ay (cf. figure 1). Given five lines on a plane, by o mitting each line in turn, we obtain five corresp ond- ing p oints Q 4 and these lie on a circle C 5 , s ay . Given six lines on a plane, we obtain six co r resp onding circ le s C 5 and these ar e concurrent in a p oint Q 6 . Etc. 2 W. K. Schief and B. G. Konop elchenko 2 l 1 4 P Q 14 P 12 P 23 P 24 P 13 P 34 l 4 l 3 l Figure 1. A ‘Menelaus configuration’ Generally , giv en n copla na r lines , we obtain n corresp onding circles C n − 1 which ar e concurrent in a po int Q n or n p oints Q n − 1 which lie on a circle C n depe nding on whether n is even or o dd resp ectively . Finally , application of an inv ers io n with res pec t to a generic p oint o n the pla ne leads to a complete and symmetric configura tion of 2 n − 1 po int s and 2 n − 1 circles with n p oints on every circle and n circles through every p oint. The firs t theorem a sso ciated with four straig h t lines immediately de mo nstrates that there e x ists a close co nnection b etw een C 4 Clifford configur ations and the ancient Theorem of Menelaus (Pedo e 19 70). The latter states that three points P 14 , P 24 , P 34 on the (extended) edges of a tria ngle with vertices P 12 , P 23 , P 13 as display ed in figure 1 are collinear if and only if the condition P 12 P 24 P 23 P 34 P 13 P 14 P 24 P 23 P 34 P 13 P 14 P 12 = − 1 (1.1) for the asso ciated directed lengths is satisfied. Accordingly , the p oints of in tersection of the fo ur lines l 1 , l 2 , l 3 , l 4 in Clifford’s fir st theorem (see figur e 1) ob ey condition (1.1). If the plane is identifi ed with the c o mplex plane then condition (1.1) con- stitutes a multi-ratio condition for the co mplex num b ers P ik which we denote by (cf. § 2 ) M ( P 14 , P 12 , P 24 , P 23 , P 34 , P 13 ) = − 1 (1.2) (mo dulo a trivial cyclic per m utation of the arg umen ts). The latter is evidently inv ar iant under the gr oup of inversiv e transfor mations (Brannan et al. 1 999) and hence the p oints P ik of a C 4 Clifford configur ation likewise satisfy the m ulti-ratio condition (1.2). In fact, it has b een p ointed out in K onop elchenk o & Sc hief (2002 ) that the latter constitutes a defining pr o p e r ty of C 4 Clifford configuratio ns. In this pap er, we inv estigate in detail the geo metric and algebr aic pro per ties of a nov el ge ner alization of Cliffor d’s C 4 configuratio n which has b een discov ered via Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 3 the theo ry of integrable systems (soliton theory) (Ablowitz & Segur 19 81; Zak harov et al. 1980). Thus, in Ko no pe lchenko & Schief (2002), it has b een shown that (1.2) interpreted as a lattice equa tion which is de fined o n the ‘o ctahedral’ vertex configuratio ns of a face-centred cubic (fcc) lattice (cf. § 7) constitutes a Sch warzian version (Dor fman & Nijhoff 19 91; Bogdanov & Konop elchenko 1 9 98) of the Hirota- Miwa eq uation, that is the discrete K adomtsev-Petviash vili (dKP) equation (Hiro ta 1981). The latter is regarded as a ‘master eq uation’ in soliton theo ry since it enco des the co mplete KP hier arch y of soliton equations via sophisticated contin uum limits. The discr ete Sch warzian KP (dSKP ) equatio n admits a natura l multi-component analogue, namely the natural matrix generaliza tion o f the multi-ratio condition (1.2) interpreted a s a lattice equation (Bo gdanov & K o nop elchenk o 199 8). In the simplest case , the qua ternionic dSKP (qdSKP) equa tion lo cally r epresents a s ix- po int relatio n for six quater nions P ik , that is , a rela tion betw een six p o int s P ik in a four - dimensional E uclidean spa ce R 4 if the s tandard identification R 4 ∼ = H with the a lgebra of quaternio ns is made. Since the mult i-ra tio condition (1.2) enco des C 4 Clifford configur ations, it is natur al to inquire a s to the geo metr ic sig nificance of its quater nionic co un terpar t. It turns out that an appro priate characterization of classical C 4 Clifford configurations giv es rise to natura l ana logues in a four- dimensional E uclidean space which are algebra ically governed b y the lo cal qdSKP equation. In the present context, the key prop erty of classica l C 4 Clifford configur ations turns out to b e the Go dt-Ziegenbein pr op erty which s tates that, in a sp ecific sense, the ang les made by four o riented cir cles passing through a p oint a re the same for all eight po ints (Go dt 1896; Ziegenbein 1941). This pro per t y is used in § 3 to define o ctahedral point-circle configuratio ns in R 4 of Clifford t yp e. In § 6, the existence of such generalized Clifford configurations is prov en and it is demonstrated tha t these are indeed gov erned b y the afore- men tioned quaternionic version of the multi-ratio condition (1.2 ). As a by-pro duct, it is shown that, for any five g eneric p oints in R 4 , there exists a pair of as so ciated gener alized Cliffo r d configura tions which are related by r e flection in the hyper s phere defined by the given five p oints. The fina l section is then devoted to the geometr y of the qdSK P (lattice) equation. 2. The classical C 4 Clifford configuration W e b egin with the class ical constructio n of C 4 Clifford c o nfigurations. Th us, co nsider a po int P on the plane and four g eneric circles S 1 , S 2 , S 3 , S 4 passing throug h P as depicted in figure 2. The six a dditional points of intersection are lab elled b y P 12 , P 13 , P 14 , P 23 , P 24 , P 34 . Here, the indices o n P ik corres p ond to those of the circles S i and S k . An y three circles S i , S k , S l int erse c t at three points and therefore define a circle S ikl passing through these po in ts. Cliffor d’s circle theorem (Clifford 187 1) then states tha t the four cir c le s S 123 , S 124 , S 134 , S 234 meet at a p oint P 1234 . Even though Clifford configur ations ( C n ) exist for a ny n umber of initial cir cles S 1 , . . . , S n passing through a p oint P , for brevity , we here ass o ciate with the term ‘Clifford configuratio n’ the ca s e n = 4. In Konop elchenk o & Sc hief (2002), as an immediate consequenc e of the cla ssical Theorem of Menelaus (Pedoe 197 0), it has been demo nstrated that any generic six po int s P 12 , P 13 , P 14 , P 23 , P 24 , P 34 on the plane regarded as complex n um b ers b elong to a Clifford configuration (with the ab ov e-mentioned com binator ics) if a nd o nly if 4 W. K. Schief and B. G. Konop elchenko 0 2 13 14 23 24 12 1 1234 34 124 123 134 234 3 4 Figure 2. A classical C 4 Clifford configuration they o b e y the m ulti-ra tio condition M ( P 14 , P 12 , P 24 , P 23 , P 34 , P 13 ) = − 1 , (2.1) where the multi-ratio of six complex num b ers P 1 , . . . , P 6 is defined by M ( P 1 , P 2 , P 3 , P 4 , P 5 , P 6 ) = ( P 1 − P 2 )( P 3 − P 4 )( P 5 − P 6 ) ( P 2 − P 3 )( P 4 − P 5 )( P 6 − P 1 ) . (2.2) In par ticula r, this c o nfirms the g eometrically evident fact that any five generic po int s o n the plane uniquely define a Clifford configuratio n. Indeed, for any g iven generic p oints P 12 , P 13 , P 14 , P 23 , P 24 , the sixth po in t P 34 is determined b y the linear equation (2.1). The la tter p oint may lie ‘at infinity’, in which ca s e the four circles passing through P 34 degenerate to str aight lines. The justification o f the o rdering of the ar guments in (2.1) is c o nsigned to § 6. The pr eceding discussio n indicates tha t one ma y think of a Clifford config u- ration as a config ur ation of six p oints P 12 , P 13 , P 14 , P 23 , P 24 , P 34 and eight circles S 1 , S 2 , S 3 , S 4 , S 123 , S 124 , S 134 , S 234 which is such that the four circle s S 1 , S 2 , S 3 , S 4 int erse c t at a p oint P or the four cir cles S 123 , S 124 , S 134 , S 234 int erse c t at a p oint P 1234 . Clifford’s circle theorem then guar antees the exis tence of the remaining po int P 1234 or P resp ectively . In this connectio n, it is no ted that the eight p oints P, . . . , P 1234 of a Clifford configuration a ppea r on an equal fo oting so that, at first sight, the ab ov e in terpre ta tion of a Cliffor d configuratio n do es not seem to b e nat- ural. How ever, it turns o ut that it is pr ecisely this p oint of view which a llows for a g eneralization of Clifford c o nfigurations in which, gener ically , the p oints P a nd P 1234 do not exist. A r emark a ble pro per ty of Clifford configurations is that the angles made b y four oriented circles passing throug h a p oint are the same for all eig h t p oints in a sense to Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 5 14 14 12 23 34 24 13 12 34 23 24 13 Figure 3. The com binatorics of an octahed ral p oint-circle configuration be sp ecified in the following s ection. Therein, it is shown that this Godt-Zie genbein prop erty (Godt 1896 ; Ziegenbein 194 1 ) constitutes a defining pro per ty of Clifford configuratio ns. In fact, it is sufficien t to demand that the Go dt-Zieg enbein prop erty holds for the p oints P 12 , P 13 , P 14 , P 23 , P 24 , P 34 . The la tter observ a tio n serves as the basis for the definition of gener alized Clifford configur a tions. 3. Oct ahedral point-circle configurations. Definitions and notation In the following, we are c oncerned with co nfigurations in a four-dimensiona l Eu- clidean s pa ce R 4 consisting of six po int s and eight circles with three p oints on every circle a nd four circles through every p oint. More precisely (cf. fig ur e 3): Definition 3.1 . (Octahedral p oint-circle configurations ) A c onfigur ation of six p oints and eight cir cles in R 4 is terme d an o ctahedra l po int-circle configur ation if the c ombinatorics of the c onfi gur ation is that of an o ct ahe dr on, that is the p oints of the c onfigur ation c orr esp ond to the vertic es of the o ctahe dr on while the cir cles c orr esp ond to t he triangular fac es. In o rder to define o ctahedral p oint-circle co nfigurations o f Cliffor d type, it is necessary to intro duce a corres po ndence betw een c ir cles which pas s through dif- ferent p oints. T o this end, we o bserve that a ny vertex of an o ctahedr on may b e asso ciated with its ‘oppos ite’ counterpart, that is the vertex which is not connected via an edge. Similarly , there ex ist four pairs of disconnected ‘opp osite’ faces . T hus, by virtue of the combinatorial cor resp ondence e mployed in the ab ove definition, any p oint P of a n o ctahedral p oint-circle configuratio n admits an ‘opp osite’ p o int P ∗ and any circle S is a s so ciated with an ‘opp osite’ cir c le S ∗ (cf. figur e 4). Definition 3.2. (Corresp ondence of circles) L et P , P 1 , P 2 and S, S 1 , S 2 b e p oints and cir cles of an o ctahe dr al p oint-cir cle c onfigur ation such that S 1 and S 2 interse ct at P 1 and P 2 and P lies on S . Then, t he cir cles S 1 and S 2 p assing thr ough P 1 ar e said to cor resp ond to the cir cles S 2 and S 1 r esp e ctively p assing thr ough P 2 and the cir cle S p assing thr ough P is s aid t o corr esp ond to the opp osite cir cle S ∗ 6 W. K. Schief and B. G. Konop elchenko * S P S P * Figure 4. ‘Opp osite’ p oints and circles p assing thr ough the opp osite p oint P ∗ : ( S 1 , S 2 ; P 1 , P 2 ) ↔ ( S 2 , S 1 ; P 2 , P 1 ) , ( S ; P ) ↔ ( S ∗ ; P ∗ ) . (3.1) Iterative applica tion of the ab ove co rresp ondence pr inciple immediately leads to the following corr esp ondence: Observ ation 3. 3. Any cir cle S 1 of an o ctahe dr al p oint-cir cle c onfigur ation p assing thr ough a p oint P 1 admits five c orr esp onding cir cles S 2 , . . . , S 6 which p ass thr ough the r emaining five p oints P 2 , . . . , P 6 . Thus, ther e exists a u nique c orr esp ondenc e b etwe en the six s et s of four cir cles S µ k , µ = 1 , 2 , 3 , 4 p assing t hr ough the p oints P k of an o ct ahe dr al p oint-cir cle c onfigur ation. Con v ent ion 3.4. (Orienta tion of circles) The orientation of t he cir cles of an o ctahe dr al p oint-cir cle c onfigur ation is chosen in such a manner that the c orr e- sp onding orientation of t he fac es of the o ctahe dr on is the same for al l fac es (when viewe d fr om ‘outside’) (cf. figu re 5). F or co mpleteness, it is r emarked that the mos t ge ner al a dmiss ible orie ntation of the circles is obtained b y simultaneously changing the or ient ation of a face of the o ctahedron and its three neighbours and then iterating this o per ation. If we now demand that an oriented c ir cle and its tangent v ectors b e of the same orientation then the following definition is natural: Definition 3 . 5. (Angles b et w een circles) The angle made by two oriente d cir- cles S and S ′ p assing thr ough a p oint P of an o ctahe dr al p oint-cir cle c onfigu r ation is that made by the t wo c orr esp onding t angent ve ctors V and V ′ at P , viz ∠ ( S, S ′ ) := ∠ ( V , V ′ ) . (3.2) W e are now in a po sition to define an a nalogue of Clifford’s classic a l configura - tion: Definition 3.6. (Generalized Cliffo rd configurations) An o ctahe dr al p oint- cir cle c onfigur ation is t erme d a gene r alized Clifford configuratio n if the six p oints P k ar e equiv alent in the sense that for any six p airs of c orr esp onding oriente d cir cles S k , S ′ k p assing thr ough P k the angle ∠ ( S k , S ′ k ) is indep endent of k . Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 7 3 1 1 3 4 4 34 3 4 2 1 14 4 2 3 12 2 3 4 1 2 1 1 2 24 2 3 4 23 13 Figure 5. A n admissible orien tation of circles and tangent vectors The following theor em demonstra tes that the ab ov e definition is natura l: Theorem 3.7. (Planar generalized Clifford configurations ) Gener alize d Clif- for d c onfigur ations on the plane c oincide with classic al Cliffor d c onfigur ations. Pr o of. The Go dt-Ziege nbein pr op erty (Go dt 1896 ; Ziegen b ein 19 41) consists of the equiv alence o f the p oints P , . . . , P 1234 of a cla ssical Clifford co nfiguration. Thus, in particular, the p oints P 12 , P 13 , P 14 , P 23 , P 24 , P 34 of any given Clifford configuratio n are equiv alent a nd he nc e the a bove definition of a generalized Clifford configuration is met. Conv ersely , let P 12 , P 13 , P 14 , P 23 , P 24 , P 34 be the six points of a plana r g ener- alized Clifford configur ation. If we set set aside the p oint P 34 , say , then it is evident that, due to the assumption of equiv alence, P 34 may b e r econstructed from the other five p o int s P 12 , P 13 , P 14 , P 23 , P 24 . Indeed, s ince the four circles passing thr ough P 12 are determined by these fiv e p oints, the a ng les made by all pair s of circles are known so that, in turn, the remaining four circles a re uniquely deter mined. On the other hand, the ab ov e five p oints als o b elong to a unique cla ssical C liffo r d c onfiguration as discusse d in the pr e v ious section. The la tter has the Go dt-Z ie g enbein pr o p e r ty and hence coincides with the given gener alized Clifford configuratio n. 4. Quaternions It is well kno wn that the group of co nformal transformations in R n , n > 2 consists o f translations, r otations, scalings a nd inv ersio ns (Dubrovin et al. 1 984). Accor ding ly , generalized Clifford configur ations ar e o b jects of conforma l geo metry since cir cles are mapp ed to circle s. In the current c ontext, it is therefor e natural to identify the four-dimensional Euclide a n spa ce R 4 with the alg ebra of q uaternions H (Ko echer 8 W. K. Schief and B. G. Konop elchenko & Remmert 199 1). Thus, we adopt the quaternio nic repre s ent ation R 4 ∋ ( a, b, c, d ) ↔ ( a 1 + b i + c j + d k ) ∈ H , (4.1) where the matrices 1 , i , j , k are defined by 1 =  1 0 0 1  , i =  0 − ı − ı 0  , j =  0 − 1 1 0  , k =  − ı 0 0 ı  . (4.2) Then, the following pro per ties and identit ies may be established. Firstly , it is r eadily verified that | X | = √ det X , X X † = det X 1 , < X , Y > = 1 2 tr ( X Y † ) . (4.3) In particula r, if, for no n- v anishing quaternino ns, we denote the cor r esp onding unit vectors b y ˆ X = X | X | (4.4) then cos ∠ ( X, Y ) = 1 2 tr ( ˆ X ˆ Y † ) = 1 2 tr ( ˆ X ˆ Y − 1 ) = 1 2 tr ( X Y † ) √ det X √ det Y . (4.5) Secondly , Cayley’s theo rem (Ko echer & Remmert 1991) states that any elemen t Ω of the or thogonal gro up O (4) is repres ent ed b y e ither X 7→ ˆ AX ˆ B or X 7→ ˆ AX † ˆ B , A, B ∈ H \{ 0 } , (4.6) depe nding on whether Ω is ‘prop er’ (det Ω = 1) or ‘improp er ’ (det Ω = − 1) r e- sp ectively . Conv ersely , any quaternionic a ction of the ab ov e type corres po nds to an orthogo nal mapping Ω. In particular, the o per ation X 7→ ˆ AX † ˆ A, A ∈ H (4.7) constitutes a reflection in the vector A . Finally , the gr oup of confor mal transforma - tions is gener ated by the or ien tation-pr eserving M¨ obius transforma tions (Ahlfors 1981) M : X 7→ ( AX + B )( C X + D ) − 1 , A, B , C , D ∈ H (4.8) and the par ticular reflectio n (conjugation) C : X 7→ X † . (4.9) 5. Quaternionic cross- and multi-ratios The r elev ance of multi-ratios in connection with classica l Cliffo r d co nfigurations has be en indica ted in § 2. It turns out that g eneralized Clifford configuratio ns may also b e describ ed alg ebraically in terms of quaternionic m ulti-ra tios. It is recalled that the cro ss-ratio of four p oints in R 4 is usually taken to be (Ahlfors 19 81) Q ( P 1 , P 2 , P 3 , P 4 ) = ( P 1 − P 2 )( P 2 − P 3 ) − 1 ( P 3 − P 4 )( P 4 − P 1 ) − 1 . (5.1) Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 9 The cro ss-ratio is ‘real’, tha t is Q ( P 1 , P 2 , P 3 , P 4 ) = a 1 , a ∈ R , (5.2) if and o nly if the four p oints P 1 , . . . , P 4 lie o n a circle. The multi-ratio of six po ints may b e defined as M ( P 1 , P 2 , P 3 , P 4 , P 5 , P 6 ) = ( P 1 − P 2 )( P 2 − P 3 ) − 1 ( P 3 − P 4 )( P 4 − P 5 ) − 1 ( P 5 − P 6 )( P 6 − P 1 ) − 1 (5.3) or, alterna tively , ˜ M ( P 1 , P 2 , P 3 , P 4 , P 5 , P 6 ) = ( P 1 − P 6 ) − 1 ( P 6 − P 5 )( P 5 − P 4 ) − 1 ( P 4 − P 3 )( P 3 − P 2 ) − 1 ( P 2 − P 1 ) . (5.4) In the scalar case, the ‘right-m ulti-ratio’ M and the ‘left-multi-ratio’ ˜ M are evi- dent ly iden tical. Howev er, in the quater nio nic case, the tw o mult i-ra tios a r e related by ˜ M ( P 1 , P 2 , P 3 , P 4 , P 5 , P 6 ) = ( P 1 − P 6 ) − 1 M ( P 6 , P 5 , P 4 , P 3 , P 2 , P 1 )( P 1 − P 6 ) . (5.5) This rela tion shows that the multi-ratio c onditions M ( P 1 , P 2 , P 3 , P 4 , P 5 , P 6 ) = − 1 (5.6) and ˜ M ( P 6 , P 5 , P 4 , P 3 , P 2 , P 1 ) = − 1 (5.7) on six p oints P 1 , . . . , P 6 coincide. F urthermore, it is readily seen that the M¨ obius transformatio ns M individua lly prese rve the multi-ratio conditions (5.6) a nd ˜ M ( P 1 , P 2 , P 3 , P 4 , P 5 , P 6 ) = − 1 , (5.8) while (5.6) and (5.8) ar e mapp ed to each other by the confor mal transfor mations C ◦ M whic h change the orientation. The g eometric s ignificance of this fact in the context of genera lized Clifford configura tio ns is discuss ed in the fo llowing section. 6. E xistence and algebraic description of generalized Clifford configurations It has b een shown in § 2 that planar generalized Clifford configur a tions a re uniquely determined b y five p oints. It turns out that the deriv ation of an analo gous sta temen t in the general case is the k ey to an alg ebraic description of genera lized Clifford configuratio ns. T o this end, it is co n venien t to make a canonical choice of the tang ent vectors to or iented circles. Thus, since four p oints X , A, B , C a r e concyclic if and only if their cross-r atio is r eal, the function X ( s ) defined by Q ( X , A, B , C ) = s 1 , s ∈ R (6.1) 10 W. K. Schief and B. G. Konop elchenko parametrizes the o riented cir cle S A,B ,C which passes through A, B , C as s increases with X (0 ) = A, X (1) = B , X ( ∞ ) = C . Differentiation and ev aluatio n a t s = 1 then r esults in the tangent v ector V A,B ,C = ( C − B )( C − A ) − 1 ( B − A ) = ( B − A )( C − A ) − 1 ( C − B ) (6.2) at the point X = B . The latter identit y is merely a prop erty of any three matrices A, B a nd C . Moreov er, if tw o oriented circles S 1 and S 2 meet a t the p oints P and P ′ with asso ciated tangent vectors V 1 and V 2 at P then the vectors V ′ 1 and V ′ 2 given by (cf. (4.7)) V ′ 1 = ( P ′ − P ) V − 1 1 ( P ′ − P ) , V ′ 2 = ( P ′ − P ) V − 1 2 ( P ′ − P ) (6.3) are tangent to the circles S 1 and S 2 at P ′ and the orientation of the tangent v ectors is preserved. In o rder to pro ceed, we int ro duce a natural lab elling of the p oints o f an o ctahe- dral p oint-circle configuratio n a nd the vertices of its underlying o ctahedro n. Th us, the s ix vertices o f the o ctahedro n ar e labe lled by ( i k ) = ( k i ), i 6 = k ∈ { 1 , 2 , 3 , 4 } in such a w ay that opp o site vertices carry complementary indices and the c o rresp ond- ing p oints of the configur ation are deno ted by Φ ik = Φ ki throughout the remainder of this pap er . Theorem 6. 1. (‘Uniqueness’) Ther e exist at most t wo gener alize d Cliffor d c on- figur ations which s har e fi ve p oints and the four asso ciate d cir cles. Pr o of. F or conv enience, we lab el the five common p o int s by Φ 12 , Φ 13 , Φ 14 , Φ 23 , Φ 24 and reg a rd the p oint(s) Φ 34 as unknown. Accordingly , only the four circles S µ 12 , µ = 1 , 2 , 3 , 4 passing thro ugh the p oint Φ 12 are known. The tangen t vectors to these circles a re linearly dep endent and may b e chosen to be V 1 12 = V 14 , 12 , 13 , V 2 12 = V 13 , 12 , 23 , V 3 12 = V 24 , 12 , 14 , V 4 12 = V 23 , 12 , 24 (6.4) as indica ted in figur e 5. Here, the notation V 14 , 12 , 13 = V Φ 14 , Φ 12 , Φ 13 etc. has b een adopted. In the gener ic case, any three of these four vectors span the three-dimen- sional tangent h yp erplane to the h yp e rsphere at Φ 12 defined by the five p oints Φ 12 , Φ 13 , Φ 14 , Φ 23 , Φ 24 . By virtue o f the corre s po ndence principle and the orienta- tion conv en tion, the orientation of the eig h t cir cles is now defined. The vectors V 1 13 = (Φ 13 − Φ 12 )( V 2 12 ) − 1 (Φ 13 − Φ 12 ) V 2 13 = (Φ 13 − Φ 12 )( V 1 12 ) − 1 (Φ 13 − Φ 12 ) (6.5) are tangent to the c ir cles S 1 13 = S 2 12 and S 2 13 = S 1 12 resp ectively at the p oint Φ 13 , while V 1 14 = (Φ 14 − Φ 12 )( V 3 12 ) − 1 (Φ 14 − Φ 12 ) (6.6) constitutes a ta ngent v ector to the cir cle S 1 14 = S 3 12 at the p oint Φ 14 . Now, in order to determine the tw o r emaining tangen t v ectors at the p oint Φ 13 , w e mak e use of the assumption that, in particular , the p oints Φ 12 , Φ 13 and Φ 14 are equiv a le n t. Thu s, the circle S 3 13 which passes thro ug h the p oints Φ 13 and Φ 14 gives r ise to the relations ∠ ( S 3 13 , S 1 13 ) = ∠ ( S 3 12 , S 1 12 ) , ∠ ( S 3 13 , S 2 13 ) = ∠ ( S 3 12 , S 2 12 ) , (6.7) Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 11 while the circle S 2 14 = S 3 13 is asso ciated with the additional r elation ∠ ( S 2 14 , S 1 14 ) = ∠ ( S 2 12 , S 1 12 ) , (6.8) where S 1 14 = S 3 12 . If the tangent vector to S 3 13 at the p oint Φ 13 is denoted by V 3 13 then the vector V 2 14 = (Φ 14 − Φ 13 )( V 3 13 ) − 1 (Φ 14 − Φ 13 ) (6.9) is tange nt to the circle S 2 14 at the p oint Φ 14 . According ly , the co nditions (6.7 ) and (6.8) tra nslate into tr [ ˆ V 3 13 ( ˆ V 1 13 ) † ] = tr [ ˆ V 3 12 ( ˆ V 1 12 ) † ] tr [ ˆ V 3 13 ( ˆ V 2 13 ) † ] = tr [ ˆ V 3 12 ( ˆ V 2 12 ) † ] tr [ ˆ V 2 14 ( ˆ V 1 14 ) † ] = tr [ ˆ V 2 12 ( ˆ V 1 12 ) † ] (6.10) which may b e written as tr [ ˆ ∆( ˆ V 2 12 ) − 1 ] = tr [ ˆ V 1 12 ( ˆ V 3 12 ) − 1 ] tr [ ˆ ∆( ˆ V 1 12 ) − 1 ] = tr [ ˆ V 2 12 ( ˆ V 3 12 ) − 1 ] tr [ ˆ ∆( ˆ V 1 12 ) − 1 ˆ V 3 12 ( ˆ V 1 12 ) − 1 ] = tr [ ˆ V 2 12 ( ˆ V 1 12 ) − 1 ] (6.11) with the definition ∆ = (Φ 13 − Φ 12 )( V 3 13 ) − 1 (Φ 13 − Φ 12 ) . (6.12) The cons traints (6.11) constitute a linear s ystem for the unit vector ˆ ∆. Hence , there are tw o ca ses to distinguish: Case 1: If the tangen t vectors V µ 12 span a tw o-dimensional plane then the five p oints Φ 12 , Φ 13 , Φ 14 , Φ 23 , Φ 24 necessarily lie o n a 2-spher e (or a plane). On use o f a con- formal transfo rmation, this 2-spher e ma y b e mapp ed to a pla ne so that w e ar e left with the co ns ideration o f g eneralized Clifford co nfigurations in a three-dimensio nal Euclidean spac e sub ject to the five p oints Φ 12 , Φ 13 , Φ 14 , Φ 23 , Φ 24 being co-plana r. If there exists a g eneralized Clifford configura tion for which Φ 34 do es not lie o n the corr esp onding pla ne Σ then reflec tio n of Φ 34 in Σ pro duces ano ther general- ized Clifford co nfiguration. F urthermor e, the classical Cliffor d configuration defined uniquely by the fiv e p oints constitutes a third (planar) g e neralized Clifford configu- ration. Thus, the num ber of distinct genera liz e d Cliffo r d configur ations sharing five co-planar p oints is o dd. On the other hand, it is readily seen that the rank of the linear system (6.11) is 2 since the tange nt vectors V 1 12 , V 2 12 , V 3 12 are linear ly depe nden t a nd hence there exist at most tw o solutions ˆ ∆. Any specific choice of ˆ ∆ de ter mines the tange nt vector V 3 13 up to its magnitude. Moreover, since the angles b etw ee n the vectors V 1 13 , V 2 13 , V 3 13 and the vector V 4 13 which is tangent to the circle S 4 13 passing through the po int s Φ 23 and Φ 13 are known and the four vectors V µ 13 m ust b e linearly dependent, the direction of the ta ng ent vector V 4 13 is uniquely determined. This , in turn, shows that the p o int Φ 34 is unique. Thus, there e xist at most t wo generalized Clifford configurations. W e therefore conclude tha t the ab ove-men tioned class ical Clifford configur ation is the only g e neralized Clifford configuratio n under the cur r ent assumption. 12 W. K. Schief and B. G. Konop elchenko Case 2: If the tangent v ectors V µ 12 span a three-dimensional vector spa ce then the tang ent vectors V 1 12 , V 2 12 , V 3 12 are linear ly independent without lo ss of gene r ality . Accordingly , the rank of the linear s ystem (6.11) is 3 and the cor resp onding tw o solutions are g iven by ˆ ∆ 1 = ˆ V 1 12 ( ˆ V 3 12 ) − 1 ˆ V 2 12 and ˆ ∆ 2 = ˆ V 2 12 ( ˆ V 3 12 ) − 1 ˆ V 1 12 . (6.13) It is noted that the a bove solutions ar e distinct since equality implies that [ ˆ V 1 12 ( ˆ V 3 12 ) − 1 , ˆ V 2 12 ( ˆ V 3 12 ) − 1 ] = 0 and hence V 1 12 , V 2 12 , V 3 12 are linear ly dependent. In fac t, since the pro jections of ˆ ∆ 1 and ˆ ∆ 2 onto V 1 12 , V 2 12 and V 3 12 coincide, the unit v ector s ˆ ∆ 1 and ˆ ∆ 2 are rela ted b y reflection in the three-dimensio nal v ector s pace spanned by V 1 12 , V 2 12 , V 3 12 . As in the prev ious c a se, any s p ecific ch oice of ˆ ∆ now determines the p oint Φ 34 uniquely so that there exis t at most tw o generalize d Clifford config- urations. Remark a bly , the ab ov e a nalysis implies that genera lized Clifford configur ations in thr e e-dimensional Euclidean s pa ces or spheres are essentially tw o-dimensiona l. Theorem 6.2. Any gener alize d Cliffor d c onfigur ation in a thr e e-dimensional Eu- clide an sp ac e R 3 or a thr e e-dimensional spher e is either planar or c onfine d to a two-dimensional spher e and may ther efor e b e mapp e d to a classic al Cliffor d c onfig- ur ation by me ans of a s u itable c onformal t r ansformation. Pr o of. Since a ny genera lized Clifford configura tion in a three-dimensional hyper - sphere may b e mapped via a confor mal transformatio n to a g eneralized Clifford configuratio n in a three-dimensio nal subspa ce of R 4 , we may co nfine our selves to the c ase o f a gener alized Clifford configura tion in R 3 which we deno te by E and regar d as b eing embedded in R 4 . As in the pr eceding pr o of, we consider the five po int s Φ 12 , Φ 13 , Φ 14 , Φ 23 , Φ 24 and set aside the po in t Φ 34 . If the tangent vectors V µ 12 span a t wo-dimensional pla ne then the generalized Cliffor d config uration is pla nar or confined to a 2 -sphere as shown ab ov e. Hence, we may assume without loss o f generality that E = s pan { V 1 12 , V 2 12 , V 3 12 } . This implies, in turn, that the tw o solu- tions ˆ ∆ 1 and ˆ ∆ 2 of the linear system (6.11) are distinct and are r e lated by reflection in E . Hence, ˆ ∆ 1 , ˆ ∆ 2 6∈ E . The key obser v ation is now the following: Theorem 6.3. (Multi-ratio de scription of generali zed Clifford configura- tions) The p oints Φ ik of a gener alize d Cliffor d c onfigu r ation ar e r elate d by either M (Φ 14 , Φ 12 , Φ 24 , Φ 23 , Φ 34 , Φ 13 ) = − 1 (6.14) or ˜ M (Φ 14 , Φ 12 , Φ 24 , Φ 23 , Φ 34 , Φ 13 ) = − 1 . (6.15) Conversely, any s ix p oints Φ ik of an o ctahe dr al p oint-cir cle c onfigura tion which ob ey either of t he mu lti-r atio c onditions ( 6.14 ) or (6.15) c onstitu te t he p oints of a gener alize d Cliffor d c onfigur ation. Before we prov e the theorem, it is observed that the ab ov e mult i-ra tio conditions are represe ntatives of tw o equiv alence clas s es o f multi-ratio co nditions which may Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 13 5 α 1 α 6 α 4 α 3 α 2 α 6 P α 5 α α 2 P P 1 P α 4 P α 3 P α Figure 6. An orien ted hexagon ( α 1 , α 2 , α 3 , α 4 , α 5 , α 6 ) asso ciated with the multi-ratio condition M ( P α 1 , P α 2 , P α 3 , P α 4 , P α 5 , P α 6 ) = − 1 be imp os ed o n a n o c tahedral p oint-circle configuratio n. Sp ecifically , let us consider an or ient ed hexag on ( α 1 , α 2 , α 3 , α 4 , α 5 , α 6 ) with fixed ‘initial’ vertex α 1 formed by six edge s o f a n o ctahe dr on with distinct vertices α i in s uch a way tha t any tw o adjacent edg es of the hexago n b elo ng to a triang ular face of the o ctahedr on (cf. figure 6) and imp os e the multi-ratio condition M ( P α 1 , P α 2 , P α 3 , P α 4 , P α 5 , P α 6 ) = − 1 (6.16) on the points P α i of an a sso ciated o ctahedral point-circle configuration. There exist 48 such hexag ons and in the planar case the asso ciated multi-ratio conditions a re equiv alent. How ever, the situation is different in the (generic) quaternionic case. Lemma 6.4. The mu lti-r atio c onditions (6.14) and (6.15 ) ar e either invariant or mapp e d to e ach other by t he asso ciate d o ctahe dr al symmetry gr oup. In p articular, the su b gr oup of symmetries which le ave the multi-r atio c onditions invariant c onsists of the p ermutations of the indic es 1 , 2 , 3 , 4 . Pr o of. On the one hand, a typical p ermutation which leads to a non-trivia l ordering of the a rguments in the multi-ratio conditions is g iven b y (1 , 2 , 3 , 4) → (4 , 2 , 3 , 1). The ass o ciated multi-ratio condition M (Φ 14 , Φ 24 , Φ 12 , Φ 23 , Φ 13 , Φ 34 ) = − 1 (6.17) may b e brought into the for m (Φ 14 − Φ 24 )(Φ 24 − Φ 12 ) − 1 (Φ 12 − Φ 23 ) + (Φ 34 − Φ 14 )(Φ 13 − Φ 34 ) − 1 (Φ 23 − Φ 13 ) = 0 . (6.18) The identities (Φ 14 − Φ 24 )(Φ 24 − Φ 12 ) − 1 (Φ 12 − Φ 23 ) = (Φ 14 − Φ 12 )(Φ 24 − Φ 12 ) − 1 (Φ 24 − Φ 23 ) + Φ 23 − Φ 14 (Φ 34 − Φ 14 )(Φ 13 − Φ 34 ) − 1 (Φ 23 − Φ 13 ) = (Φ 13 − Φ 14 )(Φ 13 − Φ 34 ) − 1 (Φ 23 − Φ 34 ) + Φ 14 − Φ 23 (6.19) 14 W. K. Schief and B. G. Konop elchenko then lea d to (Φ 14 − Φ 12 )(Φ 24 − Φ 12 ) − 1 (Φ 24 − Φ 23 ) + (Φ 13 − Φ 14 )(Φ 13 − Φ 34 ) − 1 (Φ 23 − Φ 34 ) = 0 (6.20) which, in tur n, is the original multi-ratio co ndition (6.14). On the other hand, comp o sition o f the s ubgroup o f p ermutations with any ‘ro- tation by 90 degr ees’ of the o ctahedro n gene r ates the remaining 24 discr ete sym- metries. F or instance, the rotatio n (14 , 12 , 24 , 23 , 34 , 13) → (24 , 12 , 23 , 13 , 34 , 14) pro duces the multi-ratio conditio n M (Φ 24 , Φ 12 , Φ 23 , Φ 13 , Φ 34 , Φ 14 ) = − 1 (6.21) which is, by definition, equiv alent to ˜ M (Φ 14 , Φ 24 , Φ 12 , Φ 23 , Φ 13 , Φ 34 ) = − 1 . (6.22) The latter is mer ely the tilde version of (6.1 7) a nd hence equiv alent to the multi- ratio condition (6.1 5). W e are now in a pos ition to prove theorem 6.3. Pr o of. (of theo rem 6.3) By v ir tue of theo rem 6.1 and the fact that either of the m ulti-ra tio conditions (6.14) or (6.15) may b e r e g arded as a definition of a po in t in terms of five arbitr a ry p oints, it remains to show that the multi-ratio conditions indeed give rise to gener alized Clifford c onfigurations . Thus, we here as sume that the six points Φ ik of an octa hedral point-circle configuration are constrained by one of the multi-ratio conditions and choo se the tang ent vectors V µ 12 as in the pro of of lemma 6.1. Acco rdingly , a set of corre spo nding tangent vectors V µ 13 of the corr e ct orientation at the p oint Φ 13 is given by (cf. figure 5) V 1 13 = V 23 , 13 , 12 , V 2 13 = V 12 , 13 , 14 , V 3 13 = V 14 , 13 , 34 , V 4 13 = V 34 , 13 , 23 . (6.23) The e x pressions (6.12) and (6.13) for the quantities ∆ and ˆ ∆ 1 , ˆ ∆ 2 suggest that one should consider the orientation- and a ngle-preser ving mappings O 1 : X 7→ (Φ 13 − Φ 12 )( V 2 12 ) − 1 X ( V 1 12 ) − 1 (Φ 13 − Φ 12 ) O 2 : X 7→ (Φ 13 − Φ 12 )( V 1 12 ) − 1 X ( V 2 12 ) − 1 (Φ 13 − Φ 12 ) . (6.24) Indeed, it is rea dily verified that V ν 13 = O i ( V ν 12 ) , ν = 1 , 2 (6.25) (cf. (6.5 )) and a short calculation reveals that O 1 ( V 3 12 )( V 3 13 ) − 1 = − M (Φ 13 , Φ 23 , Φ 12 , Φ 24 , Φ 14 , Φ 34 ) ( V 4 13 ) − 1 O 1 ( V 4 12 ) = − ˜ M (Φ 13 , Φ 14 , Φ 12 , Φ 24 , Φ 23 , Φ 34 ) ( V 3 13 ) − 1 O 2 ( V 3 12 ) = − ˜ M (Φ 13 , Φ 23 , Φ 12 , Φ 24 , Φ 14 , Φ 34 ) O 2 ( V 4 12 )( V 4 13 ) − 1 = − M (Φ 13 , Φ 14 , Φ 12 , Φ 24 , Φ 23 , Φ 34 ) . (6.26) Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 15 Two conclusions ma y now b e drawn from lemma 6.4. Firstly , the conditions (6.14) and (6.15) are equiv alent to M (Φ 13 , Φ 23 , Φ 12 , Φ 24 , Φ 14 , Φ 34 ) = − 1 ⇔ ˜ M (Φ 13 , Φ 14 , Φ 12 , Φ 24 , Φ 23 , Φ 34 ) = − 1 (6.27) and ˜ M (Φ 13 , Φ 23 , Φ 12 , Φ 24 , Φ 14 , Φ 34 ) = − 1 ⇔ M (Φ 13 , Φ 14 , Φ 12 , Φ 24 , Φ 23 , Φ 34 ) = − 1 (6.28) resp ectively a nd henc e it follows that V µ 13 = O 1 ( V µ 12 ) or V µ 13 = O 2 ( V µ 12 ) , µ = 1 , 2 , 3 , 4 (6.29) depe nding o n whether (6.14) or (6.15) is ass umed to hold. Thus, the p oints Φ 12 and Φ 13 are equiv alent in the sense of definition 3.6 since ∠ ( V µ 12 , V µ ′ 12 ) = ∠ ( V µ 13 , V µ ′ 13 ) , µ, µ ′ ∈ { 1 , 2 , 3 , 4 } . (6.30) Secondly , since the p oints Φ 12 and Φ 13 are equiv alent, the p oints Φ π (1) π ( 2) and Φ π (1) π ( 3) are equiv alent for any p ermutation π of the indices 1 , 2 , 3 , 4. Hence, all po int s are eq uiv alent. This completes the pro of. W e conclude this section with the remark that if a p oint of a generalized Clifford configuratio n which is not conformally equiv alent to a clas s ical Clifford config ur a- tion is inv erted with resp ect to the h yp ersphere which passe s thr ough the other five p oints then another generalized Cliffor d configuration is obtained. Thus, theo- rem 6.3 implies the following co r ollary: Corollary 6.5. If two gener alize d Cliffor d c onfigur ations define d by M (Φ 14 , Φ 12 , Φ 24 , Φ 23 , Φ 34 , Φ 13 ) = − 1 ˜ M (Φ 14 , Φ 12 , Φ 24 , Φ 23 , ˜ Φ 34 , Φ 13 ) = − 1 (6.31) ar e not c onformal ly e quivalent to a classic al Cliffor d c onfigur ation then the p oints Φ 34 and ˜ Φ 34 ar e r elate d by inversion with r esp e ct to the (wel l-define d) hyp erspher e p assing thr ough the c ommon p oints Φ 13 , Φ 14 , Φ 12 , Φ 24 , Φ 23 . Int eres ting ly , if one chooses the p oints Φ 13 , Φ 14 , Φ 12 , Φ 24 , Φ 23 in such a way that ˜ Φ 34 lies ‘at infinit y’ then the above corollar y implies tha t the p oint Φ 34 constitutes the ce ntre of the hyper sphere whic h pa sses thr ough these five points. 7. The quaternionic discrete Sc hw arzian KP equation In Ko nop elchenk o & Schief (2002 ), it has b een demonstra ted that the multi-ratio condition (2.1) in terpreted as a lattice equation is nothing but a Schw a rzian version of the disc r ete Ka do mt sev-Petviashvili (dKP) equation which constitutes a ‘master equation’ in the theory o f integrable systems (Ablowitz & Seg ur 1981 ; Z akharov et al. 19 8 0). Accor dingly , the dSK P equatio n admits a geometric in terpre tation in 16 W. K. Schief and B. G. Konop elchenko 2 1 2 3 _ _ _ 1 3 Figure 7. Tw o octahedra embedded in the fcc lattice F terms o f classica l Cliffor d config urations. Here, we present a n ana logous construc- tion of three- dimensional ‘Clifford la ttices’ in a four- dimensional Euclidea n space asso ciated with the quaternionic multi-ratio condition. Due to the absence of any additional p oints of in tersection of the circles b elo ng ing to a gener ic gener alized Clifford co nfig uration, not only is the quater nio nic case more natural but it a ls o includes the a fo re-mentioned s calar cas e. W e consider lattices o f the combinatorics of a face-centred cubic (fcc) la ttice in a four - dimensional E uclidean spa ce, that is ma ps of the form Φ : F → H , F = { ( n 1 , n 2 , n 3 ) ∈ Z 3 : n 1 + n 2 + n 3 o dd } . (7.1) An y six p oints of F which constitute the centres o f the faces of a c ub e comp osed of eight adjacent elementary cub es o f Z 3 may b e regarded as the v ertices o f an o ctahedron (cf. figure 7). In this wa y , the fcc lattice may b e iden tified with the vertices of a collection of oc tahedra which meet at common edg e s . Accor dingly , Φ( F ) constitutes a set of p oints belo nging to o ctahedr al p oint-circle configura tions. It is therefore natural to require that these p oint-circle configurations b e of gener alized Clifford type . In terms of the la b elling (7.1 ), this implies that for a n y o c ta hedron centred at ( ν 1 , ν 2 , ν 3 ) ∈ Z 3 with ν 1 + ν 2 + ν 3 even, one o f the multi-ratio conditio ns M (Φ ¯ 1 , Φ 2 , Φ ¯ 3 , Φ 1 , Φ ¯ 2 , Φ 3 ) = − 1 (7.2) and ˜ M (Φ ¯ 1 , Φ 2 , Φ ¯ 3 , Φ 1 , Φ ¯ 2 , Φ 3 ) = − 1 (7.3) obtains, wher e the arguments of Φ have b een suppressed and the notation Φ = Φ( ν 1 , ν 2 , ν 3 ) , Φ ¯ 1 = Φ( ν 1 − 1 , ν 2 , ν 3 ) , Φ 2 = Φ( ν 1 , ν 2 + 1 , ν 3 ) , . . . (7.4) has b een a dopted. Two w ell-defined maps a re obtained by demanding that the ‘same’ m ulti-ratio condition is imp ose d on all o ctahedra. Th us, we say that the map Φ defines a Cliffor d lattic e in H if either (7.2) or (7.3) regarde d as a lattice equation ho lds. It is noted that these tw o equations a re essentially identical in the Gener alize d Cliffor d c onfigur ations and the qdSKP e quation 17 sense that if Φ( n 1 , n 2 , n 3 ) is a solution of (7.2) then Φ( − n 1 , − n 2 , − n 3 ) is a solution of (7.3) and vice versa. If the fcc lattice F is mapped to a simple cubic lattice Z 3 via the rela belling F ∋ ( n 1 , n 2 , n 3 ) ↔ ( m 1 , m 2 , m 3 ) ∈ Z 3 (7.5) defined by n 1 = m 2 + m 3 − 1 , n 2 = m 1 + m 3 − 1 , n 3 = m 1 + m 2 − 1 (7.6) then the la ttice equation (7.2 ) assumes the standard for m of the quaternionic re- duction M (Φ ˜ 1 , Φ ˜ 1 ˜ 3 , Φ ˜ 3 , Φ ˜ 2 ˜ 3 , Φ ˜ 2 , Φ ˜ 1 ˜ 2 ) = − 1 (7.7) of the integrable m ulti-comp onent dSKP equation (Bogdanov & Konop elchenk o 1998), wherein the subscripts o n Φ now refer to unit increments of the v ariables m 1 , m 2 , m 3 . The co nformal geo metry of the qdSKP equation (7.7 ) and the a sso ciated (dis- crete) Sch warzian Dav ey-Stewartson I I hier arch y has b een dis cussed in detail in Konop elchenk o & Schief (200 5). There in, it has b een shown that any standard dis- crete iso thermic sur face (Bob enko & Pink all 19 96) may b e extended via a transla- tional symmetry to a three-dimensio nal lattice in suc h a manner that a (degenerate) Clifford lattice is obtained. In particular , Clifford la ttices encapsula te discrete sur- faces of constant mean curv ature and discre te minimal surfac e s . Thus, apart from its s ignificance in co nnection with (genera lized) Clifford configura tions, the qdSKP equation a lso plays an impo rtant role in the area of integrable discrete differential geometry (Bo b enko & Seiler 1 999). 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