On generalized Clifford configurations: geometry and integrability - PowerPoint PPT Presentation

On generalized Clifford configurations: geometry and integrability by W.K. Schief Technische Universit¨ at Berlin ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Australia (with B.G. Konopelchenko, Proc. R. Soc London A)

1. The origin of integrability: Incidence theorems Miquel’s theorem Pascal’s theorem ↓ ↓ discrete Lam´ e system discrete CKP equation (discrete orthogonal coordinate systems!)

2. Integrable ‘Clifford lattices’ (Konopelchenko & WKS 2002) A C 4 Clifford configuration Theorem. The six (black) points P i (regarded as ordered complex numbers) obey the multiratio condition M := ( P 1 − P 2 )( P 3 − P 4 )( P 5 − P 6 ) ( P 2 − P 3 )( P 4 − P 5 )( P 6 − P 1 ) = − 1 Idea: Extend the Clifford configuration to a ‘Clif- ford lattice’ of fcc type. Result: The above lattice equation constitutes an integrable discretisation of the Schwarzian KP equation!

3. The discrete Schwarzian KP equation The discrete Schwarzian KP (dSKP) equation M (Φ 1 , Φ 13 , Φ 3 , Φ 23 , Φ 2 , Φ 12 ) = − 1 is equivalent to the Hirota-Miwa (dKP) equation τ 1 τ 23 + τ 2 τ 13 + τ 3 τ 12 = 0 . Interpretations: • Lattice equation with Φ 1 = Φ( n 1 + 1 , n 2 , n 3 ) , etc • Superposition formula for B¨ acklund transforms of a solution of the SKP equation (KP squared eigenfunctions). • Algebraic relation between solutions of the SKP hierarchy generated by Miwa shifts, that is Φ i = T i Φ , where Φ i ( t ) = T i Φ( t ) , t = ( t 1 , t 2 , t 3 , . . . ) and t 1 + a i , t 2 + a 2 2 , t 3 + a 3 � � i i T i Φ( t ) = Φ( t + [ a i ]) = Φ 3 , . . .

4. History Longuet-Higgins (1972) in Clifford’s chain and its analogues in relation to the higher polytopes : ”... a chain of theorems ... has exerted a peculiar fascination for mathematicians since its discovery by Clifford in 1871.” Generalizations and analogues in higher dimension: de Longechamps (1877) Cox (1891) Grace (1898) (16p 10s) Brown (1954) Coxeter (1956) Longuet-Higgins (1972) (1728p 240s)

5. Clifford’s point-circle configurations (1871) Clifford’s chain of circle theorems: • Given four straight lines on a plane, the four circumcircles of the four triangles so formed are concurrent in a point Q 4 , say. [Wallace 1806] Q 4 l 3 l 4 l 1 l 2

..... • Given five lines on a plane, by omitting each line in turn, we obtain five corresponding points Q 4 and these lie on a circle C 5 , say. [Miquel] • Given six lines on a plane, we obtain six corresponding circles C 5 and these are concurrent in a point Q 6 . Etc. • Generally, given n coplanar lines, we obtain n corresponding circles C n − 1 which are concurrent in a point Q n or n points Q n − 1 which lie on a circle C n depending on whether n is even or odd respectively. • Finally, application of an inversion with respect to a generic circle on the plane leads to a complete and symmetric configuration of 2 n − 1 points and 2 n − 1 circles with n points on every circle and n circles through every point.

6. Theorem of Menelaus (100 AD; Euclid ?) Theorem of Menelaus. Three points P 14 , P 24 , P 34 on the (extended) edges of a triangle with vertices P 12 , P 23 , P 13 are collinear if and only if P 12 P 24 P 23 P 34 P 13 P 14 = − 1 . P 24 P 23 P 34 P 13 P 14 P 12 P 13 Q 4 P 14 l 3 l 4 l 1 P 34 P P 12 24 l 2 P 23

Implications • The points of intersection of the four lines l 1 , l 2 , l 3 , l 4 in Clifford’s first theorem obey the Menelaus condition. • The Menelaus condition constitutes the multi-ratio condition M ( P 14 , P 12 , P 24 , P 23 , P 34 , P 13 ) = − 1 for the complex numbers P ik with the multi-ratio 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 ) . • The multi-ratio is invariant under the group of inversive transformations. • The points P ik of a C 4 Clifford configuration likewise satisfy the above multi-ratio condition!

7. The C 4 Clifford configuration Notation: 1 2 4 0 12 14 3 13 124 23 24 1234 123 134 34 234 Theorem (Konopelchenko & WKS 2002). Any six generic points P 12 , P 13 , P 14 , P 23 , P 24 , P 34 on the plane belong to a Clifford configuration if and only if M ( P 14 , P 12 , P 24 , P 23 , P 34 , P 13 ) = − 1 .

Remarks • Ordering of the arguments? Later ... • Any five generic points on the plane uniquely define a Clifford configuration. This is both geometrically and algebraically evident! • Interpretation: A Clifford configuration is a configuration of six points P 12 , P 13 , P 14 , P 23 , P 24 , P 34 and eight circles with the four circles S 1 , S 2 , S 3 , S 4 inter- secting at a point P or the four circles S 123 , S 124 , S 134 , S 234 intersecting at a point P 1234 . Clifford’s circle theorem then guarantees the existence of the remaining point P 1234 or P respectively. It is this point of view which allows for a generalization of Clifford configurations in which, generically, the points P and P 1234 do not exist.

8. The Godt-Ziegenbein property (Godt 1896; Ziegenbein 1941) Theorem. The angles made by four oriented circles passing through a point of a Clif- ford configuration are ‘the same’ for all eight points. 4 3 2 12 1 1 13 2 3 4 Observation: It is sufficient to demand that the Godt-Ziegenbein property holds for the six points P 12 , P 13 , P 14 , P 23 , P 24 , P 34 . This is a defining property! This observation serves as the basis for the definition of generalized Clifford configura- tions.

9. Octahedral point-circle configurations � 4 consisting of six points and eight circles We are concerned with configurations in with three points on every circle and four circles through every point. � 4 is termed an octahe- Definition: A configuration of six points and eight circles in dral point-circle configuration if the combinatorics of the configuration is that of an octahedron, that is the points of the configuration correspond to the vertices of the octahedron while the circles correspond to the triangular faces. 12 12 13 13 23 23 14 24 14 24 34 34

10. Opposite points and circles Definition: P is ‘opposite’ to P ∗ and S is ‘opposite’ to S ∗ in the following sense: P S * S P * Correspondence of circles: ( S ; P ) ↔ ( S ∗ ; P ∗ ) ( S 1 , S 2 ; P 1 , P 2 ) ↔ ( S 2 , S 1 ; P 2 , P 1 ) ,

11. Orientation of and angles between circles Iterative application of the above correspondence principle: Observation: Any circle S 1 of an octahedral point-circle configuration passing through a point P 1 admits five corresponding circles S 2 , . . . , S 6 which pass through the re- maining five points P 2 , . . . , P 6 . Convention: The orientation of the circles of an octahedral point-circle configuration is chosen in such a manner that the corresponding orientation of the faces of the octahedron is the same for all faces. Definition: � ( S, S ′ ) := � ( V, V ′ )

12. Generalized Clifford configurations Definition: An octahedral point-circle configuration is termed a generalized Clifford configuration if the six points P k are equivalent in the sense that for any six pairs of corresponding oriented circles S k , S ′ k passing through P k the angle � ( S k , S ′ k ) is independent of k . 4 3 2 12 4 3 1 14 2 1 2 13 1 3 1 4 4 3 24 23 2 3 2 2 4 4 1 3 34 1

..... The above definition is natural: Theorem. Generalized Clifford configurations on the plane coincide with classical Clif- ford configurations.

13. Cayley’s theorem Identification: � 4 ∼ = algebra of quaternions � via � 4 ∋ ( a, b, c, d ) ↔ ( a ✁ + b ✂ + c ✄ + d ☎ ) ∈ � with � � � � � � � � 1 0 0 − i 0 − 1 − i 0 ✁ = , ✂ = , ✄ = , ☎ = 0 1 − i 0 1 0 0 i Cayley’s theorem. Any element Ω of the orthogonal group O (4) is represented by either AX † ˆ X �→ ˆ AX ˆ X �→ ˆ ( ˆ or A = A/ | A | ) B B, depending on whether Ω is ‘proper’ ( det Ω = 1 ) or ‘improper’ ( det Ω = − 1 ) respectively. Conversely, any quaternionic action of the above type corresponds to an orthogonal mapping Ω .

14. Conformal transformations � 4 is generated by: Group of conformal transformations in • M¨ obius transformations M : X �→ ( AX + B )( CX + D ) − 1 , A, B, C, D ∈ � • Conjugation (particular reflection) C : X �→ X †

15. Quaternionic cross- and multi-ratios Cross-ratio of four points: 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 Right multi-ratio of six points may be 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 Left multi-ratio: ˜ 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 ) . Relation: 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 ) ˜