1. Activity 2-20: The Cross-ratio www.carom-maths.co.uk

2. What happens in the above diagram if we calculate ?

3. Say A = (p, ap), B = (q, bq), C = (r, cr), D = (s, ds). So ap = mp + k, bq = mq + k, cr = mr + k, ds = ms +k..

4. Strange fact: this answer does not depend on m or k. So whatever line y = mx + k falls across the four others, the cross-ratio of lengths will be unchanged.. This is the cross-ratio of a, b, c and d.

5. This makes the cross-ratio an invariant, and of great interest in a field of maths known as projective geometry. Projective geometry might be described as ‘the geometry of perspective’. You could argue it is a more fundamental form of geometry than the Euclidean geometry we generally use. The cross-ratio has an ancient history; it was known to Euclid and also to Pappus, who mentioned its invariant properties.

6. Theorem: the cross-ratio of four complex numbers is real if and only if the four numbers lie on a straight line or a circle. Given four complex numbers z 1, z 2, z 3, z 4, we can define their cross-ratio as Task: certainly 1, i, -1 and –i lie on a circle. Show the cross-ratio of these numbers is real..

7. Proof: we can see that (z 3 -z 1 )e iα = λ(z 2 -z 1 ), and (z 2 -z 4 )e iβ = µ(z 3 -z 4 ). Multiplying these together gives (z 3 -z 1 ) (z 2 -z 4 )e i(α+β) = λµ(z 3 -z 4 )(z 2 -z 1 ), or

8. But α + β = 0 implies that α = β = 0, and z 1, z 2, z 3 and z 4 lie on a straight line, while α + β = π implies that α and β are opposite angles in a cyclic quadrilateral, which means that z 1, z 2, z 3 and z 4 lie on a circle. So the cross-ratio is real if and only if e i(α+β) is, which happens if and only if α + β = 0 or α + β = π. We are done!

9. With thanks to: Paul Gailiunas Carom is written by Jonny Griffiths, mail@jonny.griffiths.net