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Cross Ratio - an accidental discovery

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If we know nothing about this world, we should know about cross ratio. It is one of those accidents of nature that is so unbelievable, unimaginable, that we need mathematics to accept it.

Imagine that physicist telling you: if there was no air, a feather and an iron ball would hit the ground at the same time (falling from a height).

Until and unless someone takes you to Michigan’s NASA lab, and actually creates a near-vacuum chamber and actually drops a feather and an iron ball, and they actually hit the ground at the same time, you won’t be able to accept it. Seeing is believing.

This is also true about cross ratio. Geometers may think about it as the devil’s concoction. Algebrists may think about it as their victory. But no one really knows.

Choose a point of observation O and a line L outside the point. Suppose A, B, C, D are four points in the line L. We think about OA, OB, OC and OD as lines of sight from the observer point O to the observed line L. Pappus of Alexandria, about 300 years after Christ, observed something peculiar:

(AC/AD) ÷ (BC/BD) remains invariant no matter what.

This means, if another line L’ cuts the lines of sight OA, OB, OC, OD at A’, B’, C’, D’ respectively then we will have

(AC/AD) ÷ (BC/BD) = (A’C’/A’D’) ÷ (B’C’/B’D’).

(Actually one may argue that Desargues looked into this explicitly; however Pappus certainly played with these ideas first).

This is remarkable for a variety of reasons. You do not need the concept of an angle here. No matter what the inclination of L’ is with respect to L, this ratio of ratios will be preserved. Moreover it is also tied with how our brain sees things. If A, B, C, D are equally spaced on L, it will appear to the observer that A’, B’, C’, D’ are equally spaced on L’. In a way, the invariance of this cross ratio holds the secret of how are brain works!

Indeed architects and painters during the renaissance used cross ratio and allied concept of projectivity to transform their craft. Painters used it draw realistic three dimsensional painting with lines of sight meeting at infinity.

One can compare contemporary Mughal art and Italian art to observe the difference. To illustrate this I have attached two sample pieces; one of which has lines of sight meeting at an observer point. Can you tell which one it is?

A simple addition of observer point, with lines of sight meeting at it, makes all the difference. One of them appears to be flat in our brain. The other one appears to be three dimensional.

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