Cycles Cross Ratio: an Invitation

The paper introduces cycles cross ratio, which extends the classic cross ratio of four points to various settings: conformal geometry, Lie spheres geometry, etc. Just like its classic counterpart cycles cross ratio is a measure of anharmonicity between spheres with respect to inversion. It also provides a M"obius invariant distance between spheres. Many further properties of cycles cross ratio awaiting their exploration. In abstract framework the new invariant can be considered in any projective space with a bilinear pairing.