At first glance, it seems some mathematical constructions and proofs are invented by coincidence or ingenuity that is not easily attainable by most people. However, this may be a camouflage produced by the ultimate beautiful proof appearing in a tome. The real story behind the scene could be the followings.

  1. Although there is a guidance from an initial sparkling thought, enclosing the logical loop is often achieved after many assiduous tries like those hard work done by Madame Curie.

  2. Facing the beautiful results during your study, you are learning from an ensemble of mathematicians’ theories accumulated for years.

  3. The thinking process may proceed in a reverse direction compared to the presented proof.

Realizing the above facts, the “sufferings” inflicted by mathematics can be relieved and turned into aesthetic enjoyment.