Amyloids are insoluble fibrous protein aggregates which, when abnormally accumulated in the body, can result in amyloidosis and various neurodegenerative diseases. In this work, we describe a new approach to the asymptotic solution of the master equation of amyloid fiber aggregations. It is found that four distinct and successive stages (lag phase, exponential growth phase, breaking phase, and static phase) dominate the fiber formation process. On the basis of the distinctive power-law dependence of the half-time and apparent growth rate of the fiber formation on the initial protein concentration, we propose a novel classification for amyloid proteins theoretically.