• His first book,
Basics of Polytopes, was finished in 1879 and published in 1885. It offers a classification of
polytopes and derives
Fedorov polytopes, congruent polytopes that can completely fill
space. • He wrote the classic
The Symmetry of Regular Systems of Figures in 1891, which contained the first cataloging of the 230
space groups. The same year the equivalent results were presented by German mathematician
Schönflies. Fedorov and Schönflies had been intensively discussing the subject during their work, so the results can be somehow considered as joint ones, though Schönflies noted Fedorov's priority for some major ideas. • He published his classic work
The Theodolite Method in Mineralogy and Petrography in 1893. • In 1906 he published
Perfektsionizm (Perfectionism), a work he had started in the late 1870s. From a materialistic starting point he argued that natural conditions are themselves conditions of eternal change. He attacked authors claim that there is stability and equilibrium in nature. Contrariwise he claimed that evolution is the quality of living creatures rather than a tendency toward a "higher order" of stability, and equilibrium of organisms. He criticised
Herbert Spencer's views arguing that such views when applied to the evolution of natural history focus attention on the least changeable. However, he argued that such views systematically failed to take into account the fact that equilibrium is attained only at the moment of death: for a living creature, changing forms are constantly developing. Thus, Fyodorov criticised the notion that the constant and the stable are the supreme mission of life. Rather he claimed that life never finally achieves anything but always strives to achieve. For him this is where the true philosophy of nature is to be found. •
Tsarstvo kristallov (Crystal Kingdom) was first published posthumously in 1920 and featured much of the work carried out by Fyodorov and his colleagues. during the previous of forty years. Here he noted that sciences which have been fully developed not only satisfy the spiritual needs of part of mankind but also provide great power to direct the active forces of nature for man’s use. ==Legacy==