The subject of the research in this book algebraic geometry. Quadratic equality Z2=X2+Y2 and U2=X2+Y2+Z2 well known from the time of Pythagoras. However, the history of their research continues. The aim of this work is the analysis of the symmetry groups of solutions of quadratic equations on the basis of Periodical System of Natural Numbers. The book is the first to use the language of symmetry groups integers inhearent to physical phenomena and States: from the fine structure of crystals to arrangement of the periodic system of chemical elements and the solution of applied problems. Given in the book form of the calculation of the symmetry groups of quadratic identities Z2=X2+Y2 exhausts infinitely many solutions “Pythagorean” equalities and allows you to subdivide them according to the symmetry groups of numbers on 32 species - number equal to the number of point of crystallographic groups. Full package of solutions of quadratic equality U2=X2+Y2+Z2 has 230 options - known amount of all space groups of crystal symmetry.