The article is devoted to the study of the regularities of the statistical organization of nucleotide sequences of single-stranded DNAs in genomes of higher and lower organisms, as well as their connections with cyclic Gray codes and the problem of holistic structures (gestalts) in physiology. The author presents stable statistical structures of an algebraic nature, which are found in many genomic DNAs and are called algebraic gestalt-archetypes of probabilities in genomic DNAs. They are discussed as a possible basis for several genetically inherited physiological and psychophysical properties. The numerical rules of these genomic archetypes realized in nature for a representative class of genomic DNAs, whose initial data were taken by the author from the publicly available genomic data bank “GenBank”, are formulated. The analysis of single-stranded genomic DNAs was carried out using the author's method of "hierarchies of multilayer statistics", representing the nucleotide sequence of DNA as a multilayer text structure, in which each n-th layer is a sequence of n-plets (that is, of monoplets, or duplets, or triplets, etc.). In each such layer, the percentages of each of the types of its n-plets are calculated, the values of which are inserted into the so-called genetic (2n∙2n) Karnaugh matrices, whose columns and rows are numbered by n-bit Gray codes by analogy with Karnaugh maps from the Boolean algebra of logic. The data of the analysis of the nucleotide sequence of DNA of the human chromosome № 1, containing about 250 million nucleotides, are represented as an example. The obtained data are discussed in light of the problem of genetically inherited holistic structures in biology and the tasks of developing algebraic biology, genetic biomechanics, quantum bioinformatics, artificial intelligence, and genetic algorithms.