A noneuclidean Volvox, or why it is better not to teach Volvox in the course on zoology

Empty "spheroid" of Volvox is compared with biomorph "thread", "disk" and solid "sphere" using such characteristics as topological dimensionality, average distance between cells, mutual remoteness of inner and surface cells, contiguity of cells. It is usually supposed that these parameters are significant for physiological gradients that determine cell specialization. One-dimensional "thread" has the longest physiological communications between cells and the average degree of contiguity about 2 (each cell contacts two neighbors). Biological morph "disk" has a degree about 6, two-side frontal physiological gradient inside the cell, and less expressed inter-cell gradient. Biomorph designated as 3-dimensional solid "sphere" has a degree of contiguity about 12-24, strong radial inter-cell gradient (non-equal conditions for surface and inner layers) and short distances between cells. These parameters favor cell specialization and their integration in multicellular organism. The "sphere" corresponds to hypothetical ancestor of Metazoa - "Metschnikoff's Phagocytella", while the "disk" - to "Placula of Bütschli". Biomorph "spheroid" of Volvox has a degree of contiguity about 6 and continuous tangential inter-cell gradient on noneuclidean surface. Radial gradient is absent here. Due to noneuclidean nature of "spheroid" the distances between cells are longer here than in case of "disc" and "sphere". All cells are under the same conditions for specialization and multiple primary integration. The secondary integration in higher Volvocales (differentiation in somatic and generative hemispheres) was probably caused by directed movement of the whole colony. Specialization of cells in lower invertebrates develops in a way which is characteristic for biomorph "sphere" on the basis of 3-dimensionality. The differentiation of animal and vegetal poles is connected with gastrulation (but not with directed movement as in case of Volvox). Gastrulation through invagination does not comparable with inversion of plate-like embryo of Volvox into "spheroid". Invagination is the transformation of a "bent of sphere", whereas the inversion is the "bent of plate". Independently of particular mechanism gastrulation results in 3-dimensionality (as in case of "sphere"). However the integration of cells in Volvox is explained by special peculiarities of 2-dimensional noneuclidean surface. That's why Volvox cannot be considered as model of ancestor of Metazoa.