Some of the most challenging problems in Timaeus' cosmology arise from the geometry of a universe without any void. On the one hand, the universe is spherical in shape; on the other hand, it must be entirely filled with the four basic particles that make up all bodies in the universe, each shaped like one of four regular polyhedra (cubes, tetrahedra, octahedra and icosahedra). The faces of all these particles are composed of right triangles. However, this leads to two mathematical impossibilities. 1. Obtaining a spherical surface from linear surfaces, as it is impossible to create a circle from straight lines. 2. Obtaining a complete tiling of a sphere using regular polyhedra, without any voids or intersections between these polyhedra. The first problem is addressed in another article slated for publication. In the present one, our focus will be on the second problem, for which we will present a solution within the framework of Timaeus' cosmology. The crux of this solution lies in a feature of Timaeus' universe that sets it apart from almost all ancient cosmologies. Instead of being composed of rigid parts, it is a dynamic living body in which all basic components are in constant motion, continuously undergoing both destruction and reconstruction. In the first part, we examine the main features of Timaeus' cosmology relevant to our issue. In the second part, we analyze the paradox in details and its consequences for Timaeus' cosmology. We then discuss the common 'solutions' and highlight their shortcomings. Finally, we propose a solution that we believe is consistent with Plato's text and independent of the choice of the major schools of interpretation of the Timaeus.