In 100 Great Problems of Elementary Mathematics by Dorrie, it is proved that there are only five possible tessellations of the sphere using congruent regular (spherical) polygons: $4$ regular triangles, $6$ regular squares, $8$ regular triangles, $20$ regular triangles, and $12$ regular pentagons.
This is great, but the author then proceeds to say that if we connect all the vertices of the spherical polygons using straight lines, we obtain five regular solids: tetrahedron, octahedron, cube, icosahedron, and dodecahedron. Hence, there are only $5$ Platonic solids. $Q.E.D$
I feel rather uneasy about this claim. Am I missing something here? How do we know that the sphere tessellations necessarily correspond to the Platonic solids? How can we prove for sure that there doesn't exist a Platonic solid which can't be constructed by connecting the vertices of all the possible sphere tessellations?