Technologies of modern optical telescopes with large primary mirror are based on adaptive optics. These telescopes operate with many small mirror segments, so that all the segments work as a large piece of a reflective curved plate, i.e. a paraboloid. Each mirror segment is independently attached to a support structure via adjustable warping harnesses. A support structure is required to be extremely rigid in order to maintain the reflective surface. This paper describes the conceptual approach for the design of such support structures. A system proven to fulfill these requirements with efficient structural material use is a node-and-bar system, so-called space frame. The rules for geometry of space frame structures are based on the system of the five 'platonic solids': The edges of the conceptually assembled solids can be replaced by the bar members of a space frame to achieve maximum stiffness. This conceptual approach is demonstrated with examples in the paper, by illustrating the determination of the geometry and examining the deformation due to the telescope rotations during operation. This paper also demonstrates design solutions for other issues relevant to space frame geometry, such as effects of gradient thermal load and redundancy of the structures.