A new method of construction of the Hamiltonian for uniformly frustrated spin systems is proposed. In the XY case the method leads to the model which is equivalent to the well known uniformly frustrated XY model but it is more general than the latter because it makes sense for the three-dimensional spins as well. In this paper the method has been used in an atypical way because the author studies two unreal spin systems. In the first system, spins are placed on the vertices of the polytope (35) which form, in the curved space S2, a two-dimensional triangular lattice with a fivefold symmetry axis. In the second system, spins are placed on the vertices of the polytope (335) which form, in the curved space S3, a three-dimensional lattice with icosahedral symmetry. In the same way as other workers, the author interested in the polytope (335) because of a topological similarity between fragments of it and fragments of amorphous materials. Using the Monte Carlo technique the author has calculated the temperature dependence of the mean energy and specific heat for both ferromagnetic and antiferromagnetic interactions. The ground states of both systems have been found too. Moreover some low-lying metastable states have been found for the icosahedral spin system.