An approach to the Ginzburg-Landau problem for superconducting regular polygons is developed making use of an analytical gauge transformation for the vector potential A which gives A(n) = 0 for the normal component along the boundary line of an arbitrary regular polygon. As a result the corresponding linearised Ginzburg-Landau equation reduces to an eigenvalue problem in the basis set of functions obeying Neumann boundary condition. The proposed approach allows for accurate calculations of the order parameter distributions at low calculational cost (small basis sets) for moderate applied magnetic fields. This is illustrated by considering the nucleation of superconductivity in squares and equilateral triangles where novel vortex patterns containing and antivortex in the centre are obtained on the T-c-H phase boundary. The stability of these solutions against small deviations from the phase boundary line deeper into the superconducting state is investigated and the conditions for the experimental observation of the novel vortex patterns are discussed. (C) 2001 Elsevier Science B.V. All rights reserved.