In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as T1⊔T2, such that each Ti represents the lines of a copy of the Fano plane PG(2,F2). We generalize this observation by constructing, for each prime power q, a simplicial complex Xq with q2+q+1 vertices and 2(q2+q+1) facets consisting of two copies of PG(2,Fq). Our construction works for any colored k-configuration, defined as a k-configuration whose associated bipartite graph G is connected and has a k-edge coloring χ:E(G)→[k], such that for all v∈V(G), a,b,c∈[k], following edges of colors a,b,c,a,b,c from v brings us back to v. We give one-to-one correspondences between (1) Sidon sets of order 2 and size k+1 in groups with order n, (2) linear codes with radius 1 and index n in the lattice Ak, and (3) colored (k+1)-configurations with n points and n lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.