VERTEX TO VERTEX GEODESICS ON PLATONIC SOLIDS

The paper proves that no geodesic can start and end at the same vertex on the cube, tetrahedron, octahedron, or icosahedron via an unfolding argument and a refolding symmetry that yields a 180° rotational axis mapping endpoints to distinct vertices; it also notes the dodecahedron admits vertex-to-self geodesics. This is stated and sketched clearly in Theorem 1 and the subsequent case analysis for square and triangular tilings . The candidate model reproduces the correct conclusion, but its key parity–midpoint step on the triangular lattice is false: from m ≡ n (mod 2) it concludes the straight segment’s midpoint is always a lattice vertex, which is not true when both coordinates are odd (e.g., from (0,0) to (1,1) the midpoint is at half-integers). Thus the model’s proof is invalid for the triangular-lattice solids, even though the final conclusion coincides with the paper’s. The paper also correctly remarks that the dodecahedron is exceptional, admitting vertex-to-self trajectories .