Mappings which preserve regular dodecahedrons

for all x, y ∈ X . A distance r > 0 is said to be preserved by a mapping f : X → Y if ‖ f (x)− f (y)‖ = r for all x, y ∈ X whenever ‖x− y‖ = r. If f is an isometry, then every distance r > 0 is preserved by f , and conversely. We can now raise a question whether each mapping that preserves certain distances is an isometry. Indeed, Aleksandrov [1] had raised a question whether a mapping f : X → X preserving a distance r > 0 is an isometry, which is now known to us as the Aleksandrov problem. Beckman and Quarles [2] solved the Aleksandrov problem for finite-dimensional real Euclidean spaces X =Rn (see also [3–6, 10, 12–17]). Theorem 1.1 (Beckman and Quarles). If a mapping f :Rn →Rn (2≤ n <∞) preserves a distance r > 0, then f is a linear isometry up to translation.