Projecting a quantum theory onto the Hilbert subspace of states with energies below a cutoff E may lead to an effective theory with modified observables, including a noncommutative space(time). Adding a confining potential well V with a very sharp minimum on a submanifold N of the original space(time) M may induce a dimensional reduction to a noncommutative quantum theory on N. Here in particular we briefly report on our application [1, 2, 3, 4, 5] of this procedure to spheres S d ⊂ R D of radius r = 1 (D = d+1 > 1): making E and the depth of the well depend on (and diverge with) Λ ∈ N we obtain new fuzzy spheres S d Λ covariant under the full orthogonal groups O(D); the commutators of the coordinates depend only on the angular momentum, as in Snyder noncommutative spaces. Focusing on d = 1, 2, we also discuss uncertainty relations, localization of states, diagonalization of the space coordinates and construction of coherent states. As Λ → ∞ the Hilbert space dimension diverges, S d Λ → S d , and we recover ordinary quantum mechanics on S d. These models might be suggestive for effective models in quantum field theory, quantum gravity or condensed matter physics.