Lego-like spheres and tori

Given a connected surface ({mathbb {F}}^2) with Euler characteristic (chi ) and three integers (b>age 1k and only a- and b-gonal faces. The main case are (geometric) fullerenes (5, 6; 3)-({mathbb {S}}^2). By (p_a), (p_b) we denote the number of a-gonal, b-gonal faces. Call an (({a,b};k))-map lego-admissible if either (frac{p_b}{p_a}), or (frac{p_a}{p_b}) is integer. Call it lego-like if it is either (ab^f)-lego map, or (a^fb)-lego map, i.e., the face-set is partitioned into (min (p_a,p_b)) isomorphic clusters, legos, consisting either one a-gon and (f=frac{p_b}{p_a},b)-gons, or, respectively, (f=frac{p_a}{p_b},a)-gons and one b-gon; the case (f=1) we denote also by ab. Call a (({a,b};k))-map elliptic, parabolic or hyperbolic if the curvature (kappa _b=1+frac{b}{k}-frac{b}{2}) of b-gons is positive, zero or negative, respectively. There are 14 lego-like elliptic (({a,b};k))-({mathbb {S}}^2) with ((a,b)ne (1,2)). No (({1,3};6))-({mathbb {S}}^2) is lego-admissible. For other 7 families of parabolic (({a,b};k))-({mathbb {S}}^2), each lego-admissible sphere with (p_ale p_b) is (a^fb) and an infinity (by Goldberg–Coxeter operation) of (ab^f)-spheres exist. The number of hyperbolic (ab^f,({a,b};k))-({mathbb {S}}^2) with ((a,b)ne (1,3)) is finite. Such (a^f b)-spheres with (age 3) have ((a,k)=(3,4),(3,5),(4,3),(5,3)) or (3, 3); their number is finite for each b, but infinite for each of 5 cases (a, k). Any lego-admissible (({a,b};k))-({mathbb {S}}^2) with (p_b=2le a) is (a^f b). We list, explicitly or by parameters, lego-admissible (({a,b};k))-maps among: hyperbolic spheres, spheres with (ain {1,2}), spheres with (p_bin {2,frac{p_a}{2}}), Goldberg–Coxeter’s spheres and (({a,b};k))-tori. We present extensive computer search of lego-like spheres: 7 parabolic ((p_b)-dependent) families, basic examples of all 5 hyperbolic (a^fb) (b-dependent) families with (age 3), and lego-like (({a,b};3))-tori.