Given a connected surface
$${\mathbb {F}}^2$$
with Euler characteristic
$$\chi $$
and three integers
$$b>a\ge 1<k$$
, an
$$(\{a,b\};k)$$

$${\mathbb {F}}^2$$
is a
$${\mathbb {F}}^2$$
embedded graph, having vertices of degree only k and only a and bgonal faces. The main case are (geometric) fullerenes (5, 6; 3)
$${\mathbb {S}}^2$$
. By
$$p_a$$
,
$$p_b$$
we denote the number of agonal, bgonal faces. Call an
$$(\{a,b\};k)$$
map legoadmissible if either
$$\frac{p_b}{p_a}$$
, or
$$\frac{p_a}{p_b}$$
is integer. Call it legolike if it is either
$$ab^f$$
lego map, or
$$a^fb$$
lego map, i.e., the faceset is partitioned into
$$\min (p_a,p_b)$$
isomorphic clusters, legos, consisting either one agon and
$$f=\frac{p_b}{p_a}\,b$$
gons, or, respectively,
$$f=\frac{p_a}{p_b}\,a$$
gons and one bgon; 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 bgons is positive, zero or negative, respectively. There are 14 legolike elliptic
$$(\{a,b\};k)$$

$${\mathbb {S}}^2$$
with
$$(a,b)\ne (1,2)$$
. No
$$(\{1,3\};6)$$

$${\mathbb {S}}^2$$
is legoadmissible. For other 7 families of parabolic
$$(\{a,b\};k)$$

$${\mathbb {S}}^2$$
, each legoadmissible sphere with
$$p_a\le p_b$$
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
$$a\ge 3$$
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 legoadmissible
$$(\{a,b\};k)$$

$${\mathbb {S}}^2$$
with
$$p_b=2\le a$$
is
$$a^f b$$
. We list, explicitly or by parameters, legoadmissible
$$(\{a,b\};k)$$
maps among: hyperbolic spheres, spheres with
$$a\in \{1,2\}$$
, spheres with
$$p_b\in \{2,\frac{p_a}{2}\}$$
, Goldberg–Coxeter’s spheres and
$$(\{a,b\};k)$$
tori. We present extensive computer search of legolike spheres: 7 parabolic (
$$p_b$$
dependent) families, basic examples of all 5 hyperbolic
$$a^fb$$
(bdependent) families with
$$a\ge 3$$
, and legolike
$$(\{a,b\};3)$$
tori.