Let
G
G
be a finitely generated group, and let
Σ
\Sigma
be a finite generating set of
G
G
. The growth function of
(
G
,
Σ
)
(G,\Sigma )
is the generating function
f
(
z
)
=
∑
n
=
0
∞
a
n
z
n
f(z) = \sum \nolimits _{n = 0}^\infty {{a_n}{z^n}}
, where
a
n
{a_n}
is the number of elements of
G
G
with word length
n
n
in
Σ
\Sigma
. Suppose that
G
G
is a cocompact group of isometries of Euclidean space
E
2
{\mathbb {E}^2}
or hyperbolic space
H
2
{\mathbb {H}^2}
, and that
D
D
is a fundamental polygon for the action of
G
G
. The full geometric generating set for
(
G
,
D
)
(G,D)
is
{
g
∈
G
:
g
≠
1
\{ g \in G:g \ne 1
and
g
D
∩
D
≠
∅
}
gD \cap D \ne \emptyset \}
. In this paper the recursive structure for the growth function of
(
G
,
Σ
)
(G,\Sigma )
is computed, and it is proved that the growth function
f
f
is reciprocal
(
f
(
z
)
=
f
(
1
/
z
)
)
(f(z) = f(1/z))
except for some exceptional cases when
D
D
has three, four, or five sides.