Following Poincare's geometric method, we construct two new nonorientable noncompact hyperbolic space forms by the regular octahedron in Fig 1. The construction is motivated by Thurston's example [6], discussed also by Apansov [1] in details. Our new space forms will be denoted by $$ \tilde D_1= H^3/G_1\quad\text{and}\quad \tilde D_2= H^3/G_2, $$ where $\tilde D_1$ and $\tilde D_2$ are obtained by pairing faces of $D$ via isometries of groups $G_1$ and $G_2$, respectively, acting discontinuously and freely on the hyperbolic 3-space $H^3$ (Fig. 2, Fig. 3). These groups are defined by generators and relations in Sect. 3. The complete computer classification of possible space forms by our octahedron will be discussed in [4], where it turns out that our two space forms are isometric, i.e. $G_1$ and $G_2$ are conjugated by an isometry $\varphi$ of $H^3$, i.e. $G_2=\varphi^{-1}G_1\varphi$, $$ \align G_1&=(g_1,g_2,\bar g_1,\bar g_2 \raise2pt\vbox{\hrule width.5cm} g_1\bar g_1^{-1}g_2\bar g_2^{-1}=g_1g_1g_2g_2=\bar g_1\bar g_1\bar g_2\bar g_2=1),\\ G_2&=(t_1,t_2,\bar g_1,\bar g_2 \raise2pt\vbox{\hrule width.5cm} t_1\bar g_1^{-1}t_2^{-1}\bar g_2=t_1t_2t_1^{-1}t_2^{-1}=\bar g_1\bar g_1\bar g_2\bar g_2=1). \endalign $$