The special linear group $SL(2,\mathbb{R})$, the group of $2 × 2$ real matrices with determinant one, is one of the most important and fundamental mathematical objects not only in mathematics but also in physics. In this paper, we propose a three-dimensional model of $SL(2, \mathbb{R})$ in $\mathbb{R}^3,$ which is realized by embedding $SL(2,\mathbb{R})$ into the unit $3$-sphere. In this model, the set of symmetric matrices of $SL(2,\mathbb{Z})$ forms a hyperbolic pattern on the unit disk, like the islands floating on the sea named $SL(2,\mathbb{R}).$ The structure of this hyperbolic pattern is described in the upper half-plane $H.$ The upper half-plane $H$ also enables us to generate symmetric matrices of $SL(2,\mathbb{R})$ with three circles. Furthermore, the well-known fact $H = SL(2, \mathbb{R})/SO(2)$ is visualized as $S^1$ fibers of Hopf fibration in the unit $3$-sphere. With this three-dimensional model in $\mathbb{R}^3,$ we can have a concrete image of $SL(2,\mathbb{R})$ and its noncommutative group structure. This kind of visualization might bring great benefits for the readers who have background not only in mathematics, but also in all areas of science.