Three cylinders

The coordinates of the corners satisfy the system of equations $\begin{cases} x^2+y^2=R^2\\ y^2+z^2=R^2\\ z^2+x^2=R^2 \end{cases}$ , where $R$ is the radius of the cylinders. With simple manipulation the solutions of the system are $\begin{cases} x=\pm R/\sqrt{2}\\ y=\pm R/\sqrt{2}\\ z=\pm R/\sqrt{2} \end{cases}$ . Also, there're $6$ corners which are intersections of two elipses. So, there're $6+8=14$ corners.