We present the first fully spatial hopping gait of a 12 DoF tailed biped driven by only 4 actuators. The control of this physical machine is built up from parallel compositions of controllers for progressively higher DoF extensions of a simple 2 DoF, 1 actuator template. These template dynamics are still not themselves integrable, but a new hybrid averaging analysis yields a conjectured closed form representation of the approximate hopping limit cycle as a function of its physical and control parameters. The resulting insight into the role of the machine's kinematic and dynamical design choices affords a redesign leading to the newly achieved behavior.