In this paper, we explore the idea of designing non-anthropomorphic multi-fingered robotic hands for tasks that replicate the motion of the human hand. Taking as input data a finite set of rigid-body positions for the five fingertips, we develop a method to perform dimensional synthesis for a kinematic chain with a tree structure, with five branches that share three common joints. We state the forward kinematics equations of relative displacements for each serial chain expressed as dual quaternions, and solve for up to five chains simultaneously to reach a number of positions along the hand trajectory. This is done using a hybrid global numerical solver that integrates a genetic algorithm and a Levenberg-Marquardt local optimizer. Although the number of candidate solutions in this problem is very high, the use of the genetic algorithm allows us to perform an exhaustive exploration of the solution space to obtain a set of solutions. We can then choose some of the solutions based on the specific task to perform. Note that these designs match the task exactly while generally having a finger design radically different from that of the human hand.