A reservoir computer is a complex dynamical system, often created by coupling nonlinear nodes in a network. The nodes are all driven by a common driving signal. In this work, three dimension estimation methods, false nearest neighbor, covariance dimension, and Kaplan-Yorke dimension, are used to estimate the dimension of the reservoir dynamical system. It is shown that the signals in the reservoir system exist on a relatively low dimensional surface. Changing the spectral radius of the reservoir network can increase the fractal dimension of the reservoir signals, leading to an increase in a testing error.