Intraoperative monitoring of electroencephalography (EEG) data can help assess brain integrity and/or depth of anesthesia. We demonstrate a computer generated technique which provides a visually robust display of EEG data plotted as 'phase space trajectories' and a mathematically derived parameter ('dimensionality') which may correlate with depth of anesthesia. Application of nonlinear mathematical analysis, used to describe complex dynamical systems, can characterize 'phase space' EEG patterns by identifying attractors (geometrical patterns in phase space corresponding to specific ordered EEG data subjects) and by quantifying the degree of order and chaos (calculation of dimensionality). Dimensionality calculations describe the degree of complexity in a signal and may generate a clinically useful univariate EEG descriptor of anesthetic depth. In this paper we describe and demonstrate phase space trajectories generated for sine waves, mixtures of sine waves, and white noise (random chaotic events). We also present EEG phase space trajectories and dimensionality calculations from a patient undergoing surgery and general anesthesia in 3 recognizable states: awake, anesthetized, and burst suppression. Phase space trajectories of the three states are visually distinguishable, and dimensionality calculations indicate that EEG progresses from 'chaos' (awake) to progressively more 'ordered' attractors (anesthetized and burst suppression).