This paper describes how to generate a single image which, when viewed in the appropriate way, appears to the brain as a 3D scene. The image is a stereogram composed of seemingly random dots. A new, simple and symmetric algorithm for generating such images from a solid model is given, along with the design parameters and their influence on the display. The algorithm improves on previously-described ones in several ways: it is symmetric and hence free from directional (right-to-left or left-to-right) bias, it corrects a slight distortion in the rendering of depth, it removes hidden parts of surfaces, and it also eliminates a type of artifact that we call an "echo". Random dot stereograms have one remaining problem: difficulty of initial viewing. If a computer screen rather than paper is used for output, the problem can be ameliorated by shimmering, or time-multiplexing of pixel values. We also describe a simple computational technique for determining what is present in a stereogram so that, if viewing is difficult, one can ascertain what to look for.