Attempts to provide a coherent understanding of three analogies that appear at the center of Plato's Republic: The Sun and the Good, The Divided Line, and the Allegory of the Cave. The main innovation is to treat The Divided Line as an image-object metaphor illustrating the nature of mathematical reasoning. On this reading, the middle two portions of the four-part Divided Line both contain diagrams. In the lower middle portion, the diagram is treated as a physical object that can have reflected images. In the upper middle portion, the diagram is treated as a physical object, which is itself an image of a form.
Keywords: Allegory of the Cave, Plato, republic, The Divided Line, The Sun and the Good